Preface This Handbook is Volume 2 in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems help to clarify mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. The authors and editors have made an effort, however, to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopaedic completeness, but present selected paradigms. There is a web of relationships among the chapters, allowing them to be organized in different ways, each somewhat arbitrary. The chapters are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and
partial differential equations. Two of the three articles in the section onfinite-dimensional methods deal with dynamics of networks of neurons. One of them focuses on a variety of techniques, geometrical and analytical, for investigating interactions of pairs of oscillators. A complementary review uses geometrical ideas to discuss further aspects of small network interactions, and some interesting consequences for particular neural applications, including sleep rhythms and visual scene segmentation. Many of the mathematical features carry over to more general coupled oscillator networks, ranging from electronics to solid state physics, while other ingredients are specific to the particular applied field, here neurophysiology. This interplay of general mathematical structure with specific applied context is an important underlying theme of the volume, and is demonstrated by the paradigms presented. Addressing fluid dynamics on geographic scales, the third survey illustrates an application of lobe dynamics of transverse planar heteroclinic tangles to global transport phenomena by ocean currents. Methods are geometrical and are based on the analysis of invariant manifolds of hyperbolic trajectories and invariant tori. Numerical simulation has become an omnipresent- and sometimes naively appliedtool in bridging the gap between universal mathematical theory and specific applied context. It is one of the goals of the section on numerics to indicate the level of sophistication necessary to properly deal with various aspects of dynamics, numerically. We start with a thorough survey of state-of-the-art continuation methods, which seek to determine the parameter dependence of particular types of solutions, such as critical points or periodic orbits. This includes computations of normal forms, of codimension two bifurcations, and of homoclinic bifurcations. Stable and unstable manifolds, which may intersect along homoclinic curves for flows, will typically split under time discretization and exhibit exponen-
vi
Preface
tially small separation. This effect and the associated invisible chaos mark a fundamental difference between continuous time autonomous flows and their time discretizations. Both upper and lower bounds for exponentially small splitting effects are provided. More generally, another survey aims at assessing the capability of theory, algorithms, and software to elucidate the structure of dynamical models in mathematics, science, and engineering. Specific dynamical objects include classical initial value problems, periodic solutions, invariant tori, stable and unstable manifolds, and their bifurcations. In the context of chaotic dynamical systems, hyperbolicity-based shadowing estimates provide a tool to diagnose whether it is possible to achieve numerical solutions that are close, over very large time spans, to actual solutions. Finally, set oriented numerical methods are presented which provide a robust tool for the approximation of low-dimensional invariant objects, such as attracting sets or invariant manifolds. These methods are also capable to derive statistical information about the dynamical behavior via the computation of approximate SRBmeasures and almost invariant sets. Topological methods are an important tool for the analysis of dynamical systems. Conley index, in this context, stands for Conley's successful idea of extending Morse theory to the requirements and challenges posed by dynamical systems which, a priori, do not possess a variational structure. Several examples in the first article apply Conley index to corroborate numerical input into computer assisted proofs. While the exposition here, not the method, limits itself to finite-dimensional dynamics, the survey on functional differential equations demonstrates applications of topological fixed point theory to infinite-dimensional dynamics. Partial differential equations are a fascinating infinite-dimensional source o f - and challenge f o r - dynamics. They are represented by eight articles, mostly parabolic, constituting almost half of the entire volume. While three surveys are related to fluid flows, we first comment on the five surveys which address general PDE topics. The intimate relations to finite-dimensional dynamics, in particular when studying global attractors and large-time behavior, are investigated from different perspectives in two surveys. In the non-parabolic case, even mere existence of a global attractor is not always obvious. In addition to dimension estimates and inertial manifold reductions, these articles extract geometrical information about the associated global attractors from structural PDE properties like Lyapunov functions and comparison principles. Such structures can be found in a variety of equations, including reaction diffusion equations, wave equations with damping, Navier-Stokes, Cahn-Hilliard, Kuramoto-Sivashinsky equations and certain integro-differential equations. Partial differential equations involve space coordinates as well as time coordinates. This allows for richer behavior, including spatial and spatio-temporal patterns. For example, travelling waves are solutions of associated ordinary differential equations. Depending on the type of wave, the desired solutions take a particular geometric form such as homoclinic orbits (for pulses), heteroclinic orbits (for fronts), or periodic orbits, as well as more complicated objects corresponding to multiple pulses and spirals. Their bifurcations and their PDE stability analysis pose formidable challenges addressed here. These results are complemented by a survey of some first steps towards understanding the spatio-temporal complexity of higher-dimensional stable and metastable patterns in nonlinear evolution equations of gradient type. The singularities generated by finite time blow-up of solutions
Preface
vii
are yet another PDE phenomenon which illustrates the close interaction of temporal dynamics with spatial profiles. Some of these singularities can be investigated using related equations that come from changes of variables motivated by self-similar blow-up solutions. The article on blow-up emphasizes the unifying role of abstract center manifolds in this spatio-temporal analysis. Two articles in the section on partial differential equations are closely related to questions of fluid flows. We begin with a large survey on the state of the art for existence, uniqueness, and model derivation of the central Navier-Stokes equation itself. Starting from the molecular level with Hamiltonian systems, the survey proceeds to Boltzmann equations, where the original time reversibility is lost, and arrives at the Navier-Stokes equations, which themselves lead to macroscopic models for turbulent flows. This spans a large hierarchy of dynamical systems, and the various limiting processes are made precise. Ginzburg-Landau equations, on the other hand, are almost universally relevant model equations and at the same time closely related to specific fluid flow phenomena. They appear as reduced, albeit infinite-dimensional, modulation equations. In particular they describe bifurcations in unbounded domains and govern the associated dynamics of spatiotemporal patterns. Full mathematical justification of the reduction process from the underlying PDE to the Ginzburg-Landau approximation is addressed. Like the Ginzburg-Landau approximation, the nonlinear Schr6dinger equation can be viewed as an envelope equation for certain idealized fluid flows, but also possesses major applications in nonlinear optics and in plasma physics. It is surveyed here to illustrate sample behavior and phenomena for a general class of PDEs: one-dimensional, nonlinear, dispersive wave equations. These provide examples of infinite dimensional dynamical systems which exhibit diverse and fascinating phenomena, including solitary pulse waves and wave trains, the generation and propagation of oscillations, the formation of singularities, the persistence of homoclinic orbits, the existence of temporally chaotic waves in deterministic systems, dispersive wave turbulence, and the propagation of spatio-temporal chaos. Classes of nonlinear Schr6dinger equations thus provide prototypical PDEs which illustrate the use of geometrical, analytical, and computational methods to capture and to describe the rich behavior of nonlinear dispersive waves. While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to name just a few, are ubiquitous dynamical concepts throughout the articles. It is this conceptual unity, and its ability to capture and mirror an ever-changing world, which is the innermost source of strength of the dynamical systems approach. Many friends and colleagues have helped bringing this volume to life. I am indebted to Floris Takens for the original idea of the Handbook series, and, with Henk Broer, Boris Hasselblatt, and Anatol Katok, for sharing the dynamics of their volumes with me, to Stefan Liebscher for his untiring care for the web services, to Regina L6hr for expertly keeping track ever so patiently of ever so many versions and revisions, and to Arjen Sevenster at Elsevier for efficient collaboration. Nancy Kopell and G6rard Iooss have helped significantly
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Preface
with editing this volume and have very generously shared their stimulating criticism with me. Most of all, of course, I am indebted to all authors for their contributions. Berlin, August 2001
Bernold Fiedler
List of Contributors Auerbach, D., University of Maryland, College Park, MD (Ch. 7) Bardos, C., University Denis Diderot and University Pierre et Marie Curie, Paris (Ch. 11) Beyn, W.-J., Bielefeld University, Bielefeld (Ch. 4) Cai, D., New York University, New York, NY (Ch. 12) Champneys, A., University of Bristol, Bristol (Ch. 4) Dellnitz, M., University of Paderborn, Paderborn (Ch. 5) Doedel, E., Concordia University, Montreal (Ch. 4) Ermentrout, G.B., University of Pittsburgh, Pittsburgh, PA (Ch. 1) Fife, EC., University of Utah, Salt Lake City, UT (Ch. 13) Fila, M., Comenius University, Bratislava (Ch. 14) Gelfreich, V., The Steklov Mathematical Institute at St. Petersburg, Russia and Institut fiir Mathematik I, FU, Berlin (Ch. 6) Grebogi, C., Universidade de $6o Paulo, S6o Paulo (Ch. 7) Govaerts, W., University of Gent, Gent (Ch. 4) Guckenheimer, J., Cornell University, Ithaca, NY (Ch. 8) Jones, C., Brown University, Providence, RI (Ch. 2) Junge, O., University ofPaderborn, Paderborn (Ch. 5) Kopell, N., Boston University, Boston, MA (Ch. 1) Kuznetsov, Y.A., Utrecht University, Utrecht (Ch. 4) Matano, H., University of Tokyo, Tokyo (Ch. 14) McLaughlin, D.W., New York University, New York, NY (Ch. 12) McLaughlin, K.T.R., University of Arizona, Tucson, AZ (Ch. 12) Mielke, A., Universitiit Stuttgart, Stuttgart (Ch. 15) Mischaikow, K., Georgia Institute of Technology, Atlanta, GA (Ch. 9) Mrozek, M., Uniwersytet Jagiellohski, Krak6w (Ch. 9) Nicolaenko, B., Arizona State University, Tempe, AZ (Ch. 11) Nussbaum, R.D., Rutgers University, Piscataway, NJ (Ch. 10) Polfi6ik, E, Comenius University, Bratislava (Ch. 16) Poon, L., University of Maryland, College Park, MD (Ch. 7) Raugel, G., CNRS et Universit6 de Paris-Sud, Orsay (Ch. 17) Rubin, J.E., University of Pittsburgh, Pittsburgh, PA (Ch. 3) Sandstede, B., Ohio State University, Columbus, OH (Chs. 4, 18) Sauer, T., George Mason University, Fairfax, VA (Ch. 7) Terman, D., The Ohio State University, Columbus, OH (Ch. 3) Yorke, J.A., University of Maryland, College Park, MD (Ch. 7) Winkler, S., Brown University, Providence, RI (Ch. 2) ix
CHAPTER
1
Mechanisms of Phase-Locking and Frequency Control in Pairs of Coupled Neural Oscillators*
N. Kopell Department of Mathematics and Center for BioDynamics, Boston University, Boston, MA 02215, USA E-mail: nk@bn, edu
G.B. Ermentrout Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Phase oscillators and averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Derivation of phase-difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Computing frequency, phase lags and stability for pairs of coupled oscillators . . . . . . . . . . . . 2.3. Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Frequency effects and the rise/fall time of synapses . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Interaction near a Hopf bifurcation or a homoclinic orbit . . . . . . . . . . . . . . . . . . . . . . . 3. Interactions of spiking cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Spike response method and periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Networks with inhibitory connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Spike response method and electrical coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Oscillators with multiple spiking cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Long-range synchronization of the g a m m a rhythm . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. G a m m a to beta transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Long-distance synchronization of the beta rhythm . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Changing ionic currents changes synchronization properties . . . . . . . . . . . . . . . . . . . . . . 4.5. Synchronization of excitatory cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Interactions ofbursting neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Fast threshold modulation (FTM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Electrical coupling of cells or compartments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Electric coupling of heterogeneous cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 7 7 10 12 15 16 20 20 23 27 31 31 34 35 38 40 41 42 49 51 51
*Work partially supported by NIH grant RO1-MH47150 to NK and GBE, NSF grant 9706694 to NK and a NSF grant to GBE. We thank J. Ritt and S. Epstein for careful readings and helpful comments. H A N D B O O K OF D Y N A M I C A L SYSTEMS, VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved
Mechanisms of phase-locking andfrequency control
5
1. Introduction Oscillations occur in many networks of neurons, and are associated with motor behavior, sensory processing, learning, arousal, attention and pathology (Parkinson's tremor, epilepsy). Such oscillations can be generated in many ways. This chapter discusses some mathematical issues associated with creation of coherent rhythmic activity in networks of neurons. We focus on pairs of cells, since many of the issues for larger networks are most clearly displayed in that context. As we will show, there are many mechanisms for interactions among the network components, and these can have different mathematical properties. A description of behavior of larger networks using some of the mechanisms described in this chapter can be found in the related chapter by Rubin and Terman. For reviews of papers about oscillatory behavior in specific networks in the nervous system, see Gray [21], Marder and Calabrese [41], Singer [55], and Traub et al. [62,64]. The chapter is organized by mathematical structure, focusing on some of the techniques that have been found to be useful to understand the behavior of networks of neurons. The behavioral repertoire of two-component networks includes synchronized oscillations, antiphase oscillations, phase-locked oscillations with phase difference other than zero or 7r, as well as non-phase-locked solutions and steady states. We concentrate on the oscillatory phase-locked solutions, including some that occur when neither of the cells is itself an oscillator. One aim of the chapter is to provide methods for understanding what aspects of the cells and their interaction determine which of these behaviors are found, especially in complicated contexts. We pay particular attention to how changes in various time scales associated with the cells and their connections affect the behavior of the network. Though the focus is on pairs of components, we discuss implications for larger networks throughout the chapter. The component cells we have in mind are modeled by voltage-gated conductance equations, which have the form 61)
Cd-t -- -- ~
lion + IA
(1.1)
(Rinzel and Ermentrout [49]). Here lion represents an ionic current; it is the product of a driving force of the form v - VR and a conductance (inverse of resistance). Here VR denotes the "reversal potential" of the current, which depends on the ion carried by the current. The value of VR determines whether the current is depolarizing (inward, moving the voltage toward the threshold and making it easier for the cell to fire an action potential) or hyperpolarizing (outward, moving the voltage away from threshold for firing an action potential). The conductance depends in a dynamic way on the voltage, adding further differential equations; there are voltage-dependent "gates" that open or close with kinetics that also depend on the particular current. For example, for a sodium current, the conductance is usually written in the form ~,m3h, where ~ is the maximal conductance, and m and h satisfy the equations dx
at
=
-
6
N. Kopell and G.B. Ermentrout
where x = m or h. m ~ ( v ) is a monotone-increasing sigmoidal function (saturating for large and small v), while h ~ (v) is a monotone-decreasing sigmoidal function). The kinetic functions r (v) are different for h and m, and also differ between currents in the same cell; in general, the gating equations have slower kinetics than the voltage equation. Thus, the full equations are highly nonlinear, and have a range of time scales. The term IA is the applied current, sometimes referred to as the drive to the cell. It can model current injection by an experimenter, or a modulatable (time-independent) quantity representing processes slow enough to be treated as constant, and affecting the overall excitability of the cell. None of the techniques we present deals directly with the full Hodgkin-Huxley equations, though the last deals with some equations of this type. Rather, in each of the succeeding sections, we first present reduction techniques that allow us to deal with a class of simpler equations. We then present methods of analysis for that class, and some contexts in which the analysis gives answers about phase and frequency of the networks. In some cases, such as electrical coupling between cells or parts of cells, we visit the subject in several different sections, producing complementary insights with complementary methods. Section 2 concerns weakly coupled oscillators. If the oscillators have robust limit cycles, the full equations reduce (to lowest order) to ones whose interactions are through the differences of their phases. Though much has been written about such equations, e.g., (Hoppensteadt and Izhikevich [24], Kopell [31], Kopell and Ermentrout [34]), we focus here on some novel uses of these equations, including analysis of interactions with conduction delays. Another regime we discuss concerns oscillators near a Hopf bifurcation. In this case, the equations can be analyzed using normal forms that take into account amplitudes as well as phases. Using analysis and geometry, we apply these ideas to showing how interactions via diffusion can lead to non-synchrony between oscillators. Section 3 describes the interaction of spiking neurons, for which other approximations can be made that enable analysis. In this section, we treat the cells as "integrate-and-fire" (I&F) neurons. We describe the "spike-response method", a formalism that describes the time-dependent response of the voltage of the cell to the history of the pulsatile inputs that it has received. Though this method can in principle be used quite generally, it is most useful when the dynamics are fairly simple in the interval between spikes (as in I&F neurons). We use this method to see how the dynamics of the synapses affects the network behavior, including synchronization and emergent frequency; we focus mainly on inhibitory networks. We also apply the method to electrically coupled neurons to show how the shapes of the spikes can affect whether or not the electrical (diffusive) coupling is synchronizing, and how, in turn, that depends on the frequency of the network. The spike-response method can sometimes be applied to networks of excitatory and inhibitory neurons. But when there are small numbers of cells, different kinds of cells in the network, and complicated dynamical patterns, another kind of reduction, considered in Section 4, can be more effective for analysis. This reduction looks at Poincar6 maps in a neighborhood of a particular periodic solution. Though the full equations (and the Poincar6 map) can be high-dimensional, the spread of time scales in the system can allow it to be well approximated by low-dimensional maps. We use these ideas to discuss precise synchronization in the presence of conduction delays.
Mechanisms of phase-locking and frequency control
7
Section 5 deals with bursting oscillators. A bursting neuron is one that emits a fast sequence of spikes, interspersed with a well-defined inter-burst interval. Such a neuron has at least three time scales even without coupling: the time scale of the voltage change within a spike, the scale of the recovery variable within a spike, and some slower process that starts and/or ends the bursts. There has been much work on mechanisms that produce such bursting. (See Izhikevich [26] for many references, and the chapter by Rubin and Terman [52] of this volume.) In this section, we shall simplify the bursting neurons by working with the envelope of the spikes, ignoring the fast currents that produce the individual spikes. The work in this section is complementary to related work in Rubin and Terman [52]; we focus on network behavior when there are fast excitatory or inhibitory synapses, with no extra time scales associated with the coupling. We introduce geometrical ideas associated with "fast threshold modulation" and time metrics for computing synchronizing effects, and show how synaptic thresholds can affect the frequency of the coupled system. We also discuss how strong electrical coupling of "compartments" of a single cell can have some unexpected mathematical properties.
2. Phase oscillators and averaging It is not possible to analyze the general behavior of a pair of coupled nonlinear oscillators. However, if the coupling is sufficiently weak, then it is possible to reduce the behavior of a general coupled system to something much simpler. In most of this section, we work with an identical pair of oscillators that have stable limit cycles. In that case, the full system can be reduced to a single equation for the difference in the phases of the oscillators. The special structure of neural equations and the kinds of coupling between them translates into constraints on the reduced equations that can be used to understand the circumstances under which the oscillators synchronize or stabilize at other phase differences. In this section, we derive the equation, and show how to make use of the special structure. We then apply the ideas to oscillators with delays in the signals. If the limit cycle of the oscillators is only weakly stable, i.e., comparable in strength to the coupling, then the reduced equations must take into account not only the phases, but how the deviation of the trajectories from the limit cycle affects the phases. Such a situation occurs near a Hopf bifurcation. In that case, there is another reduction procedure, using a normal form, that encodes the behavior of the system. We discuss that derivation and apply it to show why coupling via discrete diffusion need not lead to stable synchrony. The analytical computations have a geometric counterpart, as we show.
2.1. Derivation o f phase-difference equations Consider a system of two similar coupled nonlinear oscillators: dXi dt
= Fi(Xi) + eGi(Xi,
Xj),
X i ~ R'", i -- 1,2, i :/: j,
(2.1)
8
N. Kopell and G.B. Ermentrout
with 0 < e << 1. Suppose dXi = Fi(Xi)
dt
has an orbitally asymptotically-stable limit cycle, X~ (t), with period Ti = T + Oi (6). (For this we assume that IF2 - Fll ~- O(e).) Then for sufficiently small e the solutions to (2.1) have the form Xi (t) = Xo(Oi) + Oi (e) and, up to lowest order in e, dO1
- - 1 + e l l 1 (02 -
dt dO2 -~ 092 -+- 8H2(01 dt
(2.2)
01),
02),
where 01,02 a r e coordinates on a toms and represent the phases of the two oscillators. 1 and 032 are the frequencies of the uncoupled oscillators, with I1 - o921 - - 0 ( 6 ) . A rigorous derivation (Ermentrout and Kopell [ 14]) uses center manifold ideas to get equations on a toms; the use of averaging then shows that the interactions are (to lowest order) through the difference of the phases. A formal method producing Equation (2.2) in a computationally simpler way is given in Ermentrout and Kopell [16]. The functions Hi are T-periodic in each of their arguments and can be explicitly computed from the original equations. Let 4~ = 02 - 01. Then
1 f o T X , (t) . Gl (Xo(t) , Xo(t + O ) ) d t . Hi (0) = -~
(2.3)
A similar formula holds for H2 with the ~b replaced by -q~. The function X* (t) is the normalized T-periodic solution to the adjoint equation"
dX* = -DF(Xo(t))TX dt
*,
X* (t) . X~o(t) -- 1.
The average in (2.3) is easy to calculate; the only difficulty lies in the computation of X* (t). Williams and Bowtell [73] devised a simple way to compute X* (t) for a stable limit cycle: Start with random initial conditions and integrate
dy dt
--
-
DF(Xo(t)) T y
backwards. Since all Floquet multipliers of the periodic solution are outside the unit circle (except for a multiplier of 1), backward integration will converge to the periodic solution. Thus, one can calculate the adjoint and the interaction function (2.3) for any given differential equation with a stable limit cycle. So far, the form of the coupling between oscillators has been very general. Coupling between neurons is typically very special and is usually through only one component in the equations, the membrane potential, Vi. The coupling is generally additive and has one
Mechanisms of phase-locking and frequency control
9
of two different forms depending on whether the synapse is chemical or electric. In the first case the coupling from cell j to cell i has the form: /coupling-- gijsj(t)(Vi(t) - Vij,syn).
The variable sj(t) is related (perhaps in a very complicated fashion) to the potential Vj (t) and gij, Vij,syn are constants. For example, sj (t) may satisfy an auxiliary equation, such as dsj
-- k ( g j ) ( 1
- sj ) - ;s--L-J.
dt
r
Let V* (t) be the voltage component of the adjoint and Vo(t) the voltage component of the periodic trajectory of the uncoupled system. We let so(t) be the periodic solution to the equation for s when V is set equal to Vo(t). Then, from (2.3), the coupling function of cell 2 to cell 1 becomes
HI (dp) = -g12 7 f0T V* (t)so(t -Jr-~b) (Vl2,syn - v o ( t ) ) d t . A similar formula holds for H2, which is a function of - r by replacing r by - r and interchanging the indices. There is a nice intuitive interpretation of the voltage component of the adjoint: it is an infinitesimal phase response curve (see Section 4.5 for more details). That is, it gives the effect on the phase of the oscillator when the latter is perturbed with a brief positive current pulse, i.e., a delta function. Thus, the function H1 (r is obtained as the convolution of the infinitesimal phase response curve with the current so(t + q~)(VlZ,syn -- Vo(t)) that carries the coupling signal. The latter can be experimentally measured so that, even in the absence of an explicit membrane model, it is possible to derive a phase equation description for weakly coupled neurons. The other kind of coupling considered here, through electrical synapses or "gap junctions", is much simpler in form. The coupling is directly dependent on the difference of the potentials, i.e., the coupling from cell j to cell i is a discretized form of the Laplacian: /coupling = gij ( Vj - Vi ).
In this case, (2.3) implies that the associated coupling function is
gij fo T V*(t)(Vo(t + r Hi (r -- --T-
- Vo(t))dt.
One important distinction between gap junctions and general synapses is that for gap junctions, H1 (0) - - 0 whereas for synapses, Hi (0) will not generally vanish. As we will see, the size and sign of Hi (0) is relevant to the frequency of the coupled system, which need not be the same as that of the uncoupled oscillators. Furthermore, if Hi (0) ~- 0, for at least one of the coupling terms, then larger arrays of oscillators need not synchronize (Kopell and Ermentrout [33,34]).
10
N. Kopell and G.B. Ermentrout
2.2. Computingfrequency, phase lags and stability for pairs of coupled oscillators The main reason for the development of the present theory is to study the synchronization properties of coupled neural oscillators. We start by considering a pair of mutually coupled identical oscillators with symmetric coupling. Then Equations (2.2) hold with H1 : H2 and col = 1. Let ~p = 02 - 01 as above. We can rewrite this system as
d~
dt
dO1 dt
=
--26Hodd(~b),
(2.4)
-- 1 + eH(q~),
where Hodd is the odd component of H. Phase-locked solutions are solutions for which the phase difference 4~ is constant at a root of Hodd(4~). Since Hodd is an odd periodic function, there are always at least two roots, synchronous (tp -- 0) and antiphase (4) = T/2). The frequency of these solutions is s = (1 + eH(~))/T. Note that, for the synchronous solution, synaptic coupling changes the frequency by ell(O)~ T, but electrical coupling keeps the frequency the same. One of the goals of the analysis is to determine the stability of the synchronous and antiphase solutions and how it depends on the parameters in the models. It follows from (2.4) that a phaselocked solution ~b is stable if and only if H~dd(tp) > 0. In particular, the synchronous solution is stable if and only if H' (0) > 0. For synaptically coupled neurons, stability of synchrony can be investigated by looking at the sign of the integral
Hi(O) = ~1 fo T V*(t)s~(t)(Vsyn- Vo(t))dt
(2.5)
For neurons coupled with gap junctions,
lf0
. ' (o) = ~
v* (t) v/~(t) dt.
The key to understanding situations that lead to synchrony for weakly coupled neuronal oscillators is found in the evaluation of these integrals. In Figure 2.1 we illustrate these integrals for excitatory synaptic coupling, in which the driving term (Vsyn - Vo(t)) is positive. (For inhibitory coupling, the driving term is negative, changing the sign of H' (0).) In Figure 2.1A, the adjoint is non-negative, as is the case in many simple cortical neural models. For an excitatory synapse that persists for a few milliseconds, the function so(t) rises quickly and then decays more slowly; thus the derivative, s~ (t) is positive only very briefly during the rise time and is then negative after that. This implies that the product, V* (t)s~ (t), is negative for most of the time (bottom of Figure 2.1A) and so the integral (2.5) is negative and synchrony is unstable. On the other hand, in cortical models that have spike-frequency adaptation (an additional outward ionic current that tends to slow the firing down), the PRC/adjoint can be negative for some time after the spike. The intuition behind this is that,
Mechanisms of phase-locking and frequency control
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S
I
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s'(t)
's~.
I I
11
I l 1
' , ~ J s'(t)
I I I
s .A
t
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A
V*(t)s'(t)
-
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A2
t
Fig. 2.1. The adjoint, V* (t) plotted along with the derivative of the synaptic conductance, s' (t) (upper panels) and the resulting product (lower panels). If the sign of the integral of the product is positive (negative) then synchrony is stable (unstable). A. A negative product in a neuron without adaptation; clearly the area A1 is larger than A2. B. A positive product in a neuron with adaptation; the adjoint has a significant negative region to the fight of the origin.
when there is such a current, stimulation after the spike creates more outward current, and thus delays the onset of the next spike. In Figure 2.1B, we depict this situation. The result is that the product V* (t)s~ (t) is now primarily positive and thus the integral (2.5) is positive and the synchronous solution stabilizes. A similar analysis can be used to show that gap junction coupling is stable for systems with a PRC shaped like V* in Figure 2.1A. PRCs of this shape occur when the spike is very narrow; V~(t) is positive for most of the cycle corresponding to a slow depolarization (increase in voltage) leading to the next spike. Thus the product of V~(t)V*(t) is mainly positive resulting in a positive integral and a stable synchronous state. Adaptation does not change this picture. The key observation is that, for very narrow spikes, the timewindow in which the potential is decreasing is small and occurs at a time when the PRC is small in magnitude. Thus, even though this derivative is quite large and negative, its effect is strongly diminished by the neuron's inability to respond to stimuli at that point in time. In Section 3.3. we use other methods to show that changing the shape of the spike can change this conclusion, especially at high frequencies. In Section 2.5 we also consider oscillations that are more gradual (less spikelike), and show why gap junctions can destabilize synchrony. If the oscillators are not identical, but are close in frequency, a phaselocked solution can be found from (2.2). We may also relax the assumption that the coupling is symmetric. If the oscillators are to be phaselocked, we must have dO]/dt = dO2/dt, so the analogue
N. Kopell and G.B. Ermentrout
12 of (2.4) is
dcb/ dt = e ( A + H2(-q~)
-
nl (~)),
where eA = 1 - 092. It can be seen from this equation that the result of unequal frequencies and/or differences in the coupling is that a phase-locked solution does not necessarily lock at synchrony, but with a phase-lag that compensates for the difference in frequency and/or coupling. In Section 4, we will contrast this behavior with that of relaxation oscillators, which can synchronize even when the oscillators are different and the coupling is asymmetric.
2.3. Delays As we mentioned above, the function sj(t) can be quite complicated in its relationship to the potential, Vj. For example, suppose that the synapse occurs on a dendrite some distance from the cell soma where the oscillations are generated. Then the effect of this distant synapse must be filtered through the dynamics of the dendrite. In the case of passive dendrites, the filtering is linear and
f0 cX3R ( t ' ) s o ( t - t ' ) d t ' ,
sR(t) =
(2.6)
where so(t) is directly related to the potential of the presynaptic neuron and R(t) is the filter function. In the simplest case, the filter can be a simple delay, R(t) = 6(t - r) so that SR (t) = so(t -- r). Because the delays are linear, it is very simple to compute the effect of these on the averaged coupling functions. Let
Ho(,)
-
l for V*(t)so(t + ~b)(Vo(t)- Vsyn) dt
be the interaction function in absence of the filter R, as in (2.3). Then, using (2.6) and changing the order of integration, we get that the full interaction function is just HR(*) --
R(t')Ho(c~-t')dt'.
f0 ~176
(2.7)
That is, the true interaction function is obtained just by averaging the restricted interaction function against the functional R. In particular, delays lead to a phase translation of the interaction function: H ~ ( O ) = n0(4~ - r ) .
In many neural systems, there exists the possibility of delays at many levels. There are delays due to axonal propagation and synaptic transmitter diffusion, and delays encountered when synaptic input occurs far from the source of the oscillation and must be communicated through dendrites. A delay, whether explicit or via a functional relationship
Mechanisms of phase-locking and frequency control
13
like (2.6) can cause the rising phase of the synapse to occur during the maximal sensitivity of the phase-response curve, and thus the integral will be positive (negative) for excitatory (inhibitory) synapses and synchrony will be stable (unstable). This is the principle that underlies the results of van Vreeswijk et al. [65], Crook et al. [ 11,12], and many others. This is illustrated in Figure 2.2 for the case of a pure delay in the synapse. As we mentioned above, one source of the delay can be the propagation time down the axon. In this case, r = d / v where d is the distance between the two neurons and v is the axonal conduction velocity. Thus, neurons that are connected at different distances can have different synchronization properties solely due to this conduction delay. These ideas were exploited in Crook et al. [ 11 ] in a one-dimensional spatial network of cells. In this model, the excitatory synapses were synchronizing with short delays but desynchronizing with long delays. Thus, if most of the connections are local, synchronized activity is stable. If connections between cells have a longer range, then the delay destabilizes the synchronous solution and waves result. Similar models involving inhibitory synapses show that delays can change synchronization properties. Another source of delay occurs due to the passive propagation of potential down the dendrite of a neuron. Crook et al. [ 12] show that the position of a synapse on a dendrite (Figure 2.3) can alter the stability of the synchronous solution. For simplicity, we assume
V (t)
T s~(t +I:) Fig. 2.2. An appropriate delay r can shift the positive part of sf(t) toward a region where the adjoint is large and positive, thus stabilizing synchrony since the area of the product will be positive (see Figure 2.1).
Fig. 2.3. Diagram of two neurons that make synaptic connections along the dendrite.
14
N. Kopell and G.B. Ermentrout
that the d e n d r i t e c a n be m o d e l e d as an infinite c a b l e satisfying: OV rm m Ot
- -
--
-~- ~ 2 02 V
V
Ox 2 '
w h e r e rm is the m e m b r a n e t i m e c o n s t a n t a n d )~ is the s p a c e c o n s t a n t o f the m e m b r a n e . To c o m p u t e the effect o f a s y n a p s e that is a d i s t a n c e d a w a y f r o m the a c t i v e l y o s c i l l a t i n g s o m a , w e use the G r e e n ' s f u n c t i o n for p a s s i v e cables: 1
G(x,t)
-
e
--t /rm--(x /Z)2/ (2t /rm)
v/4zc z z t / r m
R(t)
n o
25
40
Soma
,
,
,
,
20
-25
0 -20
-75
,
I
,
10 '
I
I
i
I
20 '
I
-50
i
30 '
I
I
i
40 '
I
-40
50
0
2
'
Dendrite (1)
10
20
,
,
1 0
-60
-1
-70
,
I
,
10 -63
.
I
,
I
,
20 ,
-64
,
I
,
30 ,
,
-2
i
40
50 ,
0
10
20
10
20
0.7
Dendrite (5)
-65
0
-66 -67 '
-68 0
10
20 t (msee)
' 30
'
J 40
'
-0.7 50
0
t (msee)
Fig. 2.4. Left: The response function R (t) for synapses on the soma or different compartments of a dendrite of the post-synaptic cell. Right: The odd part, H0, of the corresponding interaction function. Somatic synapses lead to a small stable phase-lag between identical cells and both the synchronous and anti-phase solutions are unstable. A dendritic synapse near the soma results in bistability of the synchronous and anti-phase states. If the synapse is further away, only synchrony is stable.
Mechanisms of phase-locking and frequency control
15
In this case, our filter is simply R(t) = G(d, t). Figure 2.4 illustrates R(t) for three different distances from the soma and the corresponding function H (40. From (2.7), we have
H~(O) =
R(t')H~(-t')dt'.
f0 G
Thus, the effect of the dendrite is similar to the delay. For example, suppose that H6 (0) < 0 so that synchrony of a directly coupled synapse is unstable. Then, by putting the synapse farther out on the dendrite, we can weight the above integral to a point where H~ is positive and thus stabilize the synchronized state. By adding additional structure to our simple point neurons (such as passive dendrites or axonal delay) we are able to alter the synchronization properties of a pair of mutually coupled cells. Any linear manipulation of the timing directly translates into a weighted average of the original interaction function through (2.7). Bressloff and Coombes [5] offer a complete analysis of coupled integrate-and-fire neurons with dendritic delays. We discuss the general ideas later in the chapter.
2.4. Frequency effects and the rise~fall time of synapses In the arguments above, we have altered either the phase-response curve (through adding an adaptation current) or the timing of the synapse. If we hold the timing of the synapses fixed and vary instead the frequency of the underlying oscillations, we see that the effect is almost as if the synaptic timing is varied. This is illustrated in Figure 2.5 for a biophysicallybased cortical neuron model with a fast excitatory synapse (rise time is 0.05 msec and
1
Ho(~) 0.5
0
-0.5
-1 0
0.2
0.4
0.6
0.8
Fig. 2.5. The odd part of the interaction function for a fixed synaptic time at three different frequencies. At high frequencies, only the anti-phase state is stable.
16
N. Kopell and G.B. Ermentrout
.5
o
-.5
m
m
i
m
m
i
m
m
m
=n
Frequency Fig. 2.6. Schematic bifurcation diagram with frequency as the parameter showing stability of the synchronous state at low frequencies and stability of the anti-phase state at high frequencies.
decay is 2 msec). The odd part of the interaction function is illustrated. Zeros are phaselocked solutions and those with a positive slope are stable and those with a negative slope are unstable. As the frequency decreases the antiphase state loses stability through the appearance of an intermediate locked state. This stably persists until the frequency is quite low and the synchronous state becomes stable. One can summarize this behavior via the bifurcation diagram shown in Figure 2.6. The analogous behavior is seen for inhibitory coupling but the stability is reversed. Van Vreeswijk et al. [65] obtained a similar result with I&F neurons but the frequency was held fixed and the timing of the synapse was varied. As we show in Section 3, similar results can be obtained using the spike-response methods, even when the coupling is strong. In that case, however, the frequency of the coupled system is determined only implicitly.
2.5. Interaction n e a r a H o p f bifurcation or a h o m o c l i n i c orbit Near a Hopf bifurcation, the resulting periodic orbits are only weakly stable. Thus, trajectories that have initial conditions off the limit cycle may take a significant amount of time to return to the latter; during this time, the dynamics can act on the system to stabilize or destabilize a configuration. For most kinds of neural coupling, the assumption that the system is near a Hopf bifurcation is not very useful. The reason is that supercritical (i.e., stable) Hopf bifurcations are of low amplitude, and these are not large enough to trigger chemical synaptic interactions, which are via all-or-nothing spikes. (But low-amplitude oscillations can contribute to interactions of spikes. See Section 4.5.) For coupling involving electrical synapses (gap junctions), small amplitude oscillations can be sufficient to carry coupling signals. In the rest of this subsection, we discuss the normal form associated with coupled oscillators, each near a Hopf bifurcation. We then use these and some geometric ideas to show why the discrete version of diffusion need not be stabilizing. Near a Hopf bifurcation, the equations for a single oscillator can be expressed compactly using complex notation z (instead of a 2-dimensional real system), and neglecting terms
Mechanisms of phase-locking and frequency control
17
higher than the third power. With appropriate change of coordinates, and linear coupling between the oscillators with variables z l and z2, the truncated coupled equations take the form (Aronson et al. [2]): dzl = lzI(a + bi)zl - (p + q i ) z 2 z l + v(d, + id2)(z2 - zl), dt dz2 = #2(a + bi)z2 - (p + iq)z2z2 4- v(dl + id2)(Zl 22). dt -
-
The parameters here are real, and we will consider only the case #1 = #2 - - #, a small parameter measuring the distance to the Hopf bifurcation. There are two conceptually important parameters to consider here. One is rl = q / p , which measures the "twist" in the system for an individual oscillator. This parameter describes how much the local angular velocity at any point off the limit cycle deviates from that on the limit cycle. The second important parameter is ot = d2/dl, which measures how much the coupling deviates from being scalar (a real number times the difference between the positions of the two oscillators). To understand why these parameters matter, we consider a simple case motivated by an example of Sherman and Rinzel [54], who showed numerically that weak gap-junction coupling (in cells of the pancreas) could lead to antiphase solutions. The equations have limit cycles that are relatively sinusoidal in shape, without spikes or well-defined plateaus as in the relaxation regime of the van der Pol equation. For illustration, we use the MorrisLecar equations (Morris and Lecar [46]), in the non-relaxation regime. These equations (uncoupled) have the form 61)
C-dt dw dt
=
--[gcamoc(V)(1)
-- [ w o c ( v ) - w ] / r .
--
VCa) + gKW(V -- VK) + gL(1) -- VL)],
(2.8a) (2.8b)
The following idea shows why the twist and the deviation from scalar coupling can conspire to desynchronize the synchronous state. Consider a limit cycle with two pairs of initial conditions, and the direction those trajectories are pushed by the coupling (Figure 2.7). Recall that neural oscillators interacting through gap junctions have a coupling term only for the v-component; hence the coupling pulls each trajectory horizontally. Consider the bottom pair of initial conditions, where the limit cycle has increasing v and increasing w. The leading trajectory is pulled inward and the lagging one is pulled outward. Here is where the twist is relevant: the cells are pulled somewhat off the limit cycle, and in the new position moves faster or slower compared to the limit cycle. The change in speed cannot be seen from the phase-plane diagram, but can be seen from the second equation, which says that the rate of change of w is proportional to the vertical distance to the w-nullcline. The leading oscillator is pulled inward, and hence closer to the nullcline, while the lagging oscillator is pulled outward and hence farther from the nullcline. Thus, in that portion of the trajectory, the leading oscillator slows down and the lagging one speeds up, creating a synchronizing effect. A similar analysis shows that the opposite is true for the pair of conditions in the region in which v is decreasing while w continues to increase. Thus, there
18
N. Kopell and G.B. Ermentrout
Fig. 2.7. Diffusive coupling can be synchronizing or desynchronizing in different portions of the limit cycle. For the lower pair of initial conditions, the coupling pulls nearby trajectories closer together, while the opposite is true for the upper pair. See text for explanation.
are synchronizing and desynchronizing effects along different portions of the trajectory, and whether the synchronous state is stable depends on a balance of those effects. The normal form contains the essential information about this balance. For the equations of Sherman and Rinzel, deVries et al. [66] carried out the normal form analysis. As in Aronson et al. [2], they do a change of coordinates to polar form, with
zj -- ~/lZJ 7 r ja
eiOj.
If #2 = #l = # the equations for the polar variables are given by /'1 = rl (1 -- r 2) + y (cos4~ - o t sin ~b)r2 - y r l , /'2 -- r2(1 - r 2) + y(cosq~ + c~ sin~b)rl - yr2,
-- + ~ ( r 2 . A r 2).
g ( s. i n ~ +. ot cos~b) r2
y (sin 4~ - ot c o s r rl
rl
r2
where 4' = 02 - 01 and y = v d l / # a . Note that y may be interpreted as the strength of the coupling compared with the attraction of the limit cycle. It was shown in Aronson et al. [2] that the synchronous solution is stable if and only if
•
+ (1
> o.
This implies that if c~O > 0, the synchronous solution is always stable; if this product is sufficiently negative, then y must be large enough to produce stability.
Mechanisms of phase-locking and frequency control
19
The condition for stability of the antiphase solution is ~, < 1 / 4 , 0 < V(1 +or 2) + (1 + c~r/)(2V- 1). This implies that, if (1 + or0) < 0, then for small enough coupling, the antiphase solution is always stable. DeVries et al. [66] computed the values of the parameters in the normal form, and showed that the antiphase solution is stable, and the synchronous one is unstable. We note that the parameters c~ and 0 also turn out to be important in creating spatiotemporal chaos in the complex Ginzburg-Landau equations (Kuramoto [37]) zt -- )~z - (1 + i o)z2-~ + y(1 + iot)zxx.
A situation in which the desynchronization can be seen immediately from the geometry, without a normal form analysis, is given in Han et al. [22]. In this situation, the limit cycle is near a homoclinic orbit and thus spends a great amount of time near the saddle point. Suppose that the trajectory of each oscillator comes close to the stable manifold of the saddle point (Figure 2.8). Consider initial conditions in which both trajectories are approaching the saddle point. The trailing oscillator will be pulled outside of the limit cycle while the leading oscillator will be pulled inside the limit cycle. Thus, the trailing oscillator will be pulled closer to the stable manifold of the saddle while the leading oscillator is pulled further away. The oscillator on the trajectory nearer to the stable manifold moves slower, and hence the phase-difference widens. A similar analysis holds for initial conditions in which the trajectories are moving away from the saddle point.
W
>
u
Fig. 2.8. Illustration of why diffusive coupling between voltages near a homoclinic destabilizes synchrony. The stable and unstable manifolds and the limit cycle are illustrated. On the left part of the limit cycle (near the stable manifold) diffusion pulls the slower oscillator closer to the stable manifold and the faster oscillator away from the manifold. Thus the faster cell goes even faster and the slower one goes slower, so that they separate in phase. The same reasoning holds for initial conditions on the right hand side of the rest point.
N. Kopell and G.B. Ermentrout
20
A final remark is that the destabilization of the synchronous state by twist and non-scalar coupling happens in a range well described by a normal form near a Hopf bifurcation; it fails when the oscillations are in a relaxation regime. The reason is that the trajectories are held so closely to the nullclines that the twist has too minor an effect to change stability.
3. Interactions of spiking cells 3.1. Spike response method and periodic solutions Most neurons fire all-or-nothing action potential (spikes), and the interactions of the neurons are via signals induced by the receipt of the spikes. In some simple cases, this leads to a different kind of reduction than the weak-coupling limit. This subsection of the review is taken from Chow [8]. The simplest version of such a reduction assumes that the nonlinearities of currents in (1.1) are localized in time to the action potential. That is, away from the spike, the voltage equation has the form
dl)
dt
--l--v.
As described in Section 2, this is known as an integrate-and-fire model. When the potential reaches threshold, a delta function pulse is subtracted to reset the potential to v = 0 and we say that the cell has spiked. Though this may appear too highly abstracted to be useful, it is a reasonable approximation when the behavior between the spikes is essentially linear. A biophysical model that behaves in a very similar way is one reduced from a compartmental model of Traub and Miles [61 ]. (See appendix of Ermentrout and Kopell [ 17] for the reduction.) With synaptic current, the equation for the i th cell is
dvi = Ii- Ui- Z6(tdt l
i tl)-kJ ~
S (t - t / ) ,
i=/=j,
(3.1)
l
where tIi are the times that cell i reaches threshold, J > 0 for excitatory interactions and J < 0 for inhibitory interactions. The functions 3 (the Dirac delta function) and S above are each defined only when the argument is non-negative. The synaptic current S can be modeled in various ways. One simple and explicit version has the current induced from each spike as a double exponential
S(t) - e x p ( - f l t ) - exp(-c~t),
(3.2)
where fl is the synaptic decay rate and ot is the synaptic rise rate. Note that this describes the synaptic current explicitly as a function of t, rather than obtaining it as a solution to an auxiliary differential equation as in Section 2. With ot >/3, there is a rapid rise, followed by a slower decay, as in solutions to auxiliary equations described in Section 2.
Mechanisms of phase-locking and frequency control
21
A central idea in working with some networks of spiking cells is to consider the effects of the spikes, rather than the differential equation (3.1). This leads to a formalism known as the spike-response method (Gerstner [ 19], Chow [8]). The dynamics are encoded in two sets of kernels, representing the effects on a cell of its own spikes and those of the other cell. That is 7 L ~ r l i ( t - - t l ) - + - J Zi 6 1
ui(t)--Ii
i (t --
t/).
(3.3)
l
The kernels are zero for t - t~ < 0. The kernel r/i (t) describes the recovery dynamics of neuron i after it fires, while the kernel 6i (t) represents the response to synaptic inputs (see Figure 3.1). Arguments using the kernels for integrate-and-fire, which can be explicitly computed, generalize to other kernels having similar qualitative properties. But the theory has some constraints. For example, it assumes that the form of the synaptic kernel does not depend on the time of arrival or phase of the synaptic input. Thus the interaction kernel has the form t
e(t)--
Z
L(s) S ( t - s ) d s ,
(3.4)
where S(t) is the synaptic input and L(t) is the linear response function. With a prescribed synaptic signal, it is possible to address questions about existence and stability of periodic solutions in the network. One such way is to look for self-consistent solutions, taking into account all the history of the previous spikes. Suppose that the cells have been firing with (so far unknown) period T, with neuron 2 lagging neuron 1. We pick the origin in time so that cell 1 has fired at t/ - - l T, and cell 2 has fired at t~ -- (4> - l)T, l = 1, 2 . . . . . where 0 ~< 4) < 1. Cell 1 fires again at T = 0, and then self-consistency requires that 1 -- vl (0) -- II + Z
rl(IT)+ Z
l>/1
3.0
1.5
~T).
Je(IT
(3.5a)
l>/1
-
0.3
-
0.2
"
"~
~Ib
"~ 0.1 0.0 -1.5
I 0
,
I 2 t
,
.......
I
I
4
0
,
I
,
5
I I0
,
I 15
t
Fig. 3.1. Examples of kernels O(t) and e(t). The two curves for e(t) correspond to two choices for the form of the synaptic input S (t).
22
N. Kopell and G.B. Ermentrout
Cell 2 fires again at t = r T, and self-consistency then implies that 1 = v2(r
=/2 + E
rl(1T) + E
Js(1T + ~T).
(3.5b)
The difference in the index of the second sum in (3.5a) and (3.5b) arises because cell 1 fires before cell 2. Hence, the input to cell 2 (from all negative times) includes the 1 = 0 input from cell 1, which occurs at t = 0. However, the 1 = 0 firing of cell 2 is at t = ~bT > 0, which is not included as an input to cell 1. One can derive a relationship for the phase ~b by subtracting (3.5b) from (3.5a) to get 0-= Ii - 12 - Gr(q~),
(3.6a)
GT (~b) = J ~-"~(s(IT - T + q~T) - s ( Z T - dpT)).
(3.6b)
where
l>~ l
Note that, from (3.4), s(0) = 0; this implies that G r ( 0 ) = 0 = GT(0.5). Hence, if the neurons are the same (I1 = I2), there is a synchronous solution (4~ = 0) and an antiphase solution (~b = 0.5). The difficult part is determining stability. The consistency equations do not give a map that takes a phase difference 4~ between the cells at some time to the phase difference at the next cycle, only a condition to be satisfied at steady state. Hence, other arguments are needed to deal with stability. We refer the reader to Chow [8] and Bressloff and Coombes [4] for the stability argument. This is subtle because the system is infinite-dimensional; one needs to know information about all past cycles. Chow [8] looks at small perturbations around a periodic phaselocked solution; assuming that the kernels decay very fast, so that they are negligible after some time, he derives a high-dimensional return map. The analysis shows that a necessary, but not sufficient, condition for stability is that G} (~b) > 0. In related work, van Vreeswijk et al. [65] argued heuristically that this condition was associated to stability. Gerstner et al. [20] did a related analysis with N ~ cx~ neurons; this allows the population to force a single neuron without itself being affected by that neuron, creating a somewhat easier problem. The methods of Chow [8] can also be used for larger networks. We can now see how this model is related to the weak coupling theory. Indeed, if Il = I2 = I (that is, the oscillators are identical), then we must solve a pair of equations for the ensemble period, T and the phase, r 0 = -Gr(r 1 = I + O(T) + Jg(T, dp), where O(T) and g(T, ~b) are the first and second sums in (3.5). This set of equations should be compared to phase-locked solutions to (2.4): 0 -- --28Hodd(~b),
Mechanisms of phase-locking and frequency control 1 --
T
=
23
1 + ell(g)).
Note that for weak coupling, the phase is determined independent of the period. For the spike-response model, GT (4~) changes with T, and so the non-synchronous solutions have frequency-dependent phases. (GT(0) = 0 for all T.) In the case of weak coupling for the spike-response model (J small), we can approximate T by the uncoupled period and obtain a function G T that is essentially (to lowest order) independent of T. Furthermore, since the period does not vary to lowest order, the stability is reduced to that of a 1-dimensional system, as in the first equation of (2.4). The condition G~(~b) > 0 is a sufficient condition (and, as above, necessary) for stability for the weak coupling case and, up to a scaling factor, GT -- Hodd. Bressloff and collaborators have extended the theory of coupled integrate-and-fire models to arrays and networks where they then use the results of Kopell and Ermentrout on large systems of phase models (Bressloff and Coombes [6]). This is possible since the resulting equations are functions of phase-differences only and fixed points correspond to periodic phaselocked solutions. As in the case of two oscillators, the main technical difference is that the ensemble period is implicitly defined while in the weakly coupled case, it is explicit. Bressloff and Coombes [5] have applied the techniques of this section to a pair of integrate-and-fire neurons coupled through passive dendrites in a manner similar to the work of Crook et al. [ 12]. In their work they can explicitly compute the stability boundaries for synchrony as a function of the position of the synapse on the dendrite for arbitrary coupling strength since the model is essentially linear. They have also studied the role of synaptic depression and other effects on the phase-locked behavior of coupled integrateand-fire oscillators. They have developed elegant machinery for analyzing arbitrary networks of integrate-and-fire neurons that goes beyond the special cases considered in this section.
3.2. Networks with inhibitory connections Many theoretical papers have remarked that inhibition may play a more important role in synchronization of networks than excitation (van Vreeswick et al. [65], Gerstner et al. [20], White et al. [71]). Hence we focus on inhibition; similar methods work for excitatory connections and show, as in the case of weak coupling, that such an interaction is not synchronizing. For inhibitory networks, we show below that the inhibition provides both the synchronizing signal and determines the frequency of the coupled network. The work is motivated by experimental work on a slice preparation of the hippocampus, in which the action of the excitatory cells is blocked and the relevant part of the network contains only inhibitory cells (Whittington et al. [72]). 3.2.1. Synchronization by inhibition. For inhibitory connections, we set J = - 1 . (The size of J is taken into account in the definition of e.) Starting from formula (3.6) for
24
N. Kopell and G.B. Ermentrout
the function GT(Ck) and an explicit form for the synapses, we can compute the necessary conditions for stability of synchrony, namely G~ (0) > 0. We first note that, from (3.6), G'~(O) - - T s ' (O) - 2 T ~
s' (IT).
Thus, the synchronous solution is unstable if s'(0) > - 2 Y~4>~t s'(1T). For a particular form of the synapse, such as the double exponential (3.2), the s kernel can be explicitly l computed, and used to compute G T (0). It can be shown that if the rise rate ot is infinite, then the necessary condition is violated; otherwise the necessary condition is always satisfied for integrate-and-fire neurons. It can be shown that some sufficient conditions are also satisfied except possibly for some very small periods. Van Vreeswijk et al. [65] made the same argument for the special case of the standard integrate-and-fire model with synapses of the form s(t) = at exp(-ott). The same conclusion is reached by working with weak coupling methods, as in Section 2.2. Indeed, the function s(t)(Vsyn - V) in that section plays the same role as the explicit synaptic function S(t) plays in computing s(t) in (3.2). The integrate-and-fire formalism captures some essential behavior of what has been called "type 1" neurons (Rinzel and Ermentrout [49]). These are neurons for which the transition between stable equilibrium and periodic firing (as applied current is increased) is via a saddle node. "Type 2" neurons are associated with Hopf bifurcations near the onset of periodicity with increasing drive. As we will see below, the differences between type 1 and type 2 affect synchronization properties. We remark here that type 2 cells have behavior that is not captured by the integrate-and-fire formalism; in particular, such neurons need not synchronize with inhibition. For a kernel approach to that behavior, see Kistler et al. [30]. For type 1 neurons, the stability analysis shows that the synchronous solution for homogeneous cells is stable over a very wide range of periods (changeable by modulating the applied current). However, the stability is fragile when there is heterogeneity, e.g., in the drive I. (Relevant simulations are in Wang and Buzs~iki [68] and White et al. [71 ].) For heterogeneous neurons, a similar analysis can be done (Chow [8]). When the heterogeneity is too large, there is no locked solution. For smaller heterogeneity, there is a locked but not synchronous solution. The stability of the latter turns out to depend partly on the emergent frequency of the network, which in turn depends on applied current as well as the synapses (White et al. [71 ], Chow et al. [ 10], Chow [8]). The close-to-synchronous solutions are stable only in a parameter regime in which the network frequency is governed by the decay time of the inhibition (see Section 3.2.2 below). The system can be driven so that individual cells fire at much higher frequencies, but in that case the effect of the heterogeneities is to produce phase drift between the oscillators. In this regime, the synaptic current does not have enough time to change significantly within a cycle, and hence acts effectively as a DC current. This regime has been called the "tonic" regime, for the tonic (i.e., constant) behavior of the coupling current (Chow et al. [10]). If the cells are driven at much lower levels, the effect of the heterogeneity is to cause the faster cell to suppress all firing in the slower one. The highest degree of synchrony occurs when the drive is such that the frequency of the coupled system is closely related to the decay time of the inhibition, a parameter regime
Mechanisms o f phase-locking and frequency control
25
that has been called the "phasic" regime. As described in Section 3.2.2, the dependence of the network frequency on the decay time of the synapse is different in the tonic and phasic regimes. We can get more insight into the way inhibition synchronizes if we consider the case in which only the previous spike from a pre-synaptic neuron is significant in affecting the timing of the post-synaptic neuron. This is a reasonable assumption in the phasic regime. Let tl and t2 be the times of spiking of a pair of coupled cells on some cycle. Assume t2 > tl, but sufficiently close that the firing of cell 1 does not prevent the firing of cell 2 in that cycle. We start with the analogue of (3.3), namely v i ( t ) -- Ii + ~ ( t - ti) - ~(t - t j ) ,
j =7t=i.
(This ignores the effect of cell 1 on cell 2 in the current cycle, which is permissible if the cells are close together, since ~(0) -- 0.) Cell i fires again when vi (t) -- 1, at t = i/. We wish to compute t2 - t i = A and compare this to t2 - t l -- A to check for stability of the synchronous solution. For simplicity, we again assume that the driving currents Ii a r e equal. The definitions of ti imply that _
,(i.
-,.)
-
1 -
To linearize the equation around the values associated with one cycle later, we let Tp be the period of the coupled synchronous solution, and define pi via ti -- ti -- Tp + Pi. Using the relations il - t2 - Tp + Pl - A and {2 - tl -- Tp + P2 + A, we get r l ' ( T p ) ( p 2 - P l ) -- s'(Tp) [(p2 - Pl) + 2A].
Using P2 - Pl = A -- A, we have A--
~ ' ( T p ) + r/'(Tp)
e'(Tp) - r l ' ( T p )
A.
Note that r/' (Tp) > 0 and we are assuming that Tp is such that e' (Tp) < 0, i.e., the period is high enough that the synapse is decaying when the next spike appears. (See Gerstner et al. [20].) Hence the coefficient of A is less than 1 in absolute value. Thus, if the rise of the synapse is not infinitely fast, we recover the result from the above theory that the condition e' (Tp) < 0 is a necessary condition for inhibition to be synchronizing. But the calculation also shows that the condition e'(Te) < 0 is sufficient for synchrony, and that the size of r?' (Tp) helps determine the rate of synchronization near the synchronous state; if r/' (Tp) is small, as it would be if the period is too long relative to the recovery period of the neuron, the rate of synchronization would be low. Note that the r/(t) in Figure 3.1 decays quickly; however, if the spike of the cell triggers self-inhibition (via an "autapse", or self-synapse), the resulting kernel O(t) decays over a much longer time, and the two cells synchronize much more quickly.
N. Kopell and G.B. Ermentrout
26
A related treatment is done in Chow [8], assuming that the neuron that fires second is influenced both by the last recent firing of the leading cell and the spike on the previous cycle. 3.2.2. F r e q u e n c y regulation in inhibitory networks. In the full Hodgkin-Huxley equations, there can be many different time scales associated with the synapses and the intrinsic dynamics. In order to understand how some of these may affect the p e r i o d of the network solution, we use a slightly more complicated version of the integrate-and-fire neurons, one that is not pre-scaled. It is dV C~ = I - gm(W - Wr) -- g S ( t ) , dt
(3.7)
where gm is an effective membrane recovery conductance and V,. is an effective membrane reversal potential. V (t) is reset to V0 whenever it reaches the threshold potential VT. Note that we are assuming identical cells (11 = 12) and a synchronous solution, so we may drop subscripts. This section is taken mainly from Chow et al. [ 10]. In these equations, there are three natural time scales. The first is the effective membrane time constant g m / C = rm associated with the passive return to the rest potential after a spike. The second is the decay time r of the synapse. (Though the rise time is very important for synchronization, it does not play an important role in frequency determination.) The third is the period T when two identical cells are coupled. This time scale is an emergent property of the system, and can be altered by changing the driving current I. Note that, in the absence of coupling, the frequency of an individual neuron can be very high hundreds of Hz - and the reduced spiking models in White et al. [71 ] and Ermentrout and Kopell [ 17] also behave this way. When coupled, however, the frequency is affected by time scales in the coupling. The aim of the analysis is to understand the circumstances under which different time scales in the equations dominate the determination of the network frequency. As in Section 3.2, we consider different parameter regimes, which we define asymptotically by relationships between T, r and rm. (As usual with asymptotic analysis, the conclusions hold in some regime larger than the strict definition of the regimes.) The tonic and phasic regimes have been introduced less precisely in the previous section. -
Tonic regime: Phasic regime: Fast regime:
T << rm, rm << T, r << T,
T << r. rm << r. r << rm or r ~ rm.
The phasic and tonic regimes are applicable for slowly decaying inhibition from the synapses, where "slow" here denotes the second inequality defining each of these regimes. The differences in the two are seen in the first inequality; in the tonic regime, the network period is small compared to the membrane time constant, while in the phasic regime it is large. Thus, in the tonic regime, there is significant filtering of the synaptic signals from the cell dynamics; in the phasic regime, the cell time constants are too fast to accomplish significant filtering. The fast regime denotes the parameter ranges for which the network
Mechanisms of phase-locking and frequency control
27
period is long compared to the synaptic decay time. In this regime, the intrinsic membrane time constant may be large or comparable to the synaptic decay constant; the coupling is close to pulsatile. The names "tonic", "phasic" and "fast" refer to the effects of the synaptic currents on the voltage of the post-synaptic cell. Thus, in the tonic regime, the synaptic current varies slowly compared with the other time constants, and hence acts effectively like a constant current. In the phasic regime, the current leads to significant changes in voltage within a network period, and so the signal has a significant phasic component. In the fast regime, the signal decays quickly relative to the period and the filtering properties of the network. The dependence of T on r and rm can be determined in a way similar to that used to compute the phase relationship between the coupled cells: one assumes a period T and then shows that T is determined uniquely in a self-consistent way. For the simple equations of the form (3.7), one can deduce a general but implicit formula for T. The computations are most transparent for fast synaptic rise times, in which S(t) = e x p ( - t / r ) , where r = 1//~ is the decay time of the inhibition. (See Chow et al. [10] for generalizations.) One finds that for the phasic regime one gets T = r In A, where A is a function of all the parameters, including r. Thus, the period is essentially proportional to the synaptic decay time, and weakly (logarithmically) dependent on other parameters. For the tonic regime, the dependence is quite different: the period is independent of r (or weakly dependent for other kinds of synapses). In the fast regime, T depends logarithmically on r and other parameters, and is proportional to the membrane time rm. One of the significant differences between the phasic and tonic regimes is the network behavior in the presence of inhomogeneities; it is only in the phasic regime that the network can remain coherent in the presence of even mild heterogeneities (see Section 3.2.1). Hence, when there are inhomogeneities and coherence is observed, the theory implies that the frequency should be proportional to the decay time of the inhibition. This appears true for inhibitory networks in the hippocampus (Whittington et al. [72]). The main inhibitory transmitter associated with this (and many other inhibitory networks) is GABAA, whose decay time leads to a frequency that is in the range of 30-80 Hz, depending on excitatory drive. Rhythms outside this frequency range require a different mechanism for the creation of coherence.
3.3. Spike response method and electrical coupling In the section on phase-difference coupling, the analysis was applied to spiking neurons with very thin spikes, and a heuristic explanation was given to show that electrical coupling need not be synchronizing. We now show that, with the techniques of this section, we can examine the properties of electrical coupling in more detail. The main results (Chow and Kopell [9]) are that the ability to synchronize depends on the shape of the spike, and is highly influenced by the period of the coupled network (which is in turn influenced by the drive to the cells). The reason that the spike shape matters is that the coupling current is a scaled difference of voltages, and therefore depends on the voltage during the spike as well as between spikes. In order to investigate the role of the shape of the spikes, it is necessary to include the spike shape in the equations; the standard I&F equations just reset the voltage as soon as
28
N. K o p e l l a n d G.B. E r m e n t r o u t
the threshold for spiking has been reached. Here, the spike is "pasted on" via an extra term in the voltage equation that changes the voltage in ways that depend on the rise rate of the spike, the width of the spike and a constant to calibrate the maximal voltage that the spike creates. All these parameters (or rather some combination of them) affect the voltage of the cell during a spike, which in turn affects the coupling current between the cells. The scaled equation for an uncoupled cell is du
dt
= I - v + A(t-
tt),
where A(t) -
for0 < t < A , for A ~< t.
I VA e ~-t
I -(1 + Vm)6(t-
A)
Here the dimensionless parameters are the applied current I, the spike rise rate ~, the spike width A, and the spike potential scale VA. The threshold is set at v - 1. The term Vm is computed from the parameters in A (t) in order that, just after t = A, v ( t ) is reset to v = 0. For a fast rising and narrow spike, Vm can be computed to be approximately VAe ~za/ (1 + ~) (Chow and Kopell [9]). The coupled equations have the form dvi
dt
I
(3.8)
Ii - 1)i -+- A ( t - tl) + g(1Jj -- Vi),
where g is the scaled strength of the electrical coupling (gap junction synapse). As in the case of chemical synapses, it is possible to construct an integrated version of these equations written in terms of kernels that encode the effects of the spikes on the neuron that produced that spike and on the other neuron. This is facilitated by working with the sum and difference of the two equations of (3.8). The analogue of (3.3) is then
1Ji(') -- li + Z 1
Ys(' -- 'I' ) + Z
l
Yc (, --
,I).
Here the kernel Ys refers to the effect of a spike on the cell that produced it, and is analogous to the kernel r# in (3.3); the kernel Yc encodes the effects of the coupling, and is analogous to J e . (See Figure 3.2.) Both of these are dependent on the shape of the spikes, as given in the formula for A ( t ) . The analysis is similar to that for inhibitory coupling described in earlier sections. In particular, the formula for the voltage at the next spike is given by (3.5) with r# and J e replaced by Ys and Yc. The formula for determining the phase difference 4, is again (3.6), with the same substitution. Thus, the outcome depends critically on the shape of the coupling kernel yc. The necessary condition for a solution with a phase lag 4, to be stable is that G~ (05) > 0. For a fixed T, this can be computed directly from the analogue of (3.6). As for the inhibitory coupling, Yc(0) = 0, reflecting the fact that the cells do not immediately affect one
29
Mechanisms of phase-locking and frequency control
2
0.2 t _
1
7~ 4 "
0
t
' 5
t
-o.2 ri .." 1".."
-1
-0.4 [;':
Fig. 3.2. Kernels Vs(t) and Vc(t) showing dependence on size of gap junction conductance. The dashed lines correspond to a conductance 10 times as large.
0.5
= i
i
9 i
i
i
iiiiiiiilliiiii
B
@~ @@@
0
=
a
m
m
m
m
I
m
m
m
m
Fig. 3.3. Bifurcation diagram showing dependence of stable solutions (synchrony and antiphase) on the level of drive to the cells. Higher drive corresponds to lower period.
another on firing. Thus, the necessary condition for stability of synchrony is y~ Vc(l T) > 0. Now Yc is monotone-increasing in a tiny range near t = 0, then decreasing for t in some range, and then increasing back up to zero. So if T is sufficiently large, all the terms in the sum are positive; if it is small enough (high frequency), the sum can be negative. This suggests that for high enough frequency, the synchronous solution can be unstable. (It always exists.) This is verified numerically in Chow and Kopell [9] for a biophysically-based model (White et al. [71 ]). A nonsynchronous solution, such as antiphase, may not exist if the coupling is too strong; with strong enough coupling, a spike of one cell quickly produces a spike in the other one, preventing an antiphase solution. How large "strong-enough" is depends on the frequency; for low frequency (high period), the cell receiving the coupling current is close to fully recovered when the spike arrives, and hence a small coupling strength may induce the spike. For fixed coupling strength, antiphase is not possible for low enough frequencies. Figure 3.3 summarizes the existence and stability of the synchronous and out-of-phase solutions as the drive is decreased, which, in turn increases the network period. The diagram was computed from analysis of the necessary condition for stability for the
30
N. Kopell and G.B. Ermentrout
integrate-and-fire models, and matches the behavior found for a biophysical model (White et al. [71]). At very low periods (high drive), only antiphase is stable (or neither is stable). At a higher period, the antiphase solution loses stability (to a third mode we have not discussed). At still higher periods, the synchronous state becomes stable. With drive decreasing more, antiphase becomes stable again and there is bistability of synchrony and antiphase. At the lowest drives, only synchrony is stable. We note that antiphase solutions doubles the network frequency. The frequencies that can be produced by coherent activity mediated with electrical coupling are much higher (even in the synchronous mode) than can be supported by inhibition, which is tied to the rates of the synapses. With antiphase solutions, the electrical coupling can support frequencies over 200 Hz for biologically reasonable parameters. We can summarize a role of the spike shape and the strength of the gap junctions on the network behavior by discussing how these parameters change the endpoints of the frequency regimes in which there is synchronous and antiphase behavior. Computation of GT(q~) shows that this function depends on a combination 6c of parameters including and VA; for fast rising spikes and weak coupling, it can be shown that 6c is approximately 1 + 2gym~(1 + ~), where Vm = VAexp(~A)/(1 + ~). From this, it is possible to compute how the endpoints of the stability regions change with parameters. Increasing 6c, which can be accomplished by increasing the spike amplitude or increasing the spike width A, has several effects. First, it raises the period necessary to have stable synchrony, i.e., makes it harder to have synchrony in high frequency regimes. Second, in the lower frequency regimes, increasing 3c or A increases the lower bound on period for stable anti-synchrony; this diminishes the frequency regime over which the antiphase solution is stable, thus enhancing synchrony. Thus, changing the spike shape can have different effects over different frequency regimes. One way to understand why the shape of the spikes might affect the ability to synchronize is to realize that, during the spike of one cell, the coupling current to the other cell increases the voltage, as in excitatory chemical synapses. During the reset portion and return to threshold of one cell, the current experienced by the other cell acts like an inhibition. Thus, there is tension between the effects of the two parts of the cycle. Since the excitation tends to be desynchronizing, one would expect spikes that are larger (in amplitude and duration) to be more destabilizing than smaller ones. The dependence on network period can be understood in a similar way: changing the drive to the cells mainly changes the interspike interval, rather than the spike itself. Thus, higher frequencies are associated with a large relative amount of excitation, and hence are more vulnerable to destabilization of synchrony. The spike response formalism is the same for both chemical synapses and electrical synapses (gap junctions). Because of the linearity between spikes, the formalism can also be used when there are both chemical and electrical synapses; in that case, the relevant kernels are the sum of the kernels of each kind. Though they do not interact in a nonlinear way (within this class of models), they can produce some nonintuitive effects. For example, if the individual cells are firing at frequencies too high to sustain synchrony even for electrical coupling, addition of inhibition can synchronize the cells. This does not happen by the mechanism described in the section on chemical synapses, which requires that the inhibition be strong enough to dominate the time scale of the emergent rhythm. Even inhi-
Mechanisms of phase-locking and frequency control
31
bition too weak to do the latter can help, however; at the high frequencies (the tonic regime for inhibitory coupling), the coupling term acts as a constant inhibition, pulling down the frequency enough that the gap junction can perform the synchronization (White, Kopell, Chow, unpublished).
4. Oscillators with multiple spiking cells The work in Section 3 concerns a pair of cells, interacting via spikes. The spike-response formalism allows one to deal analytically with dependence on many cycles in the past. In this section, we deal with oscillators that are composites of spiking cells, with excitatory and inhibitory interactions within a single unit oscillator. To handle this analytically, we work in frequency regimes in which the response to a cell receiving a signal depends mainly on the timing of the last cycle, not many previous cycles. In this regime, we have another technique, that of low dimensional maps. There are two different kinds of maps that we will discuss here. One is derived from the classical "phase-response curve (PRC)" theory, which describes how a limit cycle oscillator reacts to a periodic short pulse that perturbs the oscillator (Winfree [75]). The PRC gives the size of a perturbation in phase at the end of a cycle as a function of the time at which the pulse is received. From the PRC, one can read off the possible stable phase lags or leads between the forcing oscillator and the forced one. The theory depends on having the periodic solution of the forced oscillator be sufficiently attracting that the effect of the perturbation is only on the phase, and there is no effect beyond one cycle. This theory can be extended to work with a pair of oscillations whose interactions are via short pulses and is useful in describing interactions via excitatory synapses. The second kind of map is useful in situations in which the time of firing of a cell depends on decay time of the coordinating current, usually an inhibitory current; we call these "hold-and-fire" systems. For such maps, the firing time of each of the cells involved in the circuits is calculated as a function of previous firing times; since the period of the coupled system is usually significantly different from that of the uncoupled local circuit, the maps are not described in terms of changes of phase, and are calculated directly, not from a PRC. We will refer to these as spike-time maps. Though the spike times of all the cells come into the calculation, the maps can sometimes be approximated by ones that are of very low dimension, even one-dimensional. The essential reason has to do with the multiple time scales in the Hodgkin-Huxley equations: on the time scale of the signals, many intrinsic variables relax to a predictable position, decreasing the number of real degrees of freedom. This collapse is associated with different time scales in different problems, and will be illustrated below. It has not been rigorously proved that the remaining degrees of freedom correspond to a center manifold, but we conjecture that this is true.
4.1. Long-range synchronization of the gamma rhythm We now give a case study of a spike-time map analysis. We use the gamma rhythm described above in Section 3.2, as seen in a hippocampal slice. In this case, however, the
32
N. Kopell and G.B. Ermentrout
rhythm is induced by stimulation instead of pharmacological agents that block the excitatory cells; hence the network contains both excitatory and inhibitory cells. At issue is the ability to have precise synchronization of spikes over distances such that the conduction delays are a significant fraction of the period of the rhythm (e.g., 8 ms in a 25 ms period). It was noticed by Traub et al. [63], in both data from hippocampal slices and large-scale simulations with thousands of neurons (each with many compartments and conductances) that synchronization over distances was found in conjunction with a stereotypic pattern of activity: the inhibitory cells fire doublets on each cycle, the first spike (statistically) at the same time as the firing of the excitatory cells. The goal of the map analysis is to understand the role, if any, of the doublets in the synchronization process, and to use that to predict the parameter ranges in which such a mechanism could work rapidly and robustly in the presence of heterogeneity and/or noise. A minimal model has two oscillating local circuits, each composed of one excitatory (E) and one inhibitory (I) neuron (Ermentrout and Kopell [17]). Within each local circuit, the E- and I-cells have inputs to one another. Between the local circuits, the most important connection are the long-distance ones from an E-cell to its distant I-cell (Figure 4.1). Generalizations to a network with a more complex set of connections are in Ermentrout and Kopell [ 17]. Each of the cells is described by a set of Hodgkin-Huxley equations, with minimal nonlinear currents corresponding to spiking; since there are the usual three gating variables (m, n, h) of the Hodgkin-Huxley equations, each cell is four-dimensional. Furthermore, the synaptic currents have their own dynamics, described as differential equations dependent on the voltage, rather than currents with prescribed time courses. Hence the total number of equations for even a pair of 2-cell circuits is quite large. Nevertheless, under some broad set of hypotheses, it is possible to analyze the stability of the synchronous solution with a one-dimensional map. The hypotheses can be checked numerically and shown to hold for the Hodgkin-Huxley equations in some parameter regimes. Some of the hypotheses describe properties of each E-I circuit, i.e., give constraints on ranges of parameters for the Hodgkin-Huxley equations. The rhythm is driven by excitation from the E-cells, so the latter must be excitable enough to fire rhythmically; the populations of I-cells can be either excitable (do not fire without input) or oscillatory, but may not fire faster than the E cells. (If they do, they suppress the E-cell activity.) The I cells have the property (shared by some "type 1 cells") that they can have a long latency to firing after being stimulated because of the proximity of a saddle-node point. This provides
Q
Q DELAY
Fig. 4.1. Minimal model for synchronization of the gammarhythm across distances. Each E-I circuit is an oscillator, and there are long E to I connections between the two circuits.
Mechanisms o f phase-locking and frequency control
33
a form of "relative refractory period", which will be critical to the analysis. We note that this relative refractory period is not encoded in simple integrate-and-fire neurons and is not needed in the contexts discussed in the previous sections" a model that has this property is important only in contexts in which a cell receives excitation shortly after it spikes, as in the case of doublet firing. As discussed below, the second spike of the doublet is provoked by excitation from the distant circuit, arriving before the local I-cell has fully recovered. The construction of the map follows from the above hypotheses. The E-cells are able to fire when synaptic inhibition has worn off sufficiently. Thus, we treat E-cells as linear between spikes, firing when an inhibition-determined threshold has been reached. The Icells have an extra property, associated with relative refractory period. Suppose a cell fires at t - 0, receives excitation at t -- 7r > 0 and fires again at t - TI (7r). The more recovered the cell, the shorter the time to fire. Hence TI decays with 7r. For ~p very short, the cell may not fire at all. As will be seen below, the behavior of the system follows almost entirely from the properties of the map Ti. This map, and its dependence on network properties, can be found by simulation. It can be shown that TI is related to the intrinsic refractory period of the I-cells, but is influenced by the rest of the network. For example, increasing the self-inhibition in the local network increases the values of TI. We now analyze the dynamics of the network in a neighborhood of the synchronous solution. The following is a simplification of an analysis in Ermentrout and Kopell [ 17], using as long connections only excitatory synapses to the I-cells (Figure 4.1). Let tl and t2 be the times of firing of cells El and E2 on some cycle, with A -- t2 -- tl. If the initial conditions are close enough to synchrony (A - - 0 ) , we can construct a map that gives the times il and t2 of firing in the next cycle, from which we can determine the conditions under which there is a stable solution. The time at which the inhibition for an E-cell wears off enough for it to fire is approximated by the time at which the inhibitory conductance has decayed enough; the latter threshold depends on the drive to the E-cells. We assume that there is saturation of the synapse from the I- to the E-cell, so that it is only the timing of the second spike of the doublet that matters. Cell I1 is essentially recovered from firing in the previous cycle when it receives excitation from El at t l. (We assume no or minimal local conduction delay. We are also implicitly assuming that the membrane time of the inhibitory cells is shorter than the inhibitory decay time, which implies the recovery of an I-cell by the end of a cycle.) Hence it fires a short (history-independent) time tel later. (This time can be arbitrarily small, depending on the drive to the E-cells, but the latter drive may not be so large that the Icells fire before the E-cells.) Cell Il receives excitation from E2 at time t2 + 6, where is the delay time between the circuits. Cell I l then fires the second spike of its doublet at t2 + 3 + Ti(t2 § ~ - tl - tei) = TI, inducing an inhibitory current in the E-cell at that time. El fires at t -- il in the next cycle when the inhibitory conductance decays to some threshold level g,, i.e., gie e x p [ - ( { , - ~ ) / r ] -
g,,
(4.1)
where r is the time constant for decay of inhibition and gie is the maximal conductance of the inhibitory current, g, is related to the drive to the cell via the intrinsic period p• induced by that drive in an uncoupled cell, namely: gie exp[(-p~, + t e i ) / z ] = g , . From the
34
N. Kopell and G.B. Ermentrout
above, we can compute the time [l in terms of tl and t2 and similarly for [2. The ability to do the analyses given below depends on the fact that both il and i2 are closely approximated by functions only of t2 - tl. This is true for TI because the I-cells are sufficiently simple that there are no internal degrees of freedom besides the time after firing. Similarly, since the E-cells do not have currents whose gating variables have long time-constants, the time to firing of the E-cell depends only on the times of receipt of inhibition, not other historydependent variables. We now show how we can use the above to analyze the range of parameters in which the gamma configuration is stable. It is easy to solve for t-~.explicitly. Subtracting [l from t2 we get m
m
,4 ~ [2 -- t-I -- --,4 + TI(-,4 + 6 - tei) - TI(,4 -+- 6 - tei) =~ F y ( , 4 ) .
(4.2)
Standard stability analysis implies that the synchronous solution (A = 0) is stable if - 1 < F~ (0) < 1, i.e., - 1 < T{ (6 - tei) < 0. TI is a decaying function, so its derivative is negative for all 6. However, the rate at which the system synchronizes is given by IF~ (0)1 - 1, and hence becomes small as 6 becomes large. Indeed, (4.2) tells us that the conduction delay for which the speed of synchronization (near the synchronous solution) is largest is the one for which F~ (0) = 0, i.e., I { ( 6 ) = - 1 / 2 . The function TI can be numerically computed for a given biophysical model; for the reduced Traub-Miles model, T[ (6) is close to zero (and so F~ (0) is close to 1) for 6 about 8 ms or larger (Ermentrout and Kopell [17]). This effectively places a limit on the possible conduction delay that can be tolerated and still produce fast synchronization of gamma by this mechanism. We note that when the rate of convergence to the fixed point is small, small differences in the two local circuits can produce significant phase-lags between the circuits; this is another reason why longer conduction delays disrupt synchronization in this mechanism. Modifications of the parameters in the circuit can change the range of conduction delays in which there is effective synchronization, but not significantly. (See Ermentrout and Kopell [ 17] for more details.) The doublet configuration is robustly stable for the minimal circuit involving two sites, each giving excitation to each 1-cell. The mathematics describing the minimal network identifies the essential nonlinearity (the relative refractory period) involved in the synchronization, but does not address the question of how this might work in a large, distributed network in which each 1-cell gets excitation at many different times from many different E-cells. Such a distributed network was analyzed in Karbowski and Kopell [29]. Using methods of statistical physics, it was shown that the same nonlinearity works to produce synchronization with the doublet configuration, and that the latter is stable in a larger parameter region than the configuration having triplets in the I-cell within each cycle. Furthermore, the analysis showed that a small amount of disorder in the positions of the E-cells (which determine the conduction delays to the 1-cells) can increase the parameter range in which the doublet configuration is stable.
4.2. G a m m a to beta transition Traub et al. [62,64] noticed that with more stimulation given to cells in a hippocampal slice, the network first produced a gamma rhythm, and then spontaneously switched into
Mechanisms of phase-locking and frequency control
@
@
1 -(bII"
(])-
@,
35
'@
1 -(Z)II
q)-
Fig. 4.2. The change from local gamma rhythms to local beta rhythms requires adding connections between the E-cells.
a lower frequency rhythm (12-30 Hz) called beta. (See Kopell et al. [35] for references to papers that discuss the potential significance of such a switch in living animals.) In the hippocampal slice, it is possible to look at the detailed timing of the spikes in the cells, and it was discovered that the lower frequency rhythm has a different structure from that of the gamma rhythm: the inhibitory cells continue to fire at a gamma frequency, while the excitatory cells miss cycles. Furthermore, the excitatory cell population is essentially synchronous, with almost all E-cells missing the same cycles. Experimental work reveals at least two kinds of changes in the components and coupling of the network in going between the two rhythms. When the network is displaying beta, the E-cells have an extra ionic conductance, a so-called after-hyperpolarization (AHP) current that slows down the E-cells. Furthermore, the effective coupling of the network changes during the transition: the coupling between E-cells, which is sparse in the network before the large stimulation, grows stronger (Figure 4.2). The mathematical question is whether those changes are enough to account for the changes in the network behavior. It is clear that slowing down the E-cells can ensure that the latter are unable to keep up with the I-cells, and hence miss cycles. Thus, the balance of excitability of the E- and I-cells is critical to producing the beta rhythm. The question to be investigated, for a local collection of E- and I-cells, is what facilitates the synchronization of the E-cells, so they fire on the same cycles. This is not a straightforward consequence of the E - E coupling, since it is known that E-E coupling can actively prevent synchronization (see Sections 2 and 3). Indeed, in the absence of an I-cell network, typical models of cortical neurons, when coupled with typical synaptic time constants, tend to oscillate in antiphase with one another. However, sometimes slow outward currents can help with synchronization of excitatory cells. This is discussed in Section 4.5, which uses a PRC kind of analysis. 4.3. Long-distance synchronization of the beta rhythm This section is a variation on Section 4.1, showing here how a different rhythm can be analyzed in a similar way, yielding a different conclusion. The minimal circuit for considering long-distance synchronization of the beta rhythm also uses two networks, each having one E- and one I-cell; since the cells of each type are synchronous in the rhythm whose stability we wish to analyze, we can combine the cells of each type into a single cell. In this formulation, we are assuming that the E-cell represents a synchronous local population, and that this synchrony relies on local E-E connections.
36
N. K o p e l l a n d G.B. E r m e n t r o u t
50
|
m~ --T~
I
40
% 30
20
0
i
0
i
5
i
/
i
lO
i
i
15
20
8 Fig. 4.3. The maps TI and T/~ as a function of the delay.
For the long-distance synchronization of the two separated circuits, we first consider only E-to-I connections as above for gamma. The map for analysis of the synchronization of two sites, each displaying beta, is a variation of (4.2), with two modifications. The cell I i now fires three spikes per beta cycle, two during the gamma cycle in which the E-cells fire, and one during the cycle in which the E-cells are silent. The excitation from the distant cell is, as before, received by an I-cell after it has fired its first spike of the period. The map TI for the gamma rhythm is replaced by T/~, which is defined to be the interval between the time cell Il receives excitation from cell 2 and the time it fires its third (not second, as before) spike; T~ depends on the times tl, t2, defined as in Section 4.1. For a graph of T/~, see Figure 4.3. The time at which the third spike of the I-cell fires is A
t2 + 6 + T/~ (t2 + 6 -- t l -- tei ) ~ T~.
The second modification introduces another, intrinsically-based, source of inhibition, namely the slowly-decaying AHP current of the E-cells, with time constant "rahp, triggered by a spike of that cell. As in formula (4.1), the time il of the next El spike is then defined implicitly as the time at which the inhibitory conductance has decayed to some threshold level; in this case the formula takes into account both extrinsic and intrinsic inhibitors:
gieexp[-([l-
T'B)/r] + gahp e x p [ - - ( / l - tl)/rahp] = g/~,.
(4.3)
Here gahp is the maximal effective conductance of the slow current, which takes into account the actual maximal conductance gahp, but scales it using the maximal value of the gating variable for that current and the ratio of the driving force of the AHP current to that of the synaptic current. A formula for g~, analogous to the one for g, can be found in Kopell et al. [35]. The analysis of (4.3) is not as straightforward as that of (4.1) since there is no simple explicit formula for il to obtain a formula analogous to (4.2). However, it is possible to do the stability analysis with the implicit formula (4.3) by linearizing around the solution
Mechanisms of phase-locking and frequency control
37
in which the E-cells are synchronous (while missing beats of the I-cells). To do this, we define pi ---- {i -- ti -- pfi, the variation in a given cycle from the periodic rhythm with period Pt~. A is defined as before. We can rewrite (4.3) as g~, -- g i e e x p [ - ( p ~
-+- Pl - A - 6 -
T~(A -+-6- tei))/r]
-q- gahp exp[--(p/~ -+- Pl)/rahp].
(4.4)
To lowest order in the small quantities A, Pl, P2, the above can be expressed as A pl + B ( p l - A(1 + T ~ ( 6 - tei)/T))--0,
where A = (gahp/rahp)eXp(--p~/rahp) and B = ( g i e / r ) e x p ( - ( p ~ - 6 - T~(6 - t e i ) ) / r ) . A similar formula holds for p2 with A replaced by - A . Subtracting these two formulae, and noting that P2 - Pl = A -- A, we have that (A + B ) ( A -
A)---2BA(1
+ T~(6- tel)/r),
i.e., A - B - 2BT~(6 - tei)/r A =
A+B
A ---- D A .
(4.5)
Synchrony is stable if the coefficient D of A in (4.5) lies between - 1 and + 1. The fastest synchronization occurs when D - - 0 and, as above, this is also the range in which there is the most robustness to heterogeneity. As shown in Kopell et al. [35], the size of D can be analyzed as a function of the conduction delay 3, and for physiological ranges, it turns out to be significantly within the stability range [ - 1 , 1] for 6 = 20 ms or more. Thus, by combining the properties of the new slow current (encoded in the definition of A) and the extra spike of the I-cells in each cycle (affecting both the resulting period and the definition of the key non-linearity Tt~), we get that the beta rhythm can allow stable synchronization across much longer conduction delays than can the gamma rhythm in the appropriate physiological parameter regime. This has been confirmed by simulations of the Hodgkin-Huxley equations for the minimal network and for large-scale biophysically accurate representations (Kopell et al. [35]). So far, the analysis of this subsection concerns only the long-range synchronous solution, and makes no use of long-range E-E connections (though local E-E connections are critical to having a local representation using only a single E-cell). We have shown that long E-E connections (see Figure 4.4) are not necessary to the stability of the long-range synchronous solution. Nevertheless, the long E-E connections do play a critical role in the network behavior when the parameters are in the regime to create a beta rhythm. The above analysis does not rule out bistability, and does not address the issue of domain of attraction. As can be seen numerically, without the long-range E-E connections, the solution in which the two E-cells fire in antiphase is stable and has a large domain of attraction. The main function of the long E-E coupling is to insure that the antiphase solution is not a stable alternative to synchrony (Kopell et al. [35]). Adding E-E connections when the parameters
38
N. Kopell and G.B. Ermentrout I
0
I
DELAY
Fig. 4.4. Long distance synchronization of the beta rhythm requires connections between the distant E-cells to remove solutions competing with the synchronous one.
are appropriate for the gamma rhythm does not increase the range of conduction delays that are tolerated for synchrony, so it is the difference in the other biophysical factors that creates the advantage for the beta rhythm.
4.4. Changing ionic currents changes synchronization properties The beta rhythm is significantly slower than the gamma rhythm, partly due to the extra outward current in the E-cells, and partly due to the extra inhibition from the extra I-spike. One might conjecture that the greater robustness of the beta rhythm to conduction delays may depend only on the relative periods, rather than the biophysical details. The following case study shows that the latter interpretation is not correct. Again the analysis uses a spike-timing map, this time to highlight how differences in time constants other than the ones used in Sections 4.1 and 4.3 can lead to a l-dimensional map. The case study concerns the alpha rhythm (8-12 Hz) in the neocortex, which is slower than gamma or beta. This rhythm is found easily in the neocortex (indeed, much more robustly than is gamma or beta), but its cellular origins are more mysterious. It has been shown that (at least some versions of) this rhythm do not synchronize well over distances, producing variable phase lags instead of synchrony (Roelfsema et al. [50], von Stein et al. [67]). To explore how this could come about, Jones et al. [28] built a minimal model of a local alpha rhythm using ionic currents known to exist in cells in the layers of the cortex where the alpha rhythm is thought to originate. The key currents are the so-called h- and T-currents, which react differently to inhibition than do most other currents: the hcurrent is activated by inhibition (its activation current m ~ is monotone-decreasing instead of increasing), while the T-current has a voltage dependence that requires it to be at a low voltage in order to de-inactivate before firing. With these currents, it is possible to create a rhythm in which the inhibition does not determine the frequency. Instead, the time constants of the h- and T-currents of the E-cell are the critical determinants. The I-cells in the network act to provide the essential inhibition needed to turn on those currents in the E-cells. In this rhythm there is no beat skipping, and the I-cells fire almost synchronously with the E-cells. With only the long E-to-! coupling, the analysis can be done as in the case of the gamma or beta rhythms. Again there are doublets which provide phasing information; the key
Mechanisms of phase-locking and frequency control
39
question is whether the information acts to synchronize. As in the case of Section 4.1, the full equations reduce to analysis of a 1-dimensional map, but the reasons are not the same. If there is to be only one essential degree of freedom, the general hypotheses of PRC theory should be at least approximately satisfied. That is, the local oscillations, when perturbed, should return approximately to their periodic wave-forms within one cycle; also, the effect of the stereotypic signal from the other local circuit should depend only on the time in the cycle of the recipient oscillator that the perturbation is received. In the case of the circuit displaying gamma, there is no slow gating variable, and the frequency is determined by the decay of the inhibition; there is no slow variable to hold any memory beyond the end of a cycle. Furthermore, the timing of the I-cell doublet depends essentially only on the time of receipt of the excitation from the circuit and the time of the first spike in the doublet. By contrast, in the case of the alpha rhythm, some of the gating variables of the T- and h-currents are slow (the activation of the h-current, de-inactivation of the T-current). However, special properties of the h- and T-currents allow the hypotheses of the PRC theory to hold. The gating variables of the h- and T-currents are strongly voltage-dependent, and are quickly reset to almost fixed values when the E-cell spikes. Though other gating variables, such as that of the fast-spiking potassium current, do not reset with an E-cell spike, their effects are gone by the end of the long cycle. Thus, for the E-I oscillations in the alpha rhythm, the timing of the E-cell spike paces all of the other variables, and there is no history dependence in the other variables. As above, the time of the doublet spike of the I-cell depends only on time of receipt of the excitation and the time of the previous I-spike, both paced by the spike times of the E-cells. Thus, in spite of the many dimensions in the Hodgkin-Huxley equations, the only essential degree of freedom is the difference in timing of the two E-cells. The resulting map is an analogue of Fy of (4.2). In the current case, however, the properties of the analogous function are different: the slopes show that the connection is desynchronizing for some conduction delays, and essentially neutral for small enough delays. The change of sign can be traced to the different processing of inhibition in the network; in the previous rhythms, an inhibitory pulse to an E-ceU slows down the rhythm, while for this rhythm an inhibitory pulse speeds up the rhythm by turning up currents that hasten the next spike of an E-cell. Adding long E-E connections does not change this behavior, and can even exacerbate the problem. It is possible to build networks with flexible synchronization properties by taking advantage of the variation in behavior in different voltage regimes. For example, starting from the simple network model of an alpha rhythm and raising the voltage (by increasing the applied current) brings the network to a range in which the h- and T-currents no longer operate. In that parameter regime, the same network will then display gamma or beta (depending on other parameters). Raising the voltage, which switches the local behavior, then also switches the synchronization behavior: when the local behavior is beta instead of alpha, the two separated networks synchronize rapidly and robustly (Jones et al. [28]). Changes in synchronization properties can also be created by activity-dependent plasticity of the synaptic connections. In simulations with large biophysically based neurons, Bibbig et al. [3] showed that local gamma activity, in two separate networks not oscillating in phase, could give rise to changes that procedure long-range E-E connections and E-I
40
N. Kopell and G.B. Ermentrout
connections" in turn, this helps transform an antiphase relationship in the beta rhythm to synchrony, without the prior existence of adequate E - E connections.
4.5. Synchronization o f excitatory cells In previous sections, we used spike-timing maps to investigate synchrony. In this section, we use PRC-derived maps to show that some intrinsic properties of cells can lead them to synchronize when they are coupled by excitation, in contrast to integrate-and-fire cells described earlier. The PRC, to be denoted by P(t), describes the advance in time of the next spike of the receiving cell from the time it would otherwise spike next, if the input arrives at time t. Such a PRC may be found experimentally, computed numerically from biophysical equations or sometimes computed analytically from equations. We can use the function P (t) to create a map for a pair of cells that are mutually coupled. Let t l be the time at which cell 1 last fired and let t2 be the time at which cell 2 last fired. We will assume without loss of generality that t2 > tl. When t2 fires, cell 1 is advanced by an amount P (t2 - t l ) so that tl -- tl + T - P(t2 - tl), where T is the natural period of both cells. (Note that one could have different periods but this complicates the algebra considerably.) By a similar argument, it is clear that t-2 --t2 + T - P(il - t2) m
_
_
is the time that cell 2 fires again. As usual let A -- t2 - tl and A -- t2 -- tl. We then see that A-- A - P(T - P(A) - A) + P(A) = F(A)
and we have again obtained a 1-dimensional map. We now make use of the condition P (0) -- P (T) -- 0 to note that A -- 0 is a fixed point of the map F. PRCs derived from real neurons as well as those derived numerically from conductance-based models satisfy this condition, which implies that there is no effect on spike timing if the input comes at the precise time the cell is spiking. (This is not true for the well-known integrate-and-fire model. Thus, the latter requires artificial "absorption" conditions (Mirollo and Strogatz [45]) that are not necessary for more biological PRCs.) Differentiating F and setting A = 0, we find that F (0) -- (1 +
P' (0)) 2
so that a necessary and sufficient condition for stability of synchrony is that - 2 < P' (0) < 0. Crook et al. [13] showed numerically that the presence of adaptation due to a lowthreshold, slow-potassium current causes the PRC to have a negative region after spiking,
Mechanisms of phase-locking and frequency control
41
with P' (0) < 0. As long as P' is not too negative (for example, if the interaction is not too strong), then the map calculation shows how this alteration of the PRC allows synchrony to be stabilized. In contrast, without adaptation, neurons tend to have strictly positive PRCs when coupled with excitation. The effects of the change on the PRC, and the effect of the latter on stability of synchrony, mirrors changes in the adjoint V*(t); indeed V*(t) is a kind of "infinitesimal PRC" (Hansel et al. [23]). (See Figure 2.1 and Section 2.2.) More recently, Ermentrout et al. [ 18] used such methods to investigate the effects of different kinds of adaptation currents on synchrony between excitatory neurons. They showed that low threshold adaptation currents and high threshold ones can have different mechanisms for creation of synchrony. The effects work in sample model neurons capturing behavior near creation of limit cycles via saddle-node bifurcations on an invariant circle [ 15, 251. The Ermentrout et al. [ 18] treatment is based on weak interactions, using a convolution with an infinitesimal PRC to create the PRC. For larger interactions, predictions from the infinitesimal PRC can be misleading. This was shown by Ackers et al. [ 1], who constructed the PRC numerically from the Hodgkin-Huxley equations for pairs of cells in the medial entorhinal cortex, a structure that gates the inputs and outputs to and from the hippocampus. The more direct construction reveals global properties of synchronization not apparent in the infinitesimal PRC: for large values of the conductance of the low current, which is associated with prominent subthreshold oscillations for the uncoupled cell, the stable antiphase solution predicted by the infinitesimal PRC approach disappears, and the synchronous solution becomes globally stable instead of bistable with antiphase. These global changes are seen to be associated with the Hopf bifurcation that produces subthreshold oscillations in the uncoupled cells (not found in the equations in (Ermentrout et al. [ 18])). Thus, subthreshold oscillations are seen to be relevant to the domain of stability of the synchronous solution. We note that the spike-timing maps associated with the PRCs can be globally defined. This is not true of the maps from the "hold-and-fire" systems, which cease to be valid sufficiently far from the synchronous solution. The reason is that a different set of spike times must be considered far from synchrony: in the maps discussed in Sections 4.1 and 4.3, the independent variable is t2 - tl, where the ti are nearby spike times of the two cells, not dependent on which of the cells is first, provided that the "hold" due to inhibition is larger than It2 - tl [. For larger initial differences in spike time, the input most relevant to the receiving oscillator changes, changing the map. The PRCs discussed above keep the order of the relevant spikes the same from cycle to cycle, and the map does not become undefined for large values of the initial difference in spike times.
5. Interactions of bursting neurons
As described in Rubin and Terman [52] and Wang and Rinzel [70], neurons can have complicated firing patterns known as bursts, consisting of a rapid sequence of spikes followed by a quiescent period. Some biophysical models for bursting are given in those references. In studying the interactions of bursting cells, we consider the envelope of the spikes (as do Rubin and Terman); a much-used simple example of such equations are the so-called
42
N. Kopell and G.B. Ermentrout
Morris-Lecar equations (Morris and Lecar [46]), originally designed to describe a calcium spike in muscle (not nerve) tissue. These equations were introduced as (2.8) in an earlier section. In these equations, the activation w of the potassium current is slow (i.e., 1/ r << 1). The equations can be understood metaphorically as envelope equations, or as equations for a calcium plateau on which the sodium spikes would ride if the spike currents were included. The Morris-Lecar equations have phase-plane geometry similar to those of the van der Pol oscillator or its variation, the FitzHugh-Nagumo equations. The nullcline dr~ dt = 0 is cubic-shaped, and the nullcline d w / d t = 0 is monotone-increasing. Depending on the relative positions of the nullclines, the equations can have a stable fixed point on either the left-hand branch or the right-hand branch of the cubic. With the intersection in the middle (and 1/r sufficiently small), the equations have a stable limit cycle. A variation of these equations uses different currents, but has a similar geometry (Wang and Rinzel [69]). The Morris-Lecar equations and the FitzHugh-Nagumo equations have a "fast-slow" structure, with the voltage equation significantly faster than the equation for the gating variable. Many results about networks of such cells use only this and other geometric structure, not the details of the biophysical descriptions. In this section, we discuss the interaction of two such neural oscillations, giving results complementary to those in Rubin and Terman [52,51], and using the fast-slow structure in a somewhat different way.
5.1. Fast threshold modulation (FTM) Models of interacting bursting neurons generally use different models of synaptic currents than models using spiking neurons. For the latter, the spikes are very quick or instantaneous, while the currents that result from the spikes have a finite lifetime. Therefore, it is necessary to describe the currents (as above) as explicit functions of time. For bursting neurons, the active state of the cell is a significant fraction of its period. If the time span of the synaptic current from a single spike is short compared to the time of the burst, we can approximate the current by considering it a function of the voltage of the presynaptic cell; the current is on when that voltage (averaged over spikes) is sufficiently high and off when that voltage falls below some threshold. An often-used model of a synaptic current has the same form as an intrinsic current, but the gating of the current is by the voltage of the pre-synaptic cell. Thus, a term of the form --gsyn(V)(V- Vsyn)
(5.1)
is added to the fight-hand side of (2.8a), where v is the voltage of the post-synaptic cell and is the voltage of the pre-synaptic cell. Note that this is the form of the calcium current in the Morris-Lecar equations; that current has one gate (an activation gate) that is instantaneous, so there is no extra differential equation associated with this current. The constant Vsyn is the reversal potential associated with the synapse. A low value of Vsyn corresponds to an inhibitory synapse, since the current (5.2) will then be an outward (hyperpolarizing) current; a value of Vsyn that is above the voltage values produced during activity gives an excitatory synapse, producing inward (depolarizing) current.
Mechanisms of phase-locking and frequency control
43
SYNCHRONOUS ORBIT
-,c
EXCITED
Fig. 5.1. Excitation raises the nullcline of the fast variable. For the synchronous solution of the coupled system, this changes the position of the trajectory in phase space.
The function gsyn(V) has the same form as the gating in the calcium current of the Morris-Lecar equations: it is a sigmoidal function. If this function saturates for low and high values of its argument, and if the voltages in the active and inactive states correspond to saturated tails of the sigmoid, a further simplification is possible: on the left branch, the synaptic current is zero, while on the right (active) branch, gsyn(13) is a constant gsyn independent of ~3. (A limiting version of this is the Heaviside function used in Rubin and Terman [52].) Thus, the voltage equation for a coupled cell does not have ~ explicitly in the equation; one need only keep track of which branch of the O-nullcline the partner cell is on. When the partner cell is active, the effect of the synaptic term is to raise or lower the v ' = 0 nullcline with a change in shape; an excitatory synapse raises the nullcline and an inhibitory one lowers the nullcline (see Figure 5.1). This kind of coupling was called by Kopell and Somers "fast threshold modulation" (FTM). The current in (5.1) corresponds to a simple form of synapse described in Rubin and Terman [52], ones that are "direct" and rapid compared to intrinsic time scales. That chapter concentrated on the important effects of time scales of the synapses; as shown there, inhibition can be synchronizing if the synapses decay sufficiently slowly compared to some intrinsic processes, and excitation can be desynchronizing if the synapse is fast and if the active portion of the burster is much shorter than the silent portion. The techniques used in that chapter focus on behavior along slow manifolds, and make computations of synchronization or desynchronization using distance metrics within the slow manifold. In our case, the lack of extra time-scales in the synapse brings into relief other aspects of the geometric structure that are better approached using a different kind of metric. 5.1.1. Synchronization via excitatory FTM. Identical cells coupled by excitatory FTM have a synchronous trajectory; this solution is not the same as that of an uncoupled cell. The construction of the synchronous solution uses the fast-slow structure of Equations (2.8). For the coupled system, the slow equation is (2.8b), with v = v(w) determined by setting the right-hand side of (2.8a) to zero. For the unexcited cell, the equation is given by (2.8a), and for a cell receiving excitation there is an extra term -gsyn(l) - 1)syn) added to the righthand side of (2.8a). The trajectory of the synchronous solution is shown in Figure 5.1.
44
N. Kopell and G.B. Ermentrout
wl
w, w0 v
u
Fig. 5.2. The jump from the inactive branch to the active branch leaves the variable w unchanged.
(Also see Rubin and Terman [52].) Note that the period of the synchronous solution is in general longer than that of the isolated cell, because the cells have a longer stretch of nullcline to traverse. Though the speed along the raised nullcline can be larger than that of the unexcited nullcline, in most examples (including Morris-Lecar), this effect is not enough to offset the effect of changing the length of the nullcline. We now show that excitatory coupling through fast threshold modulation leads to synchronization of the cells in the limit e --+ 0. This depends critically on both the fast synapse and the fact that the cells are bursters, with a non-zero active time. (Compare with Rubin and Terman [52] and the other sections of this chapter.) We first consider two identical cells, and then discuss the effects of heterogeneity. The central idea is to use the fast-slow structure of the equations, and the time to flow between points on the slow manifold as a distance metric. Suppose that two cells start on the inactive (left) branch at some non-zero distance. When the first jumps to its active state, it changes the nullcline that governs the lagging cell. If the latter is below the knee of the raised nullcline, it will also fire, synchronously in the limit e --+ 0. Right after the jumps, the two cells are on the right-hand branch of the raised nullcline, with the values of w the same as those they had before the jump. (See Figure 5.2.) We denote those values of w by w0 and Wl. The crucial hypothesis is that the time it takes a cell to move between w0 and w l is larger on the left-hand branch than on the right-hand one. This can be easily checked for the Morris-Lecar equations, and holds for many classes of equations with similar cubic nullclines. The speed along a branch is determined by (2.8b), and hence is proportional to the distance between w and w ~ ( v ) . Note that this distance is much greater on the righthand branch than on the left one for values of w corresponding to points near the knee of the lower cubic. Thus, there is a compression in the time metric across the jump. As the trajectories of the cells move up the right-hand side of the raised cubic, they become closer to one another in the Euclidean metric, but stay the same distance in the time metric. A similar mechanism can work on the jump down from the activated to inactive branch. Each jump further compresses the distance between the cells in the time metric, synchronizing the cells.
Mechanisms of phase-locking and frequency control
45
A related analysis in Rubin and Terman [52] shows synchronization using the Euclidean distance metric. One advantage of focusing on the time metric is that it shows clearly how the rate of synchronization can depend on geometric properties of the oscillations (Somers and Kopell [58]). For example, two coupled Morris-Lecar equations converge to synchrony considerably faster than two FitzHugh-Nagumo equations ( v ' = f ( v ) - w, w' = ~v; f ( v ) -- v(1 - v2)). The essential reason is that the periodic trajectory of the ML equations is slow near the left knee, since w ~ ( v ) - w is small in absolute value in that region; the latter is much larger on the right-hand branch of the cubic slow manifold, leading to a large compression in the time-metric across the jump to the active state. By contrast, the FN equation has a much lower compression across that jump because Ivl is not that different along the two slow branches of the cubic nullcline for the same value of w. Again, this analysis is also relevant to the jump down to the inactive state. Other papers that use a time metric to analyze stability of solutions involving bursting neurons are LoFaro and Kopell [38], Terman and Wang [60] and Rubin and Terman [51]. Suppose now that the cells are not identical; we continue to assume the interaction is via fast threshold modulation. The arguments sketched above show that if the differences are not too large (but still O(1) with respect to ~), there are solutions in which the jumps are synchronous. In order for this to happen, the cell lagging on each of the branches must jump before it reaches the knee of its own (excited or unexcited) nullcline. Thus the coupled pair responds to heterogeneity in the cells by changing the wave-form of at least one of the cells, while preserving the timing of the jumps (Somers and Kopell [59]). Asymmetry in the coupling has similar consequences: if the sizes of the synaptic conductances are different in the two directions, the two cells have different excited nullclines to follow. The mathematics is identical to that in which the coupling is the same but the intrinsic dynamics are different. (See Terman and Wang [60] for related work about heterogeneity.) Note that there is a dramatic contrast in the reaction to heterogeneity (intrinsic or coupling) between bursters coupled by FTM and oscillators coupled via phase differences. (Compare to Section 2.3 of this chapter.) There is also another contrast between phase difference and FTM coupling: long chains of oscillators take a large number of cycles to settle into the phase-locked state, while long chains of FTM oscillators can lock within a couple of cycles (Somers and Kopell [58]). Related work about fast synchronization in a network of integrate-and-fire neurons is in Campbell et al. [7]. Recently, Izhikevich [27] has derived a set of phase models that show some of the properties of relaxation oscillators interacting via FTM. He considers models of the form # x ' = f (x, y),
v ' = g(x, y),
where # is a small parameter and the system has a limit cycle with discontinuities analogous to those of relaxation oscillators. For these systems, he is able to compute the singular adjoint trajectory as # --+ 0 and show that there is stable synchronization in the presence of non-uniformities in frequencies. However, the method requires that the coupling strength e satisfy e << #, so the coupling must be extremely weak. 5.1.2. Antiphase via excitatory FTM. Suppose now that the time to traverse the righthand branch in the uncoupled system is less than the time to traverse the left-hand branch.
N. Kopell and G.B. Ermenovut
46
A
B ESCAPE
,..,B,TEO---- \
UNCOUPLED
Fig. 5.3. A. When the active cell reaches the upper knee, it releases the inactive cell from inhibition. B. When the inactive cell reaches the lower knee, it escapes from the inhibition imposed by the other cell.
(A precise hypothesis is found in Kopell and Somers [36].) In the coupled system, there is a set of initial conditions for which one cell stays on the inactive branch during the entire excursion of the other cell on the active branch (see Figure 5.3). With some further conditions, it can be shown that there is a stable antiphase solution with initial conditions in that set. The mechanism for the antiphase solution is very different and more subtle than that for the synchronous solution. During the active pulse of one cell, the other cell is displaced from the nullcline associated with the uncoupled cell to the (raised) nullcline associated with the active synapse (Figure 5.1). In a large class of equations that includes the MorrisLecar equations, the movement along the displaced nullcline is slower than along the original one. Thus, the cell that stays in the inactive state experiences a "virtual" delay (the difference between the distance it actually went during the excursion minus the distance it would have gone in that time in the absence of coupling). This is true even though the coupling is excitatory, and leads to advances of the cell for other initial conditions. The analysis of the behavior is done in terms of the delay as a function of position of one cell when the other begins its active excursion. From this, it is possible to get a Poincar6 map, taking the position of cell 1 when cell 2 begins its active phase to the position of cell 2 when cell 1 is at that spot; the fixed point of this is the antiphase solution. Again, time metrics greatly facilitate the analysis of the map. 5.1.3. FTM and Frequency control in "half-center" oscillators. As shown in Rubin and Terman [52], bursting cells that are mutually inhibitory, with fast inhibition, do not stably synchronize. In this case, the same mechanism that gives rise to synchronization for fast excitatory synapses leads to antiphase. An oscillating network with the two cells in antiphase is known as a "half-center" oscillator. Such behavior is frequently found as part of networks governing locomotion or other motor patterns, especially if there is antiphase symmetry in the motor outputs (Marder et al. [42]). In Rubin and Terman [52], it was shown that periodic solutions can exist for the coupled pair even if neither is an oscillator. Here we focus on some properties of the antiphase solutions, including the control of their frequency.
Mechanisms of phase-locking and frequency control
47
When cells are locked but not synchronous, there can be different ways to trigger a switch between active and inactive states of a cell. There are two kinds of distinctions we note: which of the two cells governs the switch (Wang and Rinzel [69]), and whether the mechanism is intrinsic or synaptic (Skinner et al. [56]). We start with the intrinsic mechanisms. For these, the synaptic threshold is between the active and inactive branches of both the original and inhibited nullcline. The exact value of the threshold voltage plays no role in the behavior of the coupled network. The first of these intrinsic mechanisms has been called "release" (Wang and Rinzel [69]); it is illustrated in Figure 5.3A. In this setup, the active cell reaches the end of its active plateau and jumps down. Because its jump traverses the synaptic threshold, this changes the governing nullcline for the other cell and releases it from inhibition. As in Figure 5.2, if the position of the inactive cell is below the new nullcline, it will jump to the active state. The release-induced jump is sometimes called "post-inhibitory rebound". Note that a periodic solution with this mechanism requires that the active state be long enough to allow the inactive cell to traverse enough of the left-hand nullcline to be able to jump up at the same time as the jump down of the active cell. Figure 5.3B illustrates the mechanism known as "escape": the inactive cell, which is on the left-hand branch of the inhibited nullcline, reaches the lower knee of that nullcline, escaping the inhibition from the other cell and jumping up. This lowers the governing nullcline of the other cell, allowing it to jump down if its trajectory is high enough on this nullcline. Note that a periodic solution of this type requires the inactive state to be sufficiently long. For each of these intrinsic mechanisms, the coupled state is an oscillation with the cells in antiphase and the active and inactive states of each identical in length. Neither mechanism requires that the uncoupled cells be capable of oscillations. There are, however, some constraints on the cells for the half-center oscillator to work. For example, in the release mechanism, there may or may not be a stable critical point for the uninhibited cell and/or the inhibited one; what is critical is that the inhibited cell is able to rebound when it is released. Further mechanisms of antiphase oscillations are obtained by relaxing the restriction that the synaptic threshold lie between the active and inactive branches. If the synaptic threshold is on one of the branches, then the signal given from one cell to another changes as the former traverses that branch, not during its jump between branches. This has a large effect on the wave-form and frequency of the resulting oscillation. (See Figure 5.4A.) If the threshold is in the low-voltage (inactive) region, the inactive cell escapes when it crosses the threshold, not when it reaches the lower knee; this is "synaptic escape". In this region, increasing the synaptic threshold increases the period of the coupled system. If the threshold is in the high-voltage (active) region, the release also occurs before the releasing cell reaches its own higher knee; this is "synaptic release" (Figure 5.4B). In that region, increasing the synaptic threshold decreases the period. If the threshold is between the branches, changing the former does not affect the period of the coupled system. It is possible to probe a real network to see what mechanism it is using to switch between states. For the network controlling heartbeat in the leech, this was done using a "dynamic clamp", an apparatus that replaces the real chemical synapses between the cells by ones that are computer-generated to have prescribed characteristics. By varying the synaptic
N. Kopell and G.B. Ermentrout
48
A
B SYNAPTIC ESCAPE
SYNAPTIC RELEASE
!
_f'M
!
! !
Fig. 5.4. A. The active cell releases the other cell from inhibition when it crosses the synaptic threshold and stops inhibiting the other cell. B. The inactive cell escapes before it reaches the knee: once it crosses the synaptic threshold, it inhibits the active cell, which then jumps down and stops inhibiting the first cell.
threshold and measuring the frequency of the resulting network, it was found that the period increased with threshold in the low voltage range and decreased with it in a higher voltage range, consistent with the use of synaptic escape mechanisms in the low voltage region and synaptic release in the higher voltage range (Sharp et al. [53]). 5.1.4. Variations: coupling between dissimilar cells. In this section, we have so far focused on synchronous and anti-synchronous behavior of two bursting neurons coupled by fast excitatory or inhibitory synapses. Here we briefly mention that such synapses between two dissimilar cells can lead to new behavior; these results underscore how flexible even a two-cell network can be. LoFaro and Kopell [38], considered a two-cell subnetwork from a well-studied invertebrate network. In this network, the cells are coupled by inhibition and one of the cells has a slow inward current that turns on when the voltage is low. They showed that modulation of the slow current could lead to behavior with one cell bursting N times for each burst of the slower cell, with N adjustable by varying the magnitude or voltage activation curve (moc(v)) of the slow current; changes of kinetics of the cell (rate of reset when voltage is high) could allow the same network to produce N : M ratios of bursting instead of just N : 1. The mathematical essence is the use of time-metric techniques to reduce the full (5-dimensional) equations to a one-dimensional map, from which the network behavior could be deduced. Working with other aspects of the same cells, Nadim et al. [47] showed that, if one of the synapses is more complicated, the network can display switches in control of frequency. The complication is "depression" in the synapses: instead of (5.1), in which there is just one voltage-dependent gate, the synapse is treated as also having an inactivation gate with a monotone-decreasing hoe(v), as in the Introduction. In this network, a small increase in the size of the maximal conductance of the depressing synapse can start a regenerative
Mechanisms of phase-locking and frequency control
49
mechanism, leading to a large increase in the synaptic current and a switch from a mechanism in which the intrinsic properties of the cells control the network frequency to one in which the kinetics of the synapse controls the frequency. Even if the membrane currents are very simple (no voltage-gated membrane currents, only a leak current), a network can produce oscillations using the nonlinearities in the synapses. Manor et al. [39] modeled another subnetwork of the same invertebrate system, producing a half-center oscillation from two cells, neither of which were modeled with voltage-gated currents. The cells are different in their degree of excitability, leading one to be on and the other off in the absence of synaptic input of a third excitatory cell; the latter, turned on and off by the first two, acts more slowly to change the relative excitability of the two primary cells.
5.2. Electrical coupling of cells or compartments In Sections 2.5 and 3.3 we used phase equations and the spike response method to show that weak electrical coupling could lead to antiphase solutions; in this section, we shall be concerned with unintuitive behavior of electrical coupling in the strong coupling regime. We shall illustrate this by discussing work concerning a single cell, but one extended in space, and (conceptually) discretized into different "compartments". As in networks of spiking cells, electrical synapses are modeled as discrete diffusion in the voltage equation only. The same mathematical formalism used for coupling between compartments is also used to describe electrical coupling between cells, and we give some examples in which such coupling does not have the effect of synchronizing cells. 5.2.1. Strong coupling within a dendrite. In recent work on dopamine-releasing cells, Wilson and Callaway [74] have been investigating the hypothesis that the firing patterns of the cells depend on the differences in dynamics along the dendrites of the cell; they also have evidence for oscillations in the cell, and interpret their data to suggest that each single cell can be thought of as a chain of oscillating compartments, electrically coupled, with a gradient in frequency along the chain. Their first model used a Morris-Lecar-like set of equations for each compartment. A simpler set of equations that reproduces the numerical behavior of that model uses FitzHugh-Nagumo oscillators coupled electrically (Medvedev and Kopell [44], Medvedev [43]). For each of two compartments, the equations are ev' = f (v) - u + d(f~ - v), U t --
(5.2)
(.OU,
where v is voltage of the compartment in question, u is here the concentration of calcium in that compartment, and ~3 is the voltage in the other compartment. The function f (v) is a cubic, co governs the period in this compartment, as does ~ in the other compartment. co and ~ need not be even approximately the same. The coupling d is very strong, i.e., d >> 1. The strength of the coupling implies that the voltage in the two compartments are
N. Kopell and G.B. Ermentrout
50
almost the same, even though the natural periods of the oscillators are not necessarily close. However, the values of u in the two coupled compartments need not be close. The system turns out to have an unexpected mathematical structure: in the limit of large coupling, there is an invariant cylinder foliated by periodic orbits, with an integral that is constant on each orbit. For d large but not infinite, this integral becomes a Lyapunov function, with slow drift between orbits that are periodic when d = oo (see Figure 5.4). The analysis uses two independent small parameters, e and 1/d. The critical idea (backed up by asymptotic expansions, and then proved rigorously) is that (fi - v) is O ( 1 / d ) , so the term d(~ - v) is O(1) in 1/d. Since ~ ~ v, one can think of d(~ - v) (to lowest order) as a function of v alone and the branch on which the two trajectories lie; we denote this function c(v). One works with singular orbits, combinations of solutions to the slow and fast equations with e --+ 0. For definitions of fast and slow equations see Rubin and Terman [52,51 ]. The slow equations for (5.2) are U l = f (v) + c(v),
(5.3)
u2 -- f (v + 6 c ( v ) ) - c(v), !
U 1 ~ 0)11)1,
=
+
Here ~ = 1/d and we are using indices 1 and 2 for the compartments, with v -----vl and V2 = v + 3c(v). Note that the coupled equations (5.3) can now be treated as uncoupled within each branch, as in fast threshold modulation. From (5.3) it is possible to get a differential equation for c(v), by differentiating the last two equations and using the first two. To lowest order, one gets 0) 1 - - 0)2
co(v) -- ~
(5.4)
f (v) + K ,
0)1 + 0 ) 2
where K is an arbitrary constant. This can be used in (5.3) to get u l and U2 as functions of v to lowest order in 6. It is to next order that one sees the mathematical structure. One can compute this using asymptotic expansions, but it is conceptually clearer to see that there is a Lyapunov function for the singular orbits, which is given explicitly as
L(ul,
1 u2) -- ~
ul Li
u2 t - A
,
0)2
where
- f f (v(t)) dt. A--T ~o1~o2 Each singular periodic orbit can be associated with a value of K in (5.4) and L = L ( K ) ; there is a unique value of K for which L = 0, and L (K) > 0 otherwise. Furthermore, over
Mechanisms o f phase-locking and frequency control
51
one full singular cycle, L decreases by O(6), so L is an integral for 6 = 0 and a Lyapunov function for 6 > 0. The fact that larger coupling leads to longer-lasting transients is not a property of all relaxation oscillators coupled through the fast system. By changing the nullcline of the slow (u) variable to v : Vu, where V = O(1) in e and 6, the rate of synchronization becomes almost independent of 6. There are also slow nullclines for which the coupled system has a 1-parameter family of periodic solutions for every value of d.
5.3. Electric coupling of heterogeneous cells Strong diffusion-like coupling between identical cells tends to synchronize the cells. If the elements being coupled are very different, the effect of strong coupling is less intuitive. Manor et al. [40] and Smolen et al. [57] have shown that oscillations can emerge from the electrical coupling of cells that are not oscillators, provided that some appropriate "average" of the cells can oscillate. Kopell et al. [32] considered electrical coupling of a relaxation oscillator and a simple (one-dimensional) bistable cell. They used a new geometrical technique to transform information about the geometry of the fast equation v' = f (v, w) of the oscillator to information about the jumps between high and low voltage when the cells are coupled. The technique shows how the electrical coupling can keep the oscillator pinned at a high or low voltage level longer than the uncoupled oscillator, and that larger coupling may actually be less effective at this pinning than coupling of some intermediate size.
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[41 ] E. Marder and R.L. Calabrese, Principles of rhythmic motor pattern generation, Physiological Rev. (1996), 687-717. [42] E. Marder, N. Kopell and K. Sigvardt, How computation aids in understanding biological networks, Neurons, Networks and Motor Behavior, Stein, Selverston, Grillner, eds, MIT Press, Cambridge, MA (1998), 139-150. [43] G. Medvedev, Some problems on pattern formation and oscillations in mathematical biology, Ph.D. Thesis, Boston University (1999). [44] G. Medvedev and N. Kopell, Synchronization and transient dynamics in chains of FitzHugh-Nagumo oscillators with strong electrical coupling, SIAM J. Appl. Math. (2001) (in press). [45] R. Mirollo and S. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math. 50 (1990), 1645-1662. [46] C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophysical J. 35 (1981), 193-213. [47] E Nadim, Y. Manor, N. Kopell and E. Marder, Synaptic depression creates a switch that controls the frequency of an oscillator circuit, Proc. Nat. Acad. Sci. USA 96 (1999), 8206-8211. [48] D. Pinto, S.R. Jones and N. Kopell, Modulation of ionic currents changes rhythms expressed and synchronization properties (2001) (in preparation). [49] J. Rinzel and G.B. Ermentrout, Analysis of neural excitability and oscillations, Methods of Neuronal Modeling, C. Koch and I. Segev, eds, 2nd edn., MIT Press, Cambridge, MA (1998), 251-292. [50] P.R. Roelfsema, A.K. Engel, E K6nig and W. Singer, Visuomotor integration is associated with zero timelag synchronization among cortical areas, Nature 385 (1997), 157-161. [51] J. Rubin and D. Terman, Geometric analysis of population rhythms in synaptically coupled neuronal networks, Neural Comp. 12 (2000), 597-645. [52] J. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, Handbook of Dynamical Systems, Vol. 2, B. Fiedler, ed., Elsevier, Amsterdam (2002), 93-146. [53] A.A. Sharp, EK. Skinner and E. Marder, Mechanisms of oscillation in dynamic clamp constructed two cell half-center circuits, J. Neurophysiol. 76 (1996), 867-883. [54] A. Sherman and J. Rinzel, Rhythmogenic effects of weak electrotonic coupling in neuronal models, Proc. Nat. Acad. Sci. USA 89 (1992), 2471-2474. [55] W. Singer, Synchronization of cortical activity and its putative role in information processing and learning, Ann. Rev. Physiol. 55 (1993), 349-374. [56] EK. Skinner, N. Kopell and E. Marder, Mechanisms for oscillation and frequency control in networks of mutually inhibitory relaxation oscillators, J. Comp. Neurosci. 1 (1994), 69-87. [57] E Smolen, J. Rinzel and A. Sherman, Why pancreatic islets burst but single ~ cells do not. The heterogeneity hypothesis, Biophys. J. 64 (1993), 1668-1680. [58] D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern. 68 (1993), 393-407. [59] D. Somers and N. Kopell, Waves and synchrony in arrays of oscillators of relaxation and non-relaxation type, Phys. D 89 (1995), 169-183. [60] D. Terman and D.L. Wang, Global competition and local cooperation in a network of neural oscillators, Phys. D 81 (1995), 148-176. [61 ] R.D. Traub and R. Miles, Neuronal Networks of the Hippocampus, Cambridge Univ. Press, Cambridge, UK (1991). [62] R.D. Traub, M.A. Whittington, E.H. Buhl, J.G.R. Jeffreys and H.J. Faulkner, On the mechanism of the gamma-beta shift in neuronal oscillations induced in rat hippocampal slices by tetanic stimulation, J. Neurosci. 19 (1999), 1088-1105. [63] R.D. Traub, M.A. Whittington, I.M. Stanford and J. Jeffreys, A mechanism for generation of long-range synchronous fast oscillations in the cortex, Nature 328 (1996), 621-624. [64] R.D. Traub, J.G.R. Jeffreys and M. Whittington, Fast Oscillations in Cortical Circuits, MIT press, Cambridge, MA (1999). [65] C. van Vreeswijk, L.E Abbott and G.B. Ermentrout, When inhibition not excitation synchronizes neural firing, J. Comp. Neurosci. 1 (1994), 313-322. [66] G. de Vries, H.-R. Zhu and A. Sherman, Diffusively coupled bursters: Effects of heterogeneity, Bull. Math. Biol. 60 (1998), 1167-1200.
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CHAPTER
2
Invariant Manifolds and Lagrangian Dynamics in the Ocean and Atmosphere Christopher Jones *t and Sean Winkler* Division of Applied Mathematics, Brown University, Providence, R102912, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
2. Geometry of Lagrangian transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
2.1. Basics of transport theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Fluid exchange in the Gulf Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. History of transport studies of the Gulf Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59 60 61
3.1. Ring pinch-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.2. Kinematic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Dynamically consistent models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. A meandering jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63 63 64
4.1. An evolved jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. A periodic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64 67
4.3. Invariant manifolds
67
4.4. Lobe dynamics
............................................
..............................................
5. Finite-time transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Effective invariant manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Transport calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. An adiabatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. A theory for finite-time invariant manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. An eddy-shedding event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68 70 71 72 74 74 75
7. Geometry, statistics, and the Antarctic polar vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Relative dispersion and coherent structure boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Application to the polar vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76 78 80
8. Vorticity and viscosity
85
.............................................
8.1. Potential vorticity fluxes
.........................................
8.2. Viscosity induced transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87 88
9. Conclusions and future challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
References
90
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*Partially supported by the Office of Naval Research under grant N00014-93-1-0691. tpartially supported by the National Science Foundation under grant DMS-9704906. H A N D B O O K OF D Y N A M I C A L SYSTEMS, VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 55
56
C. Jones and S. Winkler
Abstract The ocean presents an exciting array of dynamical phenomena. The complex time dependence of the flow fields, the relevance of effects at a variety of scales, and the difficulty in collecting data all serve to make this a fertile area for the application of dynamical systems. We address here the question of Lagrangian transport in and around coherent features in the ocean. Examples of such features are the Gulf Stream and rings that detach from it. The focus is on the development of geometric techniques to track and quantify this transport. The key method is the isolation of invariant manifolds that act as boundaries of distinguished regions in the flow, such as eddies and recirculating cells, and orchestrate the transport associated with a given feature. In simple models, these are stable and unstable manifolds of fixed points. In flows with complex time dependence, such as occur in the ocean, such manifolds may not always make sense. We make the case here that these basic concepts of dynamical systems need to be adapted to accommodate such complex flows and that, moreover, this can be achieved with meaningful and significant results for the ocean and atmosphere.
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
57
1. Introduction
Despite the immensity of the ocean and the complexity of its dynamics, coherent medium and large-scale structures are clearly identifiable and persistent. The Gulf Stream in the North Atlantic and its analog in the Pacific, the Kuroshio Current, are striking examples. They can be easily distinguished from their surrounding waters through satellite data that records such properties as sea-surface height and temperature. In Figure 1, averaged sea surface temperature is shown and the sharp gradients reveal the Gulf Stream with considerable clarity. Its direction, spatial and temporal persistence, and meandering are all evident from such data. There are also eddies that detach from the Gulf Stream on a regular basis. These socalled Gulf Stream rings are oceanic versions of hurricanes that carry water through an ocean region with contrasting properties. They can survive for months while traveling
Fig. 1. Water surface temperature from AVHRRdata for a time interval of 6.76 days ending January 19, 1999. (Courtesy of the Space OceanographyGroup, Johns Hopkins UniversityApplied Physics Laboratory.)
58
C. Jones and S. Winkler
westward back toward the eastern United States. After a certain time, they are usually re-entrained into the Gulf Stream. These coherent features, currents such as the Gulf Stream and Gulf Stream rings alike, do not pass through regions of the ocean without leaving their mark. Indeed, they impact the waters through which they pass in a number of ways. One of the most evident is the straightforward process of fluid exchange between a given coherent feature and its ambient water. Indeed, such a feature does not live as exactly the same body of water but, while maintaining its overall identity, is involved in a continual swapping of water with the surrounding ocean. This process of "communication" between a coherent structure and the ambient waters has implications for the overall distribution of critical fluid properties, such as heat and salinity, as well as the distribution of biological nutrients. The consequences of changes in these processes may be far-reaching; see, for instance, the article by Calvin [ 15] for a discussion of implications for the climate. Dynamical systems techniques have recently been developed in the context of oceanic Lagrangian dynamics with the aim of elucidating this exchange process. The goal of this work has been to use these methods to track and quantify fluid redistribution in active regions, such as the Gulf Stream. The analysis promises also to expose the physical mechanisms underlying the fluid exchange processes and give an assessment of their influence on ambient waters. This article will give a description of these developments and point to some of the challenges this line of work has opened up. The key technique is the use of invariant manifolds to delineate regions of specific flow characteristics. The interactions between different manifolds then underpin the fluid exchange process. The theory of invariant manifolds is well-developed for simple flows that are steady or periodic in time, such as would occur in basic model flows. The challenge lies in the fact that dynamical systems concepts in general, and those underlying invariant manifolds in particular, rest on infinite time concepts, while in realistic flow fields the structures of interest may persist for only finite spans of time. It is not clear in this context whether the traditional concept of an invariant manifold makes sense.
2. Geometry of Lagrangian transport The motion of a fluid particle is a trajectory of a dynamical system whose vector field is the velocity field of the flow, found by solving the appropriate partial differential equations (such as the Navier-Stokes or Euler equations). The phase space is the physical space and coherent features can be understood as geometric structures familiar to researchers in dynamical systems. In the simplest models in which the flow field is periodic, these structures are invariant manifolds, stable and unstable, of saddle fixed points. The transport of fluid is effected by the transverse intersection of these stable and unstable manifolds. This mechanism of fluid transport was earlier invoked as an explanation of mixing in laboratory scale fluids, see Aref [1] and Ottino [35]. This work goes under the title of chaotic advection as it is the chaotic nature of the transport that makes the mixing so effective. For the ocean, we are not as interested in the "chaotic" nature of this effect, but rather its power to describe transport.
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
59
2.1. Basics o f transport theory A two-dimensional, incompressible flow admits a representation by a streamfunction 7z(x, y, t) in the following manner ~c = u ( x , y, t) =
O ~ ( x , y, t)
-- v(x, y, t) -- -
Oy OTt(x, y, t) Ox
where u and v are the velocities in the x and y directions respectively. From a dynamical systems point of view, this set of equations is a Hamiltonian system with the streamfunction 7t being the Hamiltonian. In a steady flow, the streamfunction is independent of time, and the dynamical system is therefore autonomous. Lagrangian particle paths will coincide with streamlines, allowing a complete description of the transport by studying level sets of the Hamiltonian. In general, the phase space will be divided into regions of recirculation around elliptic points separated by heteroclinic trajectories associated with hyperbolic fixed points. These separatrices are the invariant manifolds of the hyperbolic points and act as boundaries to fluid transport. Now, suppose the streamfunction is time-periodic with period T. To understand the Lagrangian motions of fluid particles in this case, one studies the period T Poincar6 map, denoted 79. When the time-periodic vector field results from a small periodic perturbation of a steady streamfunction, standard results from dynamical systems theory guarantee that hyperbolic fixed points of the steady system will perturb to hyperbolic fixed points of the associated Poincar6 map, and their attendant invariant manifolds perturb as well [21]. However, the perturbed invariant manifolds will, in general, intersect transversely. Such intersections are heteroclinic points, and the resulting complex manifold structure is known as a heteroclinic tangle. Suppose pl and p2 are distinct hyperbolic fixed points of the Poincar6 map, and that the unstable manifold W u (pl) intersects transversely the stable manifold W s (p2) (see Figure 2). Invariance of the manifolds and the time-periodicity of the flow imply that if the manifolds intersect once, they will in fact intersect infinitely many times. Let q be a homoclinic point, and let U [ p l , q] denote the segment of W u ( p l ) joining pl to q. Similarly, let S[p2, q] denote the segment of W s (p2) joining p2 to q. The point q is called a primary intersection point (pip) if U [ p l , q] and S[p2, q] intersect only at q. A lobe is a region bounded by the segments of W u (Pl) and W s (P2) that join an adjacent pair of pips. The invariance of the manifolds guarantees that lobes are mapped to lobes. For incompressible flows, the area of a lobe will remain constant under iteration of 79. By examining a typical heteroclinic tangle, a mechanism for transport via the dynamics of the lobes becomes apparent. Three distinct types of particle motion are evident in Figure 2. Motion in region 7~1 is left to right, or prograde, and unbounded. Similarly, motion in region 7~3 is right to left, or retrograde, and unbounded. Within the recirculation region 7~2, the fluid circulates counterclockwise. The phase space may be clearly partitioned into regions of distinct motion using the manifolds as boundaries. By choosing a pip q0, U [ p l , q0] U S[p2, q0] defines a boundary separating 7~1 from J'~2.
60
C. Jones and S. Winkler i
!
i
,
i
|
~3 -
WS(px)
i,
i 1
~w~(p~)
~
1 -
. 5 3~"~
$ 3 ..
~t~_o "',,~q-a
. . ~_g
/ f ; -" ]4_1
_
i -.~
BO x
~~....~=T~
i
ql ~ ~
~ ~
-
A1
WU(pl)~~o
-
I
t=2T
2
i
~
i
/
/
i
Fig. 2. Chaotic transport for a time-periodic flow. Transverse intersections of the stable manifold W s (P2) (dashed line) and unstable manifold W u (Pl) (solid line) result in large fluxes across the boundary separating 7~ 1 and 7~2.
Let 79k, k e Z, denote iterations of the Poincar6 map, and let qk =~ 79k(qo). Define the lobes A0 -- S[q0, q0] t_J U[q0, q0] and B0 -- S[q0, q-l] U U[q0, q-l], and Ak =- 79k(Ao), Bk -~ 79k(Bo). With these definitions, consider the action of the Poincar6 map on the lobes A0 and B0. A0 is in region 7~1, whereas A1 is in region 7~2, and B0 is in region 7~2 while B1 is in region Tr This exchange of fluid between T~l and 7~2 represents Lagrangian transport via the lobe dynamics mechanism. The lobes A0 and B0 are called turnstile lobes, since they are the lobes involved in fluid exchange from one region to the other in one iteration of the Poincar6 map. The area of Ao (respectively B0) represents the amount of fluid transported from 7~1 to 7~2 (respectively 7r to ~1) in one time period, and dividing this amount by the period defines an average flux across the boundary. This is the basic idea of lobe dynamics in time-periodic flows, with more detailed descriptions available in the works of MacKay et al. [30], Rom-Kedar et al. [42,43], Wiggins [47], Camassa and Wiggins [ 16]. Similar results may be formulated for quasiperiodic flows (see [4,5,47,19]) in terms of two-dimensional Poincar6 maps, and generalized for adiabatic flows as well (see [28,29]).
2.2. Fluid exchange in the Gulf Stream The problem of fluid exchange in the ocean can be viewed in terms of this transport theory. The first conclusion from the above considerations that is relevant here is that time dependence of the underlying flow field is critical in this process of fluid exchange. The Gulf Stream can be viewed as a snake emanating from the separation at Cape Hatteras on the East Coast of the United States (Figure 1). If the humps and troughs of the snake remained steady, at least in a moving frame, then the fluid in the jet of the Gulf Stream would flow inexorably with the current and the edges of the current would not see exchange of
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
61
Fig. 3. Cartoon of Lagrangian transport near the Gulf Stream via lobe dynamics.
fluid, only different fluid particles flowing at very different speeds. However, the humps and troughs are not steady but pulsating relative to one another (in any moving frame). A cartoon for this process can be developed by strategically placing some key dynamical features on the Gulf Stream jet. In Figure 3, fixed points of the flow are placed at the top of "hills" of the meandering jet. This is dynamically reasonable, except that the points cannot be truly fixed as the entire meander is moving to the northeast. However, by viewing the structure in a moving frame, the idea of a fixed point at each peak is more reasonable. Inside each meander will be a recirculation zone and this enforces a separation point at the peak. This stagnation point is the fixed point mentioned above. In this picture, there are now three distinct dynamic regimes: the core of the jet, the recirculation zone and the retrograde motion north of the jet (with analogous structures on the south side). The demarcation of the boundary between these three regimes is supplied by the stable and unstable manifolds of the separation point. Due to the complex time dependence and the resulting intersections of invariant manifolds, as depicted in Figure 3, water in the northern retrograde flow can be carried into the recirculation zone and, similarly, water from the recirculation zone can be carried into the jet. This cartoon of transport is definitely easiest to understand if the flow is periodic. However, any realistic fluid flow modeling the ocean will not be periodic, and there are some fundamental issues in extending these ideas to the general, non-periodic case.
3. History of transport studies of the Gulf Stream The realization that fluid trajectories in the Gulf Stream may not be regular came relatively recently. In the mid 1980s, Bower and Rossby completed the first comprehensive sub-
62
C. Jones and S. Winkler 75N
70H
.:,-...':~~ 9
. i
~
/)/ /
65H
60N
55N
40N
35N
'
35N
/
76H
70N
65H
60H
65H
S P A G H E T T I DIAGRAM OF RAFOS FLOATS 1984-1985 Fig. 4. Float trajectories from Bower and Rossby [8].
surface float study of the Gulf Stream [8]. The results were surprising in that float paths appeared to have little predictable structure. Indeed, apart from an initial period during which most floats more or less followed the Gulf Stream path, the floats tended to drift off either to the south or north. The overall result is what is known as a spaghetti plot, as the paths look as if somebody dropped a pile of spaghetti on a map of the North Atlantic (see Figure 4). This work challenged the prevailing view of the Gulf Stream as a channel that would funnel water inexorably along its northeasterly path.
3.1. Ring pinch-off Bower and Lozier [7] posed the question as to whether the complicated Lagrangian float trajectories could be explained by the pinch-off of rings from the Gulf Stream. Ring pinchoff is a mechanism by which two neighboring troughs (or peaks) of the meander come together and create a ring that detaches from the Gulf Stream. This mechanism effects the bulk transport of fluid from the South to the North (or vice-versa) and also involves the ejection of jet water as parts of the jet encircle the ring. This is however a comparatively rare event (about 10-20 per year) and would seem to be an implausible explanation of the likely ejection of any given fluid particle from the jet as indicated by the Bower-Rossby spaghetti plots.
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
63
3.2. Kinematic models From the dynamical systems perspective, it is not surprising that a flow with complex time dependence could preserve its essential character, while at the same time any individual particle path could belie this character, appearing to disobey the guide set forth by the flow. This avenue of explanation was first pursued by Bower herself, who developed a kinematic model for a meandering jet that exposed the different possible regimes of motion, including recirculation regions above the troughs and under the hills of the jet [6]. This could explain some of the swirling motions observed in the float data, but fell short of explaining the ejection of fluid parcels from the Gulf Stream as the underlying streamfunction was time-independent and thus did not allow crossing of separatrices. Samelson suggested that some simple time dependence be put into the coefficients of the streamfunction, inducing a periodic undulation of the meander [45]. The fluid exchange in such a model is effectively represented in Figure 2, which explains in a natural way the motion of fluid from one regime to another. While the analysis of kinematic models shows convincingly that time dependence of the velocity field can explain fluid exchange, it cannot offer insight into the underlying physical mechanisms for the time dependence and hence the exchange. Moreover, these models cannot be used to estimate the amount of fluid involved in the exchange as they are based on streamfunctions that are merely hypothesized and not derived from the governing equations. In other words, the only physics in these equations is the incompressibility of the fluid that leads to the existence of the streamfunction itself. The question then remained as to how much water is actually involved in the exchange process. Were it shown to be insignificant in comparison to the water transported by ring pinch-off, the oceanographers would be free to ignore it.
3.3. Dynamically consistent models A further step was taken by del-Castillo-Negrete and Morrison [ 17] and Pratt et al. [39]. In each of these papers, a dynamically consistent model was developed with qualitatively similar behavior to that of the Bower-Samelson model. The models are formed by superimposing on a base jet normal modes arising from linearizing the barotropic equations around the jet. The idea is to take two neutral modes and add them to the jet with coefficients of different magnitude. The first (larger amplitude) mode will endow the jet with a meander and the second (smaller amplitude) mode will make it time-dependent. The time dependence will be non-trivial provided the frequencies of the two modes are incommensurate. The result is behavior similar to that observed by Samelson [45], and the analysis gives further corroboration to the idea that the time-dependent variation of a meandering jet causes significant fluid exchange between different constituent parts of a jet. Since the models contain much more physics than the kinematic models, they provide a useful indicator that such effects are present. However, if one asks the question as to the quantities involved in such exchanges, their usefulness diminishes. This point is rather subtle, but the reason that they allow exchange of fluid between regions of ostensibly different motion is that they are actually solutions of a linearized equation. It follows from the observations of
C. Jones and S. Winkler
64
Brown and Samelson [14] that a comparable nonlinear solution would have no exchange. In other words, the amount of transport between regions will be dependent upon, even commensurate with, the extent to which the linearized problem differs from the full nonlinear problem. If the approximation improved then the amount of fluid involved in the transport would be decreased. If the question being asked is whether this effect of "chaotic advection" occurs then these models give a satisfactory, and affirmative, answer; but if the issue is how great an effect it is, then the model will not be useful. The most reasonable approach is then to work directly with numerical models. The next section describes an analysis of such a model. With the Gulf Stream in mind, it seemed natural to analyze a meandering jet and assess the transport between different regimes.
4. A meandering jet In the work of Flierl et al. [20], the barotropic fl-plane equations were solved numerically with the aim of determining initial data that would generate a meandering jet. Although not formulated as a serious model of any specific ocean region, the model captures the fundamental effects that generate a meandering jet such as the Gulf Stream. This section describes a precursor to the studies of this model by Miller et al. [32] and Rogerson et al. [41 ], which represent the first full Lagrangian transport analysis for a numerical model. The governing equation for the flow is the equation for the potential vorticity
Oq -+- J (7r, q) = D, Ot where 7t is the streamfunction, q = V2~ -~- fly is the potential vorticity,/3 is the variation of the Coriolis parameter with latitude,
is the Jacobian of (f, g), and D is a dissipation term. The dissipation term can be viewed in two ways: first, as an aid to avoid numerical instability due to the development of high wave number effects and, secondly, as a crude model of eddy diffusivity in the ocean. Eddy diffusivity can in turn be viewed as a regularization of small-scale (turbulent) effects. In this study, D - ~1 V 4 ~. Periodic boundary conditions in both the zonal and meridional directions are assumed, and the flow is approximated pseudospectrally in a square computational domain of non-dimensional length L D. The initial data are set as a weakly perturbed horizontal jet; in other words, the linear approximation is used as a guide to seed the equations with a meandering jet that subsequently saturates into a fully nonlinear mode with the characteristics of a meandering jet (see [20] for details on how this is achieved).
4.1. An evolved jet The evolution of the flow for the parameter setting (Re, fl, n0) = (103, 0.103, 3) is shown in Figure 5 in terms of the potential vorticity field. (The Reynolds number Re is related to
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
65
.__.__--
t=20
t=40
t=60
m
t=80 O
-4
_
I
I
4
I
I
8
I
I
12
l
I
16
I
l
20
1
l
24
Fig. 5a. Evolution of the numerically-simulated flow as represented by the potential vorticity field for (Re, j3, n0) = (103, 0.103, 3). The early nonlinear evolution of the unstable jet, t = 20, 40, 60, 80.
the dissipation term, and no sets the number of meanders in the spatial domain, see [20].) The flow is unstable and develops nonlinearly from the initial perturbed zonal jet to a finiteamplitude meandering configuration. The generation and evolution of coherent structures in the flow for times t = 0 to t ~ 100 is discussed in detail by Flierl et al. By t -- 200 (Figure 5b), the flow has saturated into a configuration characterized by a primary meander with nearly constant eastward propagation and nearly periodic meander amplitude pulsa-
66
C. Jones and S. Winkler
t=200
L=220
i
t=240
4t
I
i
/
i
I
i
I
i
I
I
I
I
t=260
0-4
I
0
I
4
I
I
8
I
I
12
I
I
16
I
i
20
I
I
24
Fig. 5b. The nearly time-periodic meandering jet flow, t = 200, 220,240,260.
tion. It is this late-stage regime, in which the large-scale flow is nearly-periodic, that is of interest in this analysis. In a broad sense, the flow is composed of three regions exhibiting qualitatively different types of motion. The jet core is the region represented by the open potential vorticity contours meandering between y ~ • in Figure 5b. To the north and south of the jet in between meanders, there are recirculation regions, and the retrograde region in the far field is characterized by a westward flow.
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
67
4.2. A periodic approximation When viewed in a stationary reference frame, the numerical solution exhibits time dependence associated with the propagation of the jet meanders. Using power spectra, two dominant time periods can be identified. The first, To, corresponds to a spatial easterly migration and the problem is cast in a moving frame that effectively brings this period to zero. The second dominant period Tl = 56.50 is used as an approximate period for the flow itself. In other words, the flow is treated as a time-periodic flow of period T1. This Tl-periodic flow agrees exactly with the full numerical solution for time t = to to t -- to + T1 - 6t but deviates from it for times t >~ to + T1. Enforcing periodicity in this manner allows constructions based on calculations as t --+ -+-~. Since stable and unstable manifolds are defined by asymptotic conditions, this infinite time construction is used in a crucial way. The manifolds indicate how the chaotic motion is geometrically constrained, so to better understand this transport mechanism in the numerically simulated meandering jet flow, an analysis is performed on the model data to construct such structures. Lagrangian trajectories are computed for the flow using a 4th-order Runge-Kutta time integration with At = 0.05 and linear interpolation to approximate the velocity at intermediate times. The time dependence in the flow is on a much longer time scale than the interval At, and using a higher-order interpolation appears to have little effect on the computed results. A spatial interpolation is also required to estimate the velocity at off-grid points, for which there are a number of approaches (see [2,32]). The method used here is a bicubic polynomial interpolation requiring values of 7t, ~Px, ~ , and 7txy on a local 2 • 2 grid. Smoothness through the first derivatives of ~(x, y, t) ensures that the interpolation preserves the Hamiltonian structure present in the 2-D incompressible flow. The necessary derivatives are precomputed spectrally and transformed back to a physical space for the interpolation. This approach is a compromise between accuracy and computational efficiency. The Hamiltonian structure of the two-dimensional flow is retained with full spectral accuracy at the grid points, and some computational efficiency is gained from employing a local interpolation scheme. The construction of Lagrangian trajectories from numericallygenerated velocity fields is described in detail in [32].
4.3. lnvariant manifolds For each parameter setting, fixed points and manifolds are calculated for a single cat's eye structure on the northern side of the jet. Two fixed points, denoted p l and p2, are located and their associated eigenvalues and eigenvectors are computed. For the Poincar6 map 79, stable eigenvalues are those satisfying 0 < IUI < 1, unstable eigenvalues satisfy I)~L'I > 1, and the incompressibility can be expressed in terms of the determinant of 79 as 1D79[ -- ,V',ks = 1. The one-dimensional stable manifold for the fixed point is approximated by the backward time iteration of a short segment of initial points aligned with the stable eigenvector. The unstable manifold is computed by iterating forwards in time a short line segment of initial points aligned with the unstable eigenvector. The invariant curves presented here are all computed to a resolution of 0.02; that is, the Euclidean distance between adjacent points on the curve does not exceed 0.02.
68
C. Jones and S. Winkler
The extreme stretching rates 0~u "~ 103) in the vicinity of the hyperbolic fixed points makes it difficult to numerically identify these points and to completely resolve their invariant manifolds. Approaches used to address these difficulties are discussed in [32]. However, as a consequence of using the bicubic spatial interpolation, the computed lobe areas remain constant to at least three decimal places over several iterations of the map, even with the strong stretching of the lobe boundaries. Moreover, the Jacobian computed at the hyperbolic fixed points is very close to the theoretical value of 1.
4.4. Lobe dynamics Figure 6 shows the stable and unstable manifolds, W s and W u, with the fixed points pl and p2 for the parameter setting (Re, fl, no) = (103, 0.103, 3); these fixed points are in the vicinity of meander crests. The transverse intersection of the stable manifold W s (pl), and the unstable manifold W u (p2), indicates that there is fluid exchange across the exterior boundary defined as the union U[p2, qe] t,3 S[pl, qe]. (Recall the notation U[a, b] to denote the segment of W u connecting points a and b, and similarly for S[a, b], a segment of W s .) There are four distinct sets of lobes that result from this transverse intersection, labeled A, B, C, and D and indexed consistently with forward iterations of the Poincar6 map 79, i.e., 7~
7~ P 7~ >Ao > AI > A2 >'", 7~ 7~> 7~> 7~ > Bo BI B2 > -".
Under forward iteration of the map, fluid in lobes A l and C1 leaves the recirculation region and enters the region of retrograde motion while fluid contained in lobes B1 and D1 enters the recirculation region from the retrograde region. That is, the exchange of fluid across the exterior boundary takes place when fluid passes from lobes A l . . . . . D1 to lobes A2 . . . . . D2. As mentioned above, the designation of turnstile lobes depends on how the boundary between regions is defined (determined by the choice of qe). However, the identification of which lobe sets are entering and leaving a particular region is not affected by the choice of qe, and therefore the resulting transport estimates (discussed below) are not affected by this choice. The transverse intersection of W u (pl) and W s (p2) indicates that there is fluid exchange across the interior boundary, defined as U[pl, qi] t,3 S[p2, qi] (Figure 6(b)). The turnstile lobes along the interior boundary are labeled E, F, G and H. Lobes Fl and Hi enter the recirculation region from the edge of the jet core while lobes El and G l are mapped from the recirculation region to a region near the edge of the jet core (lobe G2 is not shown completely in Figure 6(b)). Exchange takes place across the interior boundary when fluid is mapped from lobes E1 . . . . . HI to lobes E2 . . . . . H2. Lobe areas are calculated using Green's theorem and can be used to estimate the transport associated with the Lagrangian motion described above. The dimensional transport is simply, A
A
T -- D* A/T1,
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
69
Table 1 Summary of the transport estimates. The exterior boundary separates the recirculation region from the retrograde region. The interior boundary separates the recirculation region and the edge of the jet core. The cumulative nondimensional lobe areas are listed along with the associated dimensional transport in Sverdrups (10 6 m 3 s - l ) . The dimensional scales are described in the text. Fluid leaving vortex
Fluid entering vortex
(Re, r, no)
Bndry
Lobes
Area
Transport
Lobes
Area
Transport
(103,0.103,3)
exter inter
B,D F, H
3.031 0.724
4.69 Sv 1.12 Sv
A, C E, G
3.031 0.724
4.69 Sv 1.12 Sv
WS(pl)
_
_
w~(p~)
3
(~)
Y 2
_
A1 6
8
i0
.
12
.
.
14
q~~.
.
Pl ~ ~ ~ / / o
16
~_~
_
18
20
x
E1
5
"",,
P2
4
(b)
\\ -
\
/
\
/
\~, -|
|
1/ t
i
,
'
Fig. 6. Computed stable (dashed line) and unstable (solid line) manifolds for (Re, fl, n0) = (103, 0.103, 3) and the resulting lobes. The heavier lines define our choice of boundary as described in the text. (a) Along the exterior boundary, U[p2,qe ] U S[pl,qe], fluid in lobes A 1 and C1 moves from the recirculation region to the outer retrograde region while lobes B 1 and D 1 are mapped from the retrograde region to the recirculation region. (b) Along the interior boundary, U[p 1, qi] U S[p2, qi], lobes E 1 and G 1 transport fluid from the recirculation region to the edge of the jet core. Lobes F1 and H1 are mapped from the edge of the jet core to the recirculation region.
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C. Jones and S. Winkler
where A" is the dimensional lobe area, ~ is the dimensional period of the flow, and D* is a characteristic length scale representing the depth over which the transport takes place. Table 1 shows transport estimates for D* -- 500 m as applicable to the Gulf Stream, along with the horizontal length scale L* - 105 m and velocity scale U* = 1.75 m s - l ( u * -0.87 m s- 1) for nondimensional/3 - 0.103. The resulting transport fluxes are significant, although those associated with the exterior boundary exceed those of the interior by a factor of at least four. In comparison with the flux of water across the Gulf Stream associated with ring detachment, about 3 Sverdrup, the fluxes along the exterior boundary are the same order of magnitude. They must thus be viewed as substantial and cannot be ignored in any overall mass flux budget, see [41] for further discussion.
5. Finite-time transport While the above calculations make a striking case for the significance of Lagrangian transport in the overall budget of ocean transport, the artificial reconstruction of a flow field defined for all time by enforcing periodicity is unrealistic. As mentioned a number of times above, many structures in an ocean flow will persist for only finite spans of time. Moreover, the entire character of a flow may change and thus the templates against which the transport is being calculated may cease to be valid. Despite the temporary nature of coherent structures, they are identifiable for long time spans and they play key organizing roles in the overall flow during their lifetime. For instance, the Gulf Stream rings carry water that is quite saline to the coastal region of the East Coast. The transport of water in and out of these structures is a significant feature in the overall fluid budget and therefore a theory is needed that allows us to quantify that transport in an effective manner. Flows spanning a finite time of observation thus form the appropriate context for which a realistic theory of transport needs to be developed. To dynamicists, the limitation to finite time is a challenge, as the fundamental constructions of dynamical systems are based upon asymptotic information. Stable and unstable manifolds are defined by the behavior of solutions as t - . -+-c~, respectively. A coherent feature such as an eddy could naturally be thought of as defined by a surrounding stable manifold, but if the feature lives in a highly time-dependent flow and persists for only finite time, a stable manifold cannot be defined. A theory of "finite-time" stable and unstable manifolds is clearly needed. In the following, we outline an operational procedure for generating "effective" invariant manifolds. These play the same role in organizing the flow and orchestrating the transport as do stable and unstable manifolds of fixed points in steady or periodic flows. The underlying idea is to proceed in exactly the same way as when computing (numerically) an invariant manifold in a periodic system, or one for which infinite time data is available. The basic numerical strategy is articulated well by Nusse and Yorke [34] who adopt the term "straddling" for this procedure. The operation is most easily visualized for a saddle fixed point in the plane: a segment of initial data is picked that is transverse to the stable direction; under the evolution of the flow, or the map, it will be stretched out in the unstable directions while part of the segment close to the fixed point will be anchored by the exponential contraction along stable direction. The idea in the following is to implement this
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
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procedure in other cases where some semblance of a saddle fixed point exists dynamically, but which is perhaps time dependent in a sufficiently complex manner so as to preclude the existence of an actual fixed point. For instance, the region may have the hyperbolic character of that near a saddle fixed point for only a finite time.
5.1. Effective invariant manifolds A theory was developed in the paper by Miller et al. [32] that gives a practical approach to generating these "effective invariant manifolds". The term "effective" has both its meanings here: they are effective in the sense that they are manifestations in these complex flows of material surfaces that play the role of stable and unstable manifolds in transport studies in simpler systems, and also they are effective in that they do effect the transport under consideration. The idea is as follows. The data available and useful for the meandering jet spans over a time interval of approximately 60 time units. Before that time, the jet is forming out of the (linear) initial data and, after that, the jet is decaying due to the drain of viscosity (note that no forcing is included in that model which might have preserved the jet by balancing viscosity). However, during that time span, the characteristic structures of a meandering jet are evident. These include the central jet itself and its attendant recirculation zones lying in the troughs of the meanders and under the hills. In the analysis described in the previous section, the enforcement of periodicity allowed us to play freely with time integrations without worry as to whether we would run out of data by exiting the time window of available data. The consequence was that fixed points, or more generally trajectories, could be isolated, their stable and unstable manifolds generated, and their role in governing the dynamics determined. The natural question arises as to whether that freedom is really needed for the computations of the stable and unstable manifolds. In other words, even when the periodic approximation is being made, only a finite time span of data is used in generating the invariant manifolds. Of course, this is inevitable as in any numerical computation only a finite span of the data set is used. To answer this question, it is useful to think how an invariant manifold calculation is carried out in practice. If a saddle type fixed point is known, then an initialization of wellchosen data points near that fixed point will be stretched out along the unstable manifold by the flow and trace out the unstable manifold. The initial points can be "well-chosen" in a number of ways. They might, for instance, be chosen as a small circle surrounding the fixed point. The flow will squash the circle onto the unstable directions, the force of the unstable part will then elongate the evolved circle, and a caricature of the unstable manifold will emerge. A more refined technique can be implemented if some information is readily available about the unstable directions. In this case, an initialization of a curve of points could be chosen along an unstable direction and iterated under the flow. Various refinements of this procedure have been used, but ultimately they all largely depend on the stretching of the flow in the unstable directions. Of course, it is important to pack the initial data set densely enough so as to avoid developing gaps in the sketched manifold. The greatest danger lies near the fixed point itself as the flow carries all trajectories away from this unstable point. This potential problem can be resolved by re-seeding the initial data set if neighboring points separate more than some predetermined tolerance.
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The main interest then is whether the finite-time data sets available from the numerical runs of the meandering jet suffice for carrying out this procedure. The physics gives us a hint. A given coherent structure, such as an eddy, appears to generate a hyperbolic trajectory as a key part of what defines it. Indeed, the boundary of such a structure is reasonably viewed in a Lagrangian manner as the stable manifold of a hyperbolic point. We have come to expect these regions of strong stretching and compressing as standard accompaniments of a coherent structure. It is not unreasonable to believe that if a structure is well-defined for a sufficient interval of time, then a data set will be available to generate the invariant manifolds. At the beginning, it was an article of faith that we would be able to obtain invariant manifolds in this way. However, it has been borne out in all the examples, and it is now quite sensible to conjecture that the coherence of a structure and an attendant region of sufficient hyperbolicity to generate the invariant manifolds are really the same thing.
5.2.
Transport
calculations
In Miller et al. [32], finite time invariant manifolds were computed for the meandering jet flow of Section 4 using these ideas. The parameters used in this study were ( R e , fl, n o ) (104, 0.207, 4), and computations were carried out for the time interval 0 <~ t ~< 122, where time has been shifted by 300 units to guarantee that the flow has fully evolved from the initial conditions. T ,~ 30.5 is the dominant time period of the flow, but in this case no periodic approximation is made. The resulting manifold calculations are shown in Figure 7, with solid lines representing unstable manifolds and dashed lines representing stable manifolds. For the time interval used in this analysis, the evolution of six lobes may be followed. In Figure 7 the lobes are labeled C - H and the iteration indices 0 . . . . . 4 correspond to the time slices, t = 0, t = 30.5, t = 61.0, t = 91.5 and t = 122.0, respectively. For the aperiodic flow the exterior boundary may be changing in time, but nonetheless it is quite clear which regions of phase space are going from retrograde motion to vortex motion and which regions of fluid are leaving the vortex. That is, these finite-time stable and unstable manifolds clearly delineate regions of phase space having
Table 2 Estimated transport for the aperiodic flow covering 0 <~t ~<122. The labels vort and retr refer to the vortex and retrograderegions, respectively. "Period" indicates the time interval in which the exchange takes place. The final column 79 is the lobe area from the Poincar6 map for the corresponding time period. Lobe
Exchange
Period
Area
79
C D E F G H
vort --+ retr retr --+ vort vort --+ retr retr ~ vort vort --+ retr retr --+ vort
0.0-30.5 30.5-61.0 30.5-6 1.0 61.0-91.5 61.0-91.5 91.5-122.0
1.112 0.982 1.055 0.955 1.004 0.889
1.137 1.037 1.037 0.992 0.992 0.931
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Fig. 7. Finite time stable and unstable manifolds for the aperiodic flow. Reprinted from Physica D, Volume 110, ED. Miller, C.K.R.T. Jones, A.M. Rogerson and L.J. Pratt, Quantifying transport in numerically generated velocity fields, 105-122, 1997, with permission from Elsevier Science. qualitatively different Lagrangian motion. For example, in going from t = 30.5 to t = 61.0 fluid in lobe E1 passes out of the vortex into the retrograde region (lobe E2) and continues in retrograde motion for at least another time period (lobe E3). At the same time, fluid in lobe D1 will exit from retrograde motion during the next time period, entering the vortex (lobe D2). Although the definition of the boundary may change from slice to slice, one
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can still identify turnstile lobes and quantify a time-averaged flux between the two regions. The lobe areas are summarized in Table 2, showing that the flux is slowly decreasing with time. For comparison, the Poincar6 map was computed using a periodic truncation on each of the four time intervals. The flux estimates computed from the maps for these truncated periodic flows are included in Table 2 along with the time-dependent fluxes computed from the full aperiodic data set. A key conclusion to be drawn from this table is that the analysis of the full aperiodic flow produces fluxes comparable to those deduced from the periodic approximation. This then serves to reinforce the conclusion drawn earlier that this transport mechanism is a significant factor in the overall budget of fluid exchange into and out of the jet.
6. An adiabatic approach The work of Miller et al. [32] demonstrated that infinite amounts of data were not necessary to carry out the construction of manifolds. Prompted by this observation, Haller and Poje investigated the idea of finite time transport further [23]. Their idea was to exploit the structure that could be concluded from frozen-time slices, an Eulerian perspective, when the flow field changed (relatively) slowly in time. The underlying assumption of their work holds when the Eulerian time scale is long compared with the Lagrangian time scale. The main result in [23] gives conditions under which the presence of a stagnation point in each Eulerian data slice during a finite time interval implies the existence of an underlying saddle-type Lagrangian trajectory. Additionally, this hyperbolic trajectory will possess locally invariant stable and unstable manifolds. Due to the finite extent of the data set, the hyperbolic trajectory and its manifolds will not be unique; however, they are determined up to exponentially small errors, and will thus appear unique in numerical simulations for data sets of sufficient length. The technical conditions of the theorem may be found in the original paper, but the basic idea is that the time dependence of the velocity field should not be too fast. Let p(t) denote the hyperbolic stagnation point that is assumed present in each Eulerian time slice. Then the conditions of the theorem impose bounds on the maximal Lagrangian velocity # -- maxt~[t-,t+] I/~(t)l of the stagnation point, the rate of change of the angle formed by the eigenvectors of the stagnation point, and on the strength of the nonlinearity of the vector field near p(t). In essence, one wants an Eulerian stagnation point to manifest an underlying Lagrangian trajectory that organizes the nearby particle flow in the familiar manner of a hyperbolic saddle. It turns out that this will occur provided the Eulerian stagnation point moves sufficiently slowly enough and that its (frozen time) eigenvectors do not rapidly "scissor" or rotate. All of this makes intuitive sense and is formulated precisely in [23].
6.1. A theory for finite-time invariant manifolds There have been two contexts in which theories of finite-time invariant manifolds have been developed. The first is the adiabatic theory due to Haller and Poje, described above, and the second is due to Sandstede et al. [44]. The context of this latter development is
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
75
of flow fields that have an infinite time span as a certain parameter tends to zero. In both cases, the construction of the manifolds follows Perron's method. The data is given over a certain time interval [t-, t+]. In contrast to the assumptions used in [23] described above, for the development in [44], t - -- t - (e) and t + = t + (e) with t - --+ - c ~ as e --+ 0 and t + --+ +cx~. The vector field is first extended to all t E R in a C ~ fashion via bump functions. The extended vector field will be smooth in the domain for all time and will coincide with the time-restricted vector field in the time interval [t-, t+]. A stable manifold W s is defined by fixing a box B surrounding the hyperbolic trajectory and requiring that W s contains the set of initial conditions which never leave B in forward time. Trajectories associated with these initial conditions are shown to satisfy an integral equation that guarantees exponential attraction to a Lagrangian hyperbolic trajectory F (t) in the flow. A contraction mapping argument then verifies that the integral equation has a unique C 1 solution, and it follows that W s is a C 1 manifold. A similar argument with time reversed establishes W u. In [23], The underlying Lagrangian hyperbolic trajectory F ( t ) is C 1 close to the Eulerian hyperbolic trajectory p ( t ) . Now, any given modification of the vector field outside of the time interval [t-, t +] will yield such results. Since an infinite choice of these modifications exists, F (t) and its associated stable and unstable manifolds will not be unique. However, one can argue that for sufficiently large finite time intervals, each of these different solutions will be exponentially close to "master" manifolds W s and W u (the unique manifolds that would exist if the data were given for infinite time). The conclusion is that in practical applications, numerical resolution will render it impossible to distinguish these exponentially small errors, and the calculations will yield the desired objects as accurately as numerically possible.
6.2. A n e d d y - s h e d d i n g event Once the conditions for the existence of Lagrangian saddle-type dynamics have been verified, one may confidently proceed with manifold calculations to understand the geometry of the flow and make transport estimates. Haller and Poje applied these ideas to study eddyjet interactions during detachment and entrainment [38]. The Eulerian velocity fields are given by a reduced gravity, single layer primitive equation model that solves the shallow water equations. This leads to a double gyre circulation pattern with the gyres separated by a strong eastward flowing jet. Eddy detachment events are observed in the time-dependent Eulerian fields (see Figure 8). A hyperbolic stagnation point between the eddy and the jet is evident in the height field. As the eddy is shed from the jet, this hyperbolic stagnation point eventually loses its saddle-type structure; hence, the finite-time formulation is well suited to this application. In the analysis of the eddy-shedding event, the first step is to locate the Eulerian hyperbolic stagnation point in each time slice during the interval of interest. The relevant quantities needed to test the conditions of the theorem are estimated numerically to determine the time interval over which the theorem is satisfied. Manifolds are calculated following the straddling approach of Miller et al., with initial segments chosen along the stable and unstable eigenvectors of the Eulerian stagnation points. The geometry of the eddy-shedding event is strikingly different from that of the meandering jet model. The stable and unstable manifolds do not appear to intersect at any point
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3000 /
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Fig. 8. Topology of the double-gyre circulation. Contours of instantaneous layer depth sometime during year 5 of the simulation. Depths less than 500 m are shown dashed. Note the detaching eddy and its attendant Eulerian stagnation point (near (1200, 1500)).
other than the hyperbolic trajectory. Upon consideration of the dynamics, this makes perfect sense. In incompressible flows, lobes maintain a constant area as they are transported by the flow. In this problem, the object of interest is an eddy that grows in area as it forms and slowly moves away from the jet. Thus, the manifolds exhibit an open structure that allows fluid to flow into the growing eddy. In fact, the manifold structure reveals a long mixing channel that entrains fluid from far downstream of the eddy itself and not just from the surrounding water (see Figure 9).
7. Geometry, statistics, and the Antarctic polar vortex The numerical computations of manifolds following the methods of the preceding sections depend on some correlation between the Eulerian and Lagrangian dynamics. Since the initial segments for the manifold calculations are chosen based on considerations of the Eulerian field, this approach is difficult to apply in situations where the Eulerian picture yields little information. For example, consider again the meandering jet flow. The Eulerian streamlines do not provide the requisite hyperbolic stagnation points necessary
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere 1100
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Fig. 9. Invariant manifolds of the cold core ring at t = 95 days (top), and t -- 110 days (bottom). Stable manifold, W s solid, unstable manifold, W" dashed. The stable manifold encloses a long channel that funnels fluid into the eddy.
for the initialization of a straddling calculation. Since the hyperbolic trajectories of interest propagate eastward with the jet, they only appear in the Eulerian data when an appropriate change of reference frame is made. A priori knowledge of the physics in the meandering jet flow suggests the shift to a moving reference frame, but in general, the existence of such a frame is not likely. This exact difficulty arises in the study of the springtime atmospheric circulation near the Antarctic polar vortex. The basic flow pattern is similar to the meandering jet model; in this case, a jet encircles the polar region in the southern hemisphere, creating the polar vortex. During the southern spring, atmospheric conditions and solar radiation facilitate the photochemical destruction of ozone within the vortex, forming the infamous ozone hole. Of interest is whether the jet acts as a barrier to transport, effectively trapping the ozone depleted air within the vortex. If significant transport across the jet does occur, then current estimates of ozone depletion may be inaccurate. Transport associated with this flow has been studied extensively (see [9-12,36] and references therein). The application of finite time transport theory to this problem was first proposed by Bowman [ 13]. The velocity field is taken from the United Kingdom Meteorological Office (UKMO)
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C. Jones and S. Winkler
stratospheric assimilation for July 1, 1996 to June 30, 1997. Data is given on surfaces of constant potential temperature, on which the flow is approximately two-dimensional and incompressible. The velocity field and other relevant quantities are derived from observational data, and the complicated flow dynamics precludes any simplifying assumptions on the time dependence. The jet is composed of a near-stationary wave of wavenumber 1 and higher frequency nonstationary waves, with the largest moving component of the jet dominated by wavenumber 2. However, the erratic phase and amplitude of the wavenumber 2 component combined with the influences of higher modes defies any attempt to shift to a moving reference frame. As previously mentioned, such irregular time dependence means that any relevant hyperbolic trajectories are not likely to appear as Eulerian stagnation points, and the methods of Haller and Poje are not readily applicable to this problem.
7.1. Relative dispersion and coherent structure boundaries Bowman has proposed a method for identifying hyperbolic structures in this setting [ 13]. The idea is to infer the manifold geometry by considering the stretching associated with the hyperbolicity of these structures. Recall that segments that straddle a stable manifold will stretch in forward time in the direction of the unstable manifold. For some initial time to, consider a small segment in the phase space defined by the endpoints x(t0) and y(t0). After integrating the trajectories through these initial conditions for time T, the quantity Ix(T) - y(T)I
Ix(to)- y(to)l provides a measure of the length stretch of the segment (cf. [35] where length stretch is defined in the limit as the initial segment length goes to zero). In Bowman's finite strain method, a staggered grid of initial conditions is integrated and the resulting length stretch of each initial segment defined by this grid is assigned to the midpoint of the segment. This yields a scalar field, associated with both the initialization time to and the integration time T - to, which is then plotted against the grid of initial segment midpoints. Approximations to the stable manifolds should then appear as local maxima in the finite strain plot, while unstable manifolds will maximize the backward time finite strain calculation. The finite strain is a particular manifestation of a more general concept from Lagrangian statistics, namely relative dispersion. Given a pair of trajectories with initial conditions x(0) and y(0), the relative dispersion associated with the pair after time T is the quantity Ix(T) - y(T)I. Traditionally, this quantity is averaged over an ensemble of initial pairs to yield a single value R(T). This statistical approach is common in the study of turbulence (see [40] and the extensive references therein). However, plotting R ( T ) against the grid of initial conditions provides insight into the location of structures at time to that influence the dynamics of passive tracers in the flow. This approach has been taken by von Hardenberg et al. to study baroclinic vortices in the atmosphere [26]. In this case, the relative dispersion is defined for a grid of initial conditions by initializing eight "satellite" particles for each grid node. The relative dispersion at each node is the average square distance between the final position of the node and its eight satellites after time T. The authors also compute
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
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finite-time Lyapunov exponents to measure stretching along trajectories. This is a local calculation that has been used previously to study atmospheric mixing and transport [37]. Doerner et al. have explored a direct relationship between local maxima of the finite-time Lyapunov exponent field and stable manifolds of hyperbolic sets [ 18] (see also [22]). For short times, the relative dispersion and Lyapunov exponent plots are similar, as both calculate local stretching along trajectories. However, for longer times the relative dispersion measures global advective effects of the full nonlinear velocity field, whereas the Lyapunov exponents still measure local linear stretching along trajectories. Numerical experiments suggest that there is a correspondence between local maxima of these scalar fields and invariant manifolds. This provides a link between Lagrangian statistics and ideas from the theory of geometric dynamical systems. A key concept underlying both approaches is that of coherent structures. From the dynamical systems perspective, it is the invariant manifolds that define the boundaries of coherent structures such as cat's eye vortices, lobes, and eddies. However, the definition of a coherent structure may vary depending on the application of interest; Haller and Yuan review several approaches in the context of two-dimensional turbulence and propose a definition based on the stability of material lines [25]. In this approach, the authors measure the amount of time that a trajectory spends in a region of strong hyperbolicity, and this time is plotted against initial conditions to distinguish stable and unstable material surfaces. The common theme in all these approaches is that of measuring dynamically relevant scalar quantities associated with tracer trajectories and visualizing the spatial distribution of these scalars to determine the geometry of coherent structures. Relative dispersion is proposed here as an appropriate scalar for defining coherent structures and their boundaries. A formulation based on the integration of a single grid is used. Consider a two-dimensional grid at some initial time to with grid nodes Xi,j (to). The node's four neighbors, Xi+l,j, Xi-l,j, Xi,j+l, and Xi,j--l, a r e used to define relative dispersion: Ri,j (to, T) =
IXi+l,j(T) -- X i - l . j ( T ) ] IXi+l.j(tO) -- X i - l . j ( t o ) l
+
]xi,j+l ( T ) - xi,j_ l (T)] Ixi,j+l (to) - x i . j - I (t0)l
For each grid node, Ri.j(to, T) captures the horizontal and vertical stretching associated with the segments defined by the neighboring nodes. Using this "cross" at each gridpoint is natural for two-dimensional problems, as it accounts for stretching in any direction and associates this stretching with the intersection point of the cross (the grid node). The twopoint finite strain method is biased toward stretching normal to segments, and the eightpoint satellite method contains two superfluous segments. Local maxima of the relative dispersion field act as boundaries between regions with similar Lagrangian fates. For example, consider a vortex in a shear layer. The interior of the vortex is a coherent structure associated with a low value of R, as the relative dispersion for initial conditions inside the vortex is bounded by its width. Initial conditions on the vortex edge will exhibit large relative dispersion since neighboring interior tracers circulate in the vortex while neighboring exterior tracers move rapidly downstream in the shear flow. In general, the relative dispersion provides a hierarchy of coherent structures based on its time evolution properties. As an example, recall the geometry of the meandering jet model. Intuitively, the longest-lived coherent structures are the cat's eye vortices, whereas
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lobes are shorter-lived structures. For material volumes near the vortex core, the relative dispersion is low for long integration times, as the volume rotates en masse but does not stretch significantly. For lobes, Ri,j(to, T) is small for short times while the lobe maintains its general shape, but increases as the lobe nears a hyperbolic trajectory and elongates along the unstable manifold. As with the finite strain, considerations of stretching associated with hyperbolic invariant sets suggest that invariant manifolds manifest themselves as local maxima of the relative dispersion field. However, the relative dispersion may also detect non-hyperbolic features such as shear layers that can be useful for defining coherent structures as well. To demonstrate the kind of image this technique produces, consider the forced, damped Duffing oscillator. The structure of the invariant manifolds of the hyperbolic fixed point near the origin has been studied extensively for this system (see, for example, [21]). Figure 10(a) shows the results of a standard straddling calculation of the manifolds for a particular choice of parameters. Figure 10(b) shows the scalar field R = R ( 0 , - 1.5T) R (0, 1.5 T), the backward time relative dispersion minus the forward time relative dispersion with initial time 0 and final integration time 4-1.5 T. T is the period of the forcing, 2zr. Note the correspondence between the stable manifold in Figure 10(a) and large negative values of R (local maxima of the forward time relative dispersion calculation). A similar correspondence is evident between the unstable manifold and local maxima of the backward time relative dispersion calculation.
7.2. Application to the polar vortex Figure 11 shows the relative dispersion computation (again plotted as the difference of the backward and forward time computations) for several different initial times in the UKMO atmospheric data set. In each panel, the final integration time is -+-8 days. The cat's eye structure of the flow is apparent in these plots, with the wavenumber 2 dominance clear, though higher wavenumber effects are also apparent. Lobe-like structures similar to those found in the meandering jet problem are also visible. However, these lobes are quite large, as they are not the result of a small perturbation of a steady vector field. Filaments of high R in both forward and backward time are nearly parallel near the vortex edge, indicating the strong shear induced by the jet. The vortex edge is commonly defined as the region with the largest potential vorticity gradient [31]. In Figure 12, contour lines of the potential vorticity are overlaid on a plot of the forward and backward time relative dispersion calculation initialized on October 11, 1996. The potential vorticity is taken from the UKMO data set, and here the relative dispersion is shown with a logarithmic scaling to highlight the curve of local minima of R present at the jet core. The relative dispersion is minimized here since nearby initial conditions are translated together by the jet. A strong potential vorticity gradient is evident in the region of the jet core. Hence, the relative dispersion calculation provides dynamical support for the definition of the vortex edge as the region of the largest potential vorticity gradient. To gain further insight into the nature of the transport induced by this flow geometry, material lines may be calculated via straddling. The initial segment for the straddling
Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere
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Fig. 10. Forced, damped Duffing oscillator. (a) Invariant manifolds for the fixed point near the origin, computed with a standard straddling method. The red curve is the unstable manifold, and the blue curve is the stable manifold. (b) Relative dispersion. Pseudocolor plot of R = R ( 0 , - 1 . 5 T ) - R(0, 1.5T), where T = 2zr is the period of the forcing. Large positive values, in shades of red and yellow, indicate large stretching in backward time. Large negative values, in shades of dark blue, indicate large stretching in forward time.
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N
Fig. 11. Relative dispersion for UKMO atmospheric data. Pseudocolor plot of R = R(to,-8) - R(t0, 8) with to = (a) 26 September, (b) 1 October, (c) 6 October, (d) 11 Octoberof 1996. Coordinates are longitude, latitude.
computation is chosen by first plotting the R field and identifying the intersections of the forward and backward time relative dispersion calculations. Following ideas from the preceding sections, these intersections are treated as effective hyperbolic trajectories for the straddling computation. For example, a small segment is initialized at one such intersection near 100 ~ longitude, 45~ latitude on September 26 (see Figure 1 l(a)), chosen to lie along the red curve (maxima of the backward time relative dispersion) and straddling the blue curve (maxima of the forward time relative dispersion). This segment, referred to as an unstable material line, is then advected forward under the velocity field. Similarly, a stable material line is initialized on October 21 and advected in backward time. Figure 13 shows a time series of the evolution of these stable and unstable material lines for the intermediate time interval spanning October 1 to October 11, 1996. A comparison of Figure 13 to Figures 11 and 12 reveals that the advected material lines align with maxima of the relative dispersion throughout this time period. This consistency supports the use of relative dispersion to identify coherent structure boundaries. Five lobes, labeled A - E are identified and tracked in Figure 13 (compare to Figure 7). Transport of fluid from the edge of the jet to the surf zone occurs as long, thin lobes on the poleward side of the recirculation zone move counterclockwise, eventually reaching the boundary between the retrograde and recirculating regions (e.g., lobes C and E in
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Fig. 12. Potential vorticity contours (black) overlaid on (a) forward and (b) backward time relative dispersion calculations (reds are high values, blues are low values). A logarithmic scale is used for the plot to highlight the local minimum at the jet core, manifested by the blue curve with peaks near 60~ latitude.
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Fig. 13. Evolution of stable (green) and unstable (red) material lines between 1 October and 11 October 1996. Five lobes are identified and followed, and their areas are summarized in Table 3.
Fig. 14. Actual positions of stable (green) and unstable (red) material lines on 11 October 1996, superimposed on a map of the Southern hemisphere.
I n v a r i a n t m a n i f o l d s a n d L a g r a n g i a n d y n a m i c s in the o c e a n a n d a t m o s p h e r e
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Table 3 Estimated transport by the lobes associated with one cat's eye structure. The labels vort and retr refer to the vortex and retrograde regions, respectively. Lobe A B C D E
Exchange vort retr vort retr vort
--+ retr --+ vort -+ retr --+ vort ~ retr
Area in km 2 ( x 106) 0.704 7.64 0.717 1.07 2.39
Figure 13). They are then entrained into the retrograde region and once again stretch and fold into thin filaments, this time in the surf zone (e.g., lobe A). At the same time, lobes upstream in the surf zone are pulled into the recirculation zone and eventually stretch out along the jet edge (e.g., lobes B and D). This is the familiar lobe dynamics mechanism, only in this case the flow is completely aperiodic, so the configuration of the lobes never repeats. Comparing this picture to the case of the meandering jet model reveals a strikingly similar qualitative picture of the Lagrangian transport mechanism. The areas of the lobes may be computed using Green's Theorem (taking into account the Earth's sphericity) to obtain estimates of transport between the retrograde and recirculation zones. These calculations are summarized in Table 3. The magnitude of the transport resulting from the lobe dynamics may be better appreciated by viewing the lobes relative to their actual positions over the Earth (see Figure 14). Material lines may be followed for longer times, revealing a very complicated geometry (Figure 15). Transport between the two cat's eye structures can be seen as some of the lobes beginning near the jet are ejected from the vortex into the recirculation zone, then become re-entrained into the next vortex. Additionally, the repeated stretching and folding of the material lines gives rise to small-scale structures, leading to enhanced mixing. The stretching effect is especially pronounced near the strong hyperbolic regions at the peaks of the wave in the jet, where filamented structures appear. Hence, the manifold calculations suggest that the breaking of waves in the surf zone provides an efficient means for mixing in midlatitude regions. Note also that the material lines accumulate on the vortex edge, but no lobes enter the vortex region. This suggests little isentropic transport across the vortex edge by the lobe dynamics mechanism.
8. Vorticity and viscosity Following idealized fluid particles as in the transport studies discussed so far has a potential pitfall: if the fluid properties that are measurable and of interest physically are not preserved by the fluid motion, then their redistribution under flow evolution is not captured by this approach. A key dynamical variable in oceanographic studies is potential vorticity. Potential vorticity is made up of the usual (local) fluid vorticity and the planetary vorticity due to the rotation of the Earth. In an ideal, inviscid fluid, potential vorticity is conserved along fluid trajectories, and so potential vorticity redistribution could be concluded from
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Fig. 15. Stable (green) and unstable (red) material lines on 11 October 1996. Repeated stretching and folding is evident, but the material lines to not penetrate the vortex core.
transport studies based on fluid particle motion. However, in the ocean itself, potential vorticity is not conserved. The dissipation of potential vorticity is effected by a number of mechanisms involving a variety of scales, many of which remain mysterious. In fact, dissipation remains largely resistant to modeling. Eddy diffusivity is a simplistic, but common way, of parameterizing small-scale effects that promote the dissipation of potential vorticity. The idea is that turbulent motion at small scales creates a drain on potential vorticity by transferring it to sub-grid scales that will not be reflected in the large-scale motion. Eddy diffusivity is simply the parameterization of this effect through a viscous term in the equations. In numerical models, such a viscous term is usually included for reasons of stability of the calculations. A natural correspondence is then made between this viscous term and eddy viscosity. In cases where potential vorticity is conserved, a constraint is imposed on the Lagrangian dynamics that has significant implications, see Brown and Samelson [ 14] for further discussion. In particular, the conservation of potential vorticity has implications for the transport itself. The dissipation of potential vorticity by viscosity, or other effects, therefore raises two interesting questions: (1) Can Lagrangian transport studies accurately reflect the redistribution of potential vorticity?
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(2) What role does viscosity play in promoting transport that may not be present in the inviscid limit?
8.1. Potential vorticity fluxes It is natural to suspect that in numerical models in which potential vorticity is dissipated, a characteristic of all numerical models discussed here, the relationship between potential
Fig. 16. Potential vorticity difference q ( x ( t f ) , y ( t f ) ) - q(x(to),y(to) ) plotted against initial conditions (x(to), y(to)). A stable manifold of the hyperbolic trajectory near the island boundary is shown in black.
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vorticity fluxes and material transport will be broken. As part of the work on recirculation past an island, Miller et al. [33] have exhibited a striking relationship between potential vorticity change and the effective invariant manifolds. Based on an idea in [24], Figure 16 shows the color-coded potential vorticity difference over two sets of 30 time units, with a corresponding stable manifold of an appropriate hyperbolic point on the island. The picture reveals an extraordinary correlation between boundaries of contrasting regions of potential vorticity differences and the stable manifold. The right hand-column shows an analogous picture over a backward time interval. The point to be concluded from these pictures is that the dominant potential vorticity changes are a result of the advective fate of a fluid particle. If a particle is dragged into the boundary layer near the island, for instance, then whether it moves north or south will have an enormous impact on the change in potential vorticity it experiences. In turn, whether it is a fluid particle destined to move north or south along the island boundary is information encoded in the lobe structures and revealed by the invariant manifolds. A key problem in oceanography is to measure potential vorticity fluxes in active regions. A critical element of any such analysis is choosing the boundaries through which the flux is measured. A standard approach is to use boundaries of coherent structures based on time averaged flow fields, see [27]. Such definitions may lead to deceptive results if the structure actually deviates from its time-averaged version significantly. A Lagrangian definition of the boundary, of the kind discussed here, is more appropriate. The picture discussed above shows the strong correspondence between the manifolds and the change in potential vorticity and suggests the use of effective invariant manifolds for measuring potential vorticity fluxes. A systematic study of the different ways this can be done for the island problem is given in [33 ].
8.2. Viscosity induced transport The second question raised by taking viscosity into account is whether the viscosity itself induces transport between the constituent parts of a given coherent structure and between the structure and ambient waters. This question was taken up by Balasuriya et al. [3]. A simple flow for an inviscid fluid can be imagined that is steady, possibly in a moving frame, and which exhibits a cat's eye or eddy-like structure in which a heteroclinic loop or a homoclinic orbit surround a recirculating region. Since potential vorticity is conserved in an inviscid flow field, such a steady flow is a natural model. An important question is then whether the addition of viscosity, which would lead to a perturbed flow field, would render a flow field with intersecting stable and unstable manifolds. This is a daunting question as the perturbed flow field is not known, only that it satisfies a certain partial differential equation. Nevertheless, a surprisingly strong result is shown in [3]. Indeed, a Melnikov calculation is carried out and an expression derived for it that depends only on the inviscid flow field. The overall conclusion is that viscosity can promote transport both into and out of a coherent structure and that the flux can be estimated through the Melnikov function, which affords an expression in terms of the inviscid flow field. One issue that arose through this work was whether the result could hold for perturbed flow fields that stayed close to the inviscid field for only finite times. A complete theory
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would require an affirmative answer to this question as there are no guarantees that the viscous field will stay close to its inviscid limit for infinite time. Prompted by this question, Sandstede et al. [44] developed the theory of finite-time invariant manifolds discussed above. The extraordinary result is that the modifications outside of the fixed time interval contribute only at higher order in the calculation of the distance function, and so the Melnikov function remains valid.
9. Conclusions and future challenges The pictures that usually reveal a coherent structure in the ocean are Eulerian in nature. Contours of temperature or sea surface height from satellite data depict the Gulf Stream clearly as in, for instance, Figure 1. In numerical models, streamfunction contours or contours of potential vorticity at frozen times are often shown to exhibit coherent structures, such as vortices, jets etc. While these are satisfying pictures of the effect under consideration, they contain little dynamic information as they depend purely on information at a fixed time. Additionally, viewing the contours at different fixed times is not illuminating since the contours do not stack up in general to form material surfaces. The concept being put forward here is that an improved understanding of these phenomena is obtained through depictions of Lagrangian structures that delineate the coherent features. In contrast to Eulerian characterizations that represent features in terms of distributions of scalars at frozen times, Lagrangian characterizations contain dynamic information about the fluid, such as which particles will be entrained into the feature and which will leave the feature. In this article and the papers to which it refers, it is shown that such a Lagrangian map of coherent features can be formed by generating what we have called "effective invariant manifolds". These are slices of material surfaces that are tied to regions of strong hyperbolicity (stretching and compressing). These regions themselves are attendant to the coherent features under consideration. An operational strategy is given by the work of Miller et al. [32], which depends on the idea of "straddling" the stable directions to pin the manifold down, and extending it under the expansion of the flow in the unstable directions. These manifolds lead to naturally defined boundaries of the coherent feature and also contain all the information concerning transport into and out of the feature, as well as between its constituent parts. It should be kept in mind that any curve generated this way will amount to a slice of some invariant surface in the three-dimensional phase space (two space dimensions + time), a so-called "material surface" in the terminology of a fluid dynamicist. This can be achieved by seeding any initial curve at a fixed time and applying the evolution of the flow. The point is, however, that the initial seeding should be chosen to generate a material surface that reveals information about the transport in the flow field. The key in any given problem is how to find the "hyperbolic" region to initially seed the material surface with a "straddling" segment. We have seen three different approaches to this above: (1) Periodic approximation: If the flow field is close to having a dominant time period, then an initial approximation can be made by enforcing periodicity and repeating the velocity data from one time interval, based on that dominant period. The hyperbolic fixed
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points, and their stable and unstable directions, of this periodic flow can be used as a first approximation. Judiciously chosen initial segments to be used in the Miller et al. routine can then easily be developed. These are then used to generate effective invariant manifolds for the full flow field. (2) Eulerian approximation: If the Eulerian time scale is long compared with the Lagrangian time scale, then it is reasonable to use the frozen-time Eulerian fields as first approximations. The idea is to use the saddle points of, say, streamfunction contours as initial guesses for the "hyperbolic" regions. This is the basis for the Haller-Poje approach and is shown by them to be very effective in the eddy-shedding events. (3) Relative dispersion: In cases of extremely complex time dependence, neither of the above methods will work well and a direct method is needed to locate the regions of strong hyperbolicity. Such a method is supplied by the dispersion computations, described above in the context of the Antarctic polar vortex. One of the main challenges is to extend these transport ideas to three-dimensional flows. Little is known in this context, even for tame steady or periodic flows. Effects of vertical flows are important, but a fully three-dimensional theory is not as important as might be initially thought. Indeed, the ocean is well stratified and even multi-layer computations trusted by oceanographers may involve only a small number of layers. The difficulties surrounding the collection of data in the ocean is part of what makes oceanography so challenging. For sub-surface data, only sparse Lagrangian data can be collected using floats that render information only about their own tracks. Since the floats are individually quite expensive and their placement a significant logistical operation, the amount of data available now, or in the future, is not great. It is thus a fundamental issue to design optimal experimental systems that maximize the information resulting from any given float experiment. The dynamical systems analysis discussed in this paper promises to delineate the regions in physical space with specific fates, such as entrainment into an eddy, faithful following of a jet, etc. It is natural therefore to incorporate this insight into an optimal design for a float experiment system. This issue has become the focus of much current research. There is a danger of a circular argument as the above shows how to glean dynamical information from observational data, but here it is being suggested that the observational data should be based on the results of the dynamical systems analysis. The work by Toner et al., discussed in [46], is aimed at breaking this circle through the use of reconstructions of approximate velocity fields and numerical models. This area of observational design is one where dynamical systems can have tremendous impact in a truly practical fashion. Its potential remains to be fulfilled, but the prospects are good and the results so far are promising.
References [1] H. Aref, Integrable, chaotic, and turbulent vortex motion in two-dimensional flows, Ann. Rev. Fluid Mech. 15 (1983), 345-389. [2] S. Balachandar and M.R. Maxey, Methods for evaluating fluid velocities in spectral simulations of turbulence, J. Comp. Phys. 83 (1989), 96-125. [3] S. Balasuriya, C.K.R.T. Jones and B. Sandstede, Viscous perturbations of vortici~-conserving flows and separatrix splitting, Nonlinearity 11 (1998), 47-77.
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[41 D. Beigie, A. Leonard and S. Wiggins, Chaotic transport in the homoclinic and heteroclinic tangle regions of quasi-periodically forced two-dimensional dynamical systems, Nonlinearity 4 (1991), 775-819. [5] D. Beigie, A. Leonard and S. Wiggins, The dynamics associated with the chaotic tangles of two-dimensional quasiperiodic vector fields: theoo, and applications, Nonlinear Phenomena in Atmospheric Oceanic Sciences, EG. Carnevale and R. Pierrehumbert, eds, IMA Vol. Math. Appl., Vol. 40, Springer, Berlin (1992), 47-138. [61 A.S. Bower, A simple kinematic mechanism for mixing fluid parcels across a meandering jet, J. Phys. Ocean. 21 (1991), 173-180. [7] A.S. Bower and M.S. Lozier, A closer look at particle exchange in the Gulf Stream, J. Phys. Ocean. 24 (1994), 1399-1418. [81 A.S. Bower and H.T. Rossby, Evidence of cross-frontal exchange processes in the Gulf Stream based isopycnal RAFOSfloat data, J. Phys. Ocean. 19 (1989), 1177-1190. [9] K.P. Bowman, Evolution of the total ozone field during the breakdown of the antarctic circumpolar vortex, J. Geophys. Res. 95 (1990), 16529-16543. [lO] K.E Bowman, Barotropic simulation of large-scale mixing in the antarctic polar vortex, J. Atmospheric Sci. 50 (1993), 2901-2914. [ll] K.P. Bowman and N.J. Mangus, Observations of deformation and mixing of the total ozone field in the antarctic polar vortex, J. Atmospheric Sci. 50 (1993), 2915-2921. [121 K.E Bowman, Rossby wave phase speeds and mixing barriers in the stratosphere. Part I: Observations, J. Atmospheric Sci. 53 (1996), 905-916. [13] K.P. Bowman, Manifold geometry, and mixing in observed atmospheric flows, Preprint (1999). [14] M.G. Brown and R.M. Samelson, Particle motion in vorticity-conserving, two-dimensional incompressible flows, Phys. Fluids 6 (1994), 2875-2876. [15] W.H. Calvin, The great climate flip-flop, Atlantic Monthly 281 (1) (1998), 47-64. [16] R. Camassa and S. Wiggins, Chaotic advection in a Rayleigh-Bdnardflow, Phys. Rev. A 43 (1991), 774797. [171 D. del-Castillo-Negrete and P.J. Morrison, Chaotic transport by Rossby waves in shear flow, Phys. Fluids A 5 (1993), 938-965. [18] R. Doerner, B. Htibinger, W. Martienssen, S. Grossman and S. Thomae, Stable manifolds and predictability' of dynamical systems, Chaos, Solitons, & Fractals 10 (1999), 1759-1782. [19] J.Q. Duan and S. Wiggins, Fluid exchange across a meandering jet with quasi-periodic variabili~, J. Phys. Ocean. 26 (1996), 1176-1188. [20] G.R. Flied, E Malanotte-Rizzoli and N.J. Zabusky, Nonlinear waves and coherent vortex structures in barotropic fl-plane jets, J. Phys. Ocean. 17 (1987), 1408-1438. [21] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1983). [22] G. Haller, Distinguished material surfaces and coherent structures in 3D fluid flows, Preprint (2000). [23] G. Haller and A.C. Poje, Finite time transport in aperiodic flows, Phys. D 119 (1998), 352-380. [24] G. Haller and A.C. Poje, Lagrangian transport and diffusion in two-dimensional aperiodic flows, Preprint
(2OOO). [25] G. Hailer and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Phys. D 147 (2000), 352-370. [26] J. von Hardenberg, K. Freadrich, F. Lunkeit and A. Provenzale, Transient chaotic mLving during a baroclinic life cycle, Chaos 10 (2000), 122-134. [27] W.R. Holland and EB. Rhines, Example of eddy-induced ocean circulation, J. Phys. Ocean. 10 (1980), 1010-1031. [281 T.J. Kaper and S. Wiggins, Lobe area in adiabatic Hamiltonian systems, Phys. D 51 (1991), 205-212. [291 T.J. Kaper and G. Kovarir, A geometric criterion for adiabatic chaos, J. Math. Phys. 35 (1994), 1202-1218. [301 R.S. MacKay, J.D. Meiss and I.C. Percival, Transport in Hamiltonian systems, Phys. D 13 (1984), 329-354. [311 M.E. McIntyre, On the Antarctic ozone hole, J. Atmospheric Terr. Phys. 51 (1989), 29-43. [32] P.D. Miller, C.K.R.T. Jones, A.M. Rogerson and L.J. Pratt, Quantifying transport in numerically generated velocity'fields, Phys. D 110 (1997), 105-122. [331 P.D. Miller, L.J. Pratt, K.R. Helfrich and C.K.R.T. Jones, Material transport and potential vorticity flux in a model of recirculation past an island, Preprint (2000).
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[34] H.E. Nusse and J. Yorke, Dynamics: Numerical Explorations, 2nd edn., Appl. Math. Sci., Vol. 101, Springer, Berlin (1998). [35] J.M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge Univ. Press, Cambridge (1989). [36] E Paparella, A. Babiano, C. Basdevant, A. Provenzale and P. Tanga, A Lagrangian study of the Antarctic polar vortex, J. Geophys. Res. 102 (1997), 6765-6773. [37] R.T. Pierrehumbert and H. Yang, Global chaotic mixing on isentropic surfaces, J. Atmospheric Sci. 50 (1993), 2462-2480. [38] A.C. Poje and G. Haller, Geometry of cross-stream mixing in a double-gyre ocean model, J. Phys. Ocean. 29 (1999), 1649-1665. [39] L.J. Pratt, M.S. Lozier and N. Beliakova, Parcel trajectories in barotropic jets: Neutral modes, J. Phys. Ocean. 25 (1995), 1451-1466. [40] A. Provenzale, Transport by coherent barotropic vortices, Ann. Rev. Fluid Mech. 31 (1999), 55-93. [41] A.M. Rogerson, P.D. Miller, L.J. Pratt and C.K.R.T. Jones, Lagrangian motion and fluid exchange in a barotropic meandering jet, J. Phys. Ocean. 29 (1999), 2635-2655. [42] V. Rom-Kedar, A. Leonard and S. Wiggins, An analytical study of transport, mixing, and chaos in an unsteady vortical flow, J. Fluid Mech. 214 (1990), 347-394. [43] V. Rom-Kedar and S. Wiggins, Transport in two-dimensional maps, Arch. Rational Mech. Anal. 109 (1990), 239-298. [44] B. Sandstede, S. Balasuriya, C.K.R.T. Jones and P. Miller, Melnikov theory for finite-time vector fields, Nonlinearity 13 (2000), 1357-1377. [45] R.M. Samelson, Fluid exchange across a meandering jet, J. Phys. Oceanography 22 (1992), 431-440. [46] M. Toner, A.C. Poje, A.D. Kirwan Jr., B. Lipphardt and C. Grosch, Reconstructing basin-scale Eulerian velocity fields from simulated drifter data, J. Phys. Ocean. (to appear). [47] S. Wiggins, Chaotic Transport in Dynamical Systems, Springer, New York (1992).
CHAPTER
3
Geometric Singular Perturbation Analysis of Neuronal Dynamics Jonathan E. Rubin Department of Mathematics, Universib' of Pittsburgh, Pittsburgh, PA 15260, USA
David Terman Department of Mathematics, The Ohio State Universi~, Columbus, OH 43210, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which display bursting oscillations. There are, in fact, several different classes of bursting solutions; these have been classified by the geometric properties of how solutions evolve in phase space. We describe several of the bursting classes and then review related rigorous mathematical analysis. We then discuss the dynamics of small networks of neurons. We are primarily interested in whether excitatory or inhibitory synaptic coupling leads to either synchronous or desynchronous rhythms. We demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. Finally, we discuss larger networks of neuronal oscillators involving two distinct cell populations. In particular, we demonstrate how dynamical systems methods can be used to analyze recent models for sleep rhythms and other oscillations generated in the thalamus. The analysis helps to explain the generation of the different thalamic rhythms and the transitions between them, in both of which inhibition plays a crucial role.
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1. Introduction Oscillations arise throughout the central nervous system [47,38,79,89,9]. These oscillations have been implicated in the generation of sleep rhythms, epilepsy, Parkinsonian tremors, sensory processing, and learning [30,39,76,79,40]. Oscillatory behavior also arises in such activities as respiration, movement, and secretion [ 13,10]. Models for the relevant neuronal networks often exhibit a rich structure of dynamic behavior. The behavior of even a single cell can be quite complicated [11,13,82,83,31]; an individual cell may, for example, fire repetitive action potentials or bursts of action potentials that are followed by a silent phase of near quiescent behavior [57,58,96]. The bursting behavior may wax and wane on a slower time scale [18,3,19]. Examples of population rhythms include synchronous behavior, in which every cell in the network fires at the same time, and clustering [27,28,44], in which the entire population of cells breaks up into subpopulations or blocks; every cell within a single block fires synchronously and different blocks are desynchronized from each other. Of course, much more complicated population rhythms are also possible [89,86,44]. The activity can also propagate through the network in a wave-like manner [41,19,88,29,61 ]. In this article, we review numerous recent results in which geometric singular perturbation methods have been used to analyze the dynamics of neuronal networks. Much of the analysis presented is done in the singular limit. Nonetheless, the results are extremely useful for elucidating the roles of network parameters and components in generating various activity patterns. We begin in Section 2 by considering models for single cells which exhibit bursting oscillations. There are, in fact, several different types of bursting oscillations, and there has been considerable effort in trying to classify the underlying mathematical mechanisms responsible for these oscillations [58,4,37]. Models for bursting oscillations contain multiple time scales and this often leads to very interesting issues related to the theory of singular perturbations [46]. Even the simplest bursting models may exhibit exotic behavior, including chaotic dynamics, for certain parameter ranges [ 11,83]. In Section 3, we consider models for a pair of mutually coupled cells. The results in this section are complementary to related work in [43], Chapter 1 in this Handbook. Each cell will be modeled as a relaxation oscillator; one can view this as a simple depiction of a bursting neuron in which the active phase corresponds to the envelope of a burst's rapid spikes. We only consider a form of coupling between cells that represents chemical synapses, since this is the primary means by which neurons in the central nervous system communicate with each other. There are, however, many forms of synaptic coupling [40]. It may be excitatory or inhibitory and it may exhibit either fast or slow dynamics. We will be primarily interested in whether excitatory or inhibitory coupling leads to either synchronous or desynchronous rhythms. We will demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. In Section 4, we consider a larger network of relaxation oscillators with inhibitory and excitatory synaptic coupling, referred to as a globally inhibitory network [65,63]. This network may exhibit both synchronous and clustered solutions and we give sufficient conditions for when each of these rhythms arises. These results help to clarify how the intrinsic properties of individual cells interact with the synaptic properties to produce the emergent
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population rhythm. Inhibition, for example, may play multiple roles in producing different rhythms. Which rhythms arise depends on how inhibition interacts with intrinsic properties of the neurons; the nature of these interactions depends on the underlying architecture of the network. Finally, in Section 5, we consider a detailed model for certain sleep rhythms produced in the thalamus. Many of the results in this paper, as well as the particular form of the globally inhibitory network, are motivated by experimental, modeling, and computational studies of these rhythms (see [28,22], which provides a review of related works). After describing the rhythms, we discuss in some detail how one models these networks. The model for each cell is based on the Hodgkin-Huxley formalism [33]. We then discuss how the geometric analysis helps to explain the generation of different thalamic rhythms and the transitions between them.
2. Bursting oscillations 2.1. Introduction Certain neurons and other excitable cells exhibit bursting oscillations; this behavior is characterized by a silent phase of near steady state resting behavior alternating with an active phase of rapid, spike-like oscillations, as shown in Figure 1. Examples of biological systems which display bursting oscillations include the Aplysia R-15 neuron, insulin secreting pancreatic beta cells, and neurons in the hippocampus, cortex and thalamus. For reviews, see [96,67,37]. Figure 1 shows several different types of bursting oscillations. Figure 1A displays an example of square-wave bursting. This is characterized by abrupt periodic switching between the quiescent, or silent, phase and the active phase of repetitive firing. Note that the frequency of spikes decreases at the end of the active phase. Figure 1B illustrates elliptic bursting. Small amplitude oscillations occur during the silent phase and the amplitude of spikes gradually waxes and wanes. Finally, Figure 1C displays parabolic bursting. The spike rate first increases and then decreases in a parabolic manner. The mathematical mechanisms responsible for each class of bursting oscillation are described in terms of geometric properties of the corresponding phase space dynamics. The model for each involves multiple time scales and can be written as !
x =f(x,y), y' = eg(x, y).
(2.1)
Here, x 6 R n represents fast variables, while y 6 IR'n represents slow variables; e is a small singular perturbation parameter. We refer to the first equation in (2.1), with y considered as constant, as the fast subsystem (FS). The silent phase of the bursting solution corresponds to the passage of a trajectory of (2.1) near a manifold of fixed points of (FS). The active phase of repetitive spikes corresponds to the passage of the trajectory near a manifold of periodic solutions of (FS). It is the slow processes which modulate the fast dynamics between these two phases. Different classes of bursting oscillations are distinguished by
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the mechanisms by which the bursting trajectories switch between the silent and active phases. This is closely related to the global bifurcation structure of the fast subsystem with the slow variables treated as parameters. We note that models for bursting oscillations may exhibit other types of periodic solutions, as well as more exotic behavior including chaotic dynamics [ 11,82,83]. The models contain multiple time scales and this often leads to very interesting issues related to the theory of singular perturbations. For example, homoclinic orbits usually play an important role in the generation of these rhythms; the active phase of rapid oscillations may either begin or end (or both) as the bursting trajectory crosses a homoclinic point. At these points, standard singular perturbation methods may break down, so more delicate analysis is required. Moreover, the homoclinic orbits are often directly responsible for the generation of chaotic dynamics [82,83]. In the next section, we describe the mathematical mechanisms responsible for the three classes of bursting oscillations shown in Figure 1. The description is quite heuristic; however, the formal constructions are extremely useful in understanding what geometric ingredients are needed to generate a particular class of bursters and how the dynamics changes with respect to parameters.
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Rigorous results related to square-wave bursting are presented in Section 2.3. These results demonstrate that even the simplest models for bursting oscillations lead to very interesting problems associated with the geometric theory of singular perturbations. It need not be true, for example, that the bursting solution is uniquely determined for all e sufficiently small. Moreover, there are multiple ways in which chaotic dynamics may arise as parameters in the model are varied.
2.2. Bursting mechanisms We now describe the mathematical mechanisms responsible for the three classes of bursting oscillations shown in Figure 1. This description follows closely the pioneering work of Rinzel [58]. We only consider the simplest, lowest dimensional models which generate these solutions. As we shall see, square-wave and elliptic bursting can arise in threedimensional systems, while parabolic bursting requires at least four dimensions. In each case, we consider a system of the form (2.1) and make geometric assumptions concerning the set of fixed points and periodic solutions of (FS). Some assumptions are then needed on the equations governing the slow variables. These assumptions can usually be verified for a specific system by using numerical software. 2.2.1. Square-wave bursting.
Consider a three-dimensional version of (2.1) of the form
v ' = f (v, w, y), w ' = g(v, w, y),
(2.2)
y' = eh(v, w, y, ~). A concrete system which exhibits square-wave bursting is given in Remark 2.6. The fast subsystem (FS) now consists of the first two equations in (2.2) with the slow variable y considered as constant. In the third equation, )~ represents a fixed parameter. Later, we discuss complex bifurcations that arise when )~ is varied. The primary assumptions on (2.2) concern the bifurcation structure of the fast subsystem with y treated as a parameter. This structure appears in Figure 2, which also shows the projection of the square-wave bursting solution onto the corresponding bifurcation diagram. The set of fixed points of (FS) is assumed to be a Z-shaped curve in the (v, w, y) phase space. We denote this curve by S; only a portion of this Z-shaped curve is shown in Figure 2. The fixed points along the lower branch of S are stable solutions of (FS), while the fixed points on the middle branch of S are saddles. Fixed points along the upper branch of S may be stable or unstable. We also assume that there exists a one-parameter family of periodic solutions of (FS), denoted by P. These limit cycles originate at a (subcritical) Hopf bifurcation along the upper branch of S and terminate along a solution of (FS) that is homoclinic to one of the fixed points on the middle branch of S. Assumptions are also needed about the slow dynamics. We assume that the y-nullsurface {h = 0} defines a two-dimensional manifold that intersects S at a single point. This point lies on the middle branch of S between the homoclinic point and the left knee of S. Finally, h > 0 above {h = 0} and h < 0 below {h = 0}.
Geometric singular perturbation analysis of neuronal dynamics 20
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We now give a heuristic explanation for why this system generates a square-wave bursting solution. Suppose that e > 0 is small and consider a solution that begins close to the lower branch. Because this branch consists of stable fixed points of (FS), the trajectory quickly approaches a small neighborhood of the lower branch. The trajectory tracks leftward along the lower branch according to the slow dynamics, until it passes the left knee. This portion of the solution corresponds to the silent phase. Once past the left knee, the trajectory is attracted to near P, the branch of periodic solutions of (FS). This generates the fast repetitive spikes of the bursting solutions. The trajectory passes near P, with increasing y, until it reaches a neighborhood of the homoclinic orbit of (FS). Once it passes the homoclinic orbit, the fast dynamics eventually forces the trajectory back to near the lower branch of S and this completes one cycle of the bursting solution. This description is formal. It is not at all clear that if the system (2.2) satisfies the above assumptions, then, for all s sufficiently small, there exists a unique periodic solution corresponding to a bursting oscillation. In Section 2.3, we show, in fact, that such a result cannot
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be true, in general. We describe a result, proved in [46], that the bursting solution will be uniquely determined for all e sufficiently small, except for those e that lie in a certain very small set. REMARK 2.1. A crucial ingredient for square-wave bursting is bistability. This allows for a hysteresis loop between a lower branch of stable fixed points and an upper branch of stable limit cycles. It is also very important that the slow nullsurface {h = 0} lies between these two branches. If this last condition is not satisfied, then the system may exhibit other types of solutions. For example, suppose that {h = 0} intersects the lower branch of S. This point of intersection will then be a globally stable fixed point of (2.2). If, on the other hand, {h = 0} intersects S along its middle branch above the homoclinic point, then (2.2) may give rise to a stable limit cycle which always remains in the active phase near P. This type of solution is referred to as continuous spiking. Rigorous results concerning the existence of continuous spiking are presented in [82,83]. REMARK 2.2. Square-wave bursting arises in models for electrical activity in pancreatic /3-cells. It is believed that this activity plays an important role in the release of insulin from the cells. The first mathematical model for this bursting was due to Chay and Keizer [ 10]. There have been numerous related models, based on experimental data, since then. A review of these models, along with a detailed description of the more biological issues, is given in [67]. Square wave bursting also arises in recent models for respiratory CPG neurons [7] and models for pattern generation based on synaptic depression [81 ]. REMARK 2.3. It is still not clear what underlying biological mechanisms cause bursting oscillations in pancreatic/3-cells [ 15]. In fact, it is possible that bursting may not be generated by the intrinsic properties of a single cell; rather, the bursting may be a network property generated by the coupling between cells. Computational studies in [69] have demonstrated that a population of cells with electrical, or gap-junction, coupling can exhibit bursting oscillations although the parameters of each cell are set so that the cells do not oscillate without any coupling. Detailed analysis of two mutually coupled square-wave bursters is given in [66]. REMARK 2.4. Very complicated (global) bifurcations can take place as the parameters e or )~ are varied in (2.2). The singular perturbation parameter e controls the rate at which a bursting trajectory passes through the silent and active phases. In particular, the number of spikes per burst is O(1/e) and becomes unbounded as e --+ 0. It is demonstrated in [82] that Smale horseshoe chaotic dynamics can arise during the transition of adding a spike. For example, the solution shown in Figure 3B exhibits a somewhat random variation between 3 and 4 spikes per burst. In Section 2.3, we describe how spikes are added to the bursting solution as e decreases. Perhaps even more interesting is the bifurcation structure of (2.2) as )~ is varied. In the /3-cell models, ~, is related to the glucose concentration. As the glucose level gradually increases, the cells exhibit resting behavior, then bursting oscillations, and then continuous spiking. This is consistent with behavior exhibited by the model. As )~ increases, the
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Fig. 3. Chaotic dynamics from the square-wave bursting model. Note that v represents a rescaled voltage. A. (top) Chaotic solution generated by a bifurcation in X. B. (bottom) Chaotic solution generated by a bifurcation in e.
y-nullsurface {h = 0} intersects the lower branch of S, then the middle branch of S below the homoclinic point, and then the middle branch of S above the homoclinic point. Numerical studies [11] and rigorous analysis [83] have shown that as )~ varies between the bursting and continuous spiking regimes, the bifurcation structure of solutions must be very complicated. A Poincar6 return map defined by the flow from a section transverse to the homoclinic orbit of (FS) will exhibit Smale-horseshoe dynamics for a robust range of parameter values. This leads to solutions in which the number of spikes per burst varies considerably, as shown in Figure 3A. REMARK 2.5. Some phenomenological, polynomial models for square-wave bursting have been proposed. See, for example, [32,54,16]. Analysis of models with two slow variables which exhibit square-wave bursting is given in [70]. REMARK 2.6. A system of equations which give rise to square-wave bursting is [59]: v
!
-
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-
vc~) + g ~ w ( v - v~)
+ g l ( v - vl) + g ~ c a Z ( y ) ( v -- V~)) + I,
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,,_/=
,,,)/,-(,,),
y' = 2 0 e ( - l Z g c a m ~ ( v ) ( v -
Vca) - Y),
where, gca - 4.0, gk = 8.0, gl -- 2.0, Vk -- --84, Vl -- --60, l)ca -- 120.0, I -- 45, gkca = 0.25, r = 0.23, e -- 0.005, and/z -- 0.02. The nonlinear functions are given by moo (v) 0 . 5 ( 1 . 0 + t a n h ( ( v + 1.2)/18)), w c ~ ( v ) = 0 . 5 ( 1 . 0 + t a n h ( ( v - 12)/17.4)), z ( y ) = y / ( 1 + y ) and r ( v ) - c o s h ( ( v - 12.0)/34.8).
2.2.2. E l l i p t i c bursting. Elliptic bursting can also arise in a system of the form (2.2) in which there are two fast variables and one slow variable. The bifurcation diagram of (FS) for an elliptic bursting scenario, with the slow variable treated as a parameter, is shown in Figure 4, which includes the projection of the elliptic bursting solution. Bistability is crucially important for the generation of elliptic bursting, just as it is for square-wave bursting. An important difference, however, is that for elliptic bursting, the
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Geometric singular perturbation analysis of neuronal dynamics
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curve S of fixed points of (FS) need not be Z-shaped; there may be only one fixed point of (FS) for each value of y. The branch of periodic solutions P now originates at a subcritical Hopf bifurcation along S. To obtain bursting, we hypothesize that the slow variable decreases along S; that is, h < 0 near S. During the silent phase, the bursting solution evolves near the stable portion of S until it passes the Hopf point, beyond which the fixed points of S are no longer stable. Note, however, that the trajectory does not jump up to the active phase immediately after crossing the Hopf point. The slow variable y may traverse a distance that is O(1) with respect to e past the Hopf point before jumping up. This type of delayed behavior or slow-passage past a Hopf point has been studied extensively in the singular perturbation literature [53,2]. We must also hypothesize that there is a net increase in the slow variable as the bursting trajectory passes near P. More precisely, let (Vy(t), Wy(t)) denote the periodic solutions along the outer branch of P and suppose that their periods are T (y). Let
h- ( y ) -
r ( 1y )
~0 T(y) h(vy(t),
Wy(t), y ) d t
denote the average of h along these limit cycles. We assume that/t(y) > 0 for each y. Thus during the active phase, y increases until the bursting solution passes the turning point, or knee, of P. The fast dynamics then forces the trajectory back towards S and a new silent phase begins. REMARK 2.7. Both square-wave and elliptic bursting depend on bistability and hysteresis. An important difference between these two classes of bursting is how the active phase terminates. Square-wave bursting ends at a homoclinic bifurcation; the period of oscillation, therefore, increases at the end of each burst. For elliptic bursting, the active phase ends at a saddle node of periodic orbits. There is no pattern for the spike frequencies in general. REMARK 2.8. Elliptic bursting arises in models for thalamic neurons [21], rodent trigeminal neurons [52], and 40 Hz oscillations [48,92]. REMARK 2.9. Here we assumed that the branch of periodic orbits of (FS) originate at a subcritical Hopf bifurcation. Hoppensteadt and Izhikevich [34] show that elliptic bursting is also possible even if the Hopf bifurcation is supercritical. REMARK 2.10. A system of equations which give rise to elliptic bursting is [59]:
v' - -(gcamc~(v)(v - Vca) + gkw(v -- l)k) + gl(v - Vl) + gkcaZ(y)(v - vk)) + I, y' -- e ( - # g c a m ~ ( v ) ( v -
Y e a ) - Y),
where, gca -= 4.4, gk = 8.0, gt = 2.0, Vk = --84, vt = --60, V c a = 120.0, I = 120, gkca "- 0.75, ~b -- 1.2, e -- 0.04, and # = 0.016667. The nonlinear functions
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are given by mc~(v) = 0.5(1.0 + tanh((v + 1.2)/18)), w ~ ( v ) = 0.5(1.0 + tanh((v 2.0)/30.0)), z(y) = y/(1 + y) and r(v) = cosh((v - 2.0)/60.0). 2.2.3. Parabolic bursting. Both square-wave and elliptic bursting can be achieved in a system with only one slow variable. Moreover, both depend on bistability of the fast dynamics. Parabolic bursting, on the other hand, requires at least two slow variables and does not arise from a hysteresis phenomenon. A geometric model for parabolic bursting is the following. Consider a system of the form (2.1) where x 6 ~2 and y - (yl, Y2) E ~2. There are now two slow variables, namely yl and y2. We first describe the bifurcation structure of the fast subsystem with both slow variables considered as parameters. This is illustrated in Figure 5, where we plot one component of the fast variable, corresponding to the membrane potential, along with both slow variables. We also show the projection of a parabolic bursting solution onto this figure. We hypothesize that the set of fixed points of (FS) forms a Z-shaped surface. The fixed points along the lower branch of this surface are assumed to be stable fixed points of (FS). There is a curve along the upper branch of fixed points where Hopf bifurcations occur. Periodic solutions arise at these Hopf points and terminate along orbits homoclinic to the
Fig. 5. Bifurcation structure for parabolic bursting. Here, x denotes one component of the fast variable. The meshed surface represents the maximumand minimumx-values of periodic orbits of (FS) while the solid surface consists of fixed points of (FS). The solid trajectory is the projection of a parabolic bursting solution.
Geometric singular perturbation analysis of neuronal dynamics
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fixed points along the lower fold of the fixed point surface. In Figure 5, we show the maximum and minimum values of the fast variable along each of these periodic solutions. The existence of a parabolic bursting solution also requires hypotheses on the slow dynamics. To write reduced equations which determine the evolution of the slow variables in the silent phase, denote the lower branch of fixed points by x = q~(y). The silent phase evolution of the slow variables is then given by the reduced equations
y - g(~(y), y),
(2.3)
where differentiation is now with respect to r = et. One obtains reduced equations for the evolution of the slow variables in the active phase using averaging. Suppose that y = (yl, y2) lies in the region where there exists a stable limit cycle of (FS). Let Xy(t) be the corresponding periodic solution of (FS) with period T (y) and consider the averaged quantity
~'(Y) -
r ( y1)
f0 r(y) g(xy(t),
y) dt.
The active phase evolution of the slow variables is then given by the averaged equations y -- ~(y).
(2.4)
Parabolic bursting corresponds to the existence of a closed curve in the slow (yl, y2) phase plane which passes through both the region of stable fixed points of (FS), where it satisfies (2.3), and the region of stable limit cycles of (FS), where it satisfies (2.4). This is illustrated in Figure 6. Note that the active phase of the bursting solution both begins and ends along a curve of homoclinic bifurcations. Since the periods of the limit cycles tend to infinity at the homoclinic bifurcations, the interspike interval is longer at both the beginning and end of each burst. This accounts for the parabolic nature of the period of fast oscillations. REMARK 2.1 1. Parabolic bursting is found in the Aplysia R-15 neuron [ 1]. Rinzel and Lee [60] considered a model due to Plant [56] and were the first to give a detailed analysis of parabolic bursting by dissecting the equations into fast and slow subsystems as described here. REMARK 2.12. Rigorous results related to parabolic bursting are given in [42,73,34]. REMARK 2.13. A system of equations which give rise to parabolic bursting is [59]:
v' - --(i~a(V) + i~(v) + i~(v) + i~ca(V, c) + icas(V, s)) + I,
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0.2
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where, ica(V) = g c a m ~ ( v ) ( v - Yea), ik(V) = gkW(V-- Vk), il(V) = g l ( v - - Vl), ikca(V, C) = gkcaZ(C)(V -- Vk), and icas(V,S) = gcasS(V - Yea). The constants are given by gca = 4.0, gk - 8.0, gl = 2.0, Vk = --84, Vt -- --60, Yea -- 120.0, I = 65, gkca - 1.0, ~b -1.333, e = 0.002, /z = 0.025, rs -- 0.05, and gcas = 1.0. The nonlinear functions are given by m ~ ( v ) = 0.5(1.0 + tanh((v + 1.2)/18)), w ~ ( v ) = 0.5(1.0 + tanh((v - 12.0)/17.0)), rw(V) = cosh((v - 12.0)/34.0), z(c) = c/(1 + c), and s o = 0.5(1 + tanh((v - 12)/24)).
2.3. Rigorous results f o r square-wave bursting solutions Here we present a theorem that addresses the issue of when, and in what sense, the heuristic description of a square-wave bursting trajectory given in Section 2.2.1 can be rigorously justified. The theorem implies that there may exist a small range of values of e for which the bursting trajectory is not uniquely determined; moreover, for exactly these values, the bursting solution does not closely follow the heuristically defined orbit. The reason why such values of e exist is related to the mechanism by which spikes are added as e decreases, as we describe here. A more complete discussion is given in [46]. For the following theorem, we consider (2.2) and assume that the geometric assumptions described in Section 2.2.1 are satisfied. We also need some technical assumptions concerning the bounded solutions of the fast subsystem (FS). We assume that each of the stable fixed points along the lower branch of S is hyperbolic, the manifold P of periodic solutions is normally hyperbolic, the right knee of S is nondegenerate, and the homoclinic
Geometric singular perturbation analysis of neuronal dynamics
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orbit arises from the transverse intersection of stable and unstable manifolds. Finally, we assume that S, P, and orbits heteroclinic to S and P represent all the bounded solutions of (FS). THEOREM 2.1. The periodic bursting solution is uniquely determined and asymptotically stable f o r all values o f e > 0 sufficiently small except f o r those in a set o f the f o r m O. Moreover, U i % 1(ei -- 3i, ei + 6i) 9 The F~i and 3i can be chosen so that limi__+e~ei 6i - - 6 i + 1 >
C182
and
~i ~ C2 e -k/ei
f o r some positive constants C1, C2 and k.
The proof of Theorem 2.1 is given in [46]. An important step in the proof is to explain how the bursting solution adds spikes as e varies. It is during these transitions that the heuristic construction is not justified. Here we give a geometric description of how these transitions take place. The number of spikes is determined by how many times the bursting trajectory spirals around in phase space near P. The active phase terminates when the bursting trajectory passes near the homoclinic orbit of (FS) and jumps down to the lower branch of S. The key to understanding how spikes are added, therefore, is to understand what determines when the trajectory jumps down to the silent phase. As we describe below, this crucially depends on the center-unstable and center-stable manifolds of the fixed points along the middle branch of S. The center-stable manifolds serve to separate those trajectories which continue to spiral in the active phase and those which jump down to the silent phase. Note that, when e = 0, there are two trajectories in the unstable manifold of each of the fixed points along the middle branch of S. The union of these trajectories forms the centerunstable manifold of the middle branch. One of these trajectories evolves towards the active phase, looping around the upper branch. Suppose that the homoclinic orbit of (FS) is at Y = Yh. If y < Yh, then this trajectory approaches one of the periodic solutions along P, while if y > Yh, then it ultimately approaches a stable fixed point along the lower branch. The other unstable trajectory evolves directly towards the silent phase and approaches the stable fixed point along the lower branch. Now the stable manifolds to the fixed points along the middle branch separate the two branches of unstable trajectories. Hence, if a trajectory lies close to the middle branch, it will either give rise to a spike or jump down to the silent phase depending on which side of the appropriate stable manifold it lies on. What we have described so far holds for e = 0, however, this all carries over for small e > 0. To make this more precise, let W~ and W~ be the union of all the stable and unstable manifolds to the fixed points along the middle branch when e = 0. (We exclude small neighborhoods of the left and right knees.) These are both smooth, two-dimensional, invariant manifolds. For e > 0, these manifolds perturb to manifolds Ws and W~ (see [23]), which are also both smooth, two-dimensional, invariant, and lie a Cl-distance O(e) close to W~ and W~ near the middle branch. If we let W~ = Ws n W~, then We, Ws, and We are the center, center-stable, and center-unstable manifolds corresponding to the middle branch, respectively. As discussed before, W~ divides Wff into two pieces; one piece 'points' towards the active phase, while the other piece 'points' towards the silent phase.
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We now can see the significance of this separation to bursting solutions. We begin a bursting solution in the active phase near the branch of periodic orbits P. The orbit gives rise to spikes as it tracks near P moving slowly to the right. As the orbit approaches the homoclinic orbit, it passes close to the middle branch. As long as the orbit lies on the 'jump-up' side of Ws , it will keep spiking. Once it crosses over to the other side of Ws, it will jump down to the silent phase. It is possible, however, that the orbit lies precisely on W~, and it is important to understand the fate of the trajectory in this case. If the orbit lies on Ws, then it must track close to the middle branch (actually we), slowly moving to the fight. The orbit eventually jumps down near the fight knee. Note that if we start (exponentially) close to W~, then the trajectory will track close to the middle branch for some finite distance before it either jumps up or jumps down. If the bursting solution behaves in this way, then it will not lie close to the heuristically defined bursting orbit. It is precisely this mechanism (lying close to W~) that can destroy the uniqueness and stability of the bursting solution. Now consider the transition of adding a spike as e decreases. Suppose, for concreteness, that when e = e2, (2.2) exhibits a solution with 2 spikes per burst and when e = e3, there are 3 spikes per burst. When e = e2, the bursting solution winds around P two times. After the second cycle, it lies on the jump-down side of Ws so it falls down to the silent phase. When e = e3, however, the bursting solution winds around P three times before jumping down. After two cycles, the solution still lies on the jump-up side of Ws, and thus it returns to the active phase. There must exist, therefore, some e* 6 (e3, e2) for which a burstinglike solution lies precisely in W~. It is for values of e very close to e* that the singular construction breaks down.
3. Two mutually coupled cells 3.1. Introduction In this section, we consider a network consisting simply of two mutually coupled cells. By considering such a simple system, we are relatively easily able to describe how we model networks of oscillators, the types of behavior that can arise in such systems and the mathematical techniques we use for the analysis of the behavior. For this discussion, we assume that each cell, without any coupling, is modeled as the relaxation oscillator v' = f (v, w), w' = eg(v, w).
(3.1)
Here e is assumed to be small; that is, w represents a slowly evolving quantity. We assume that the v-nullcline, f (v, w) = 0, defines a cubic-shaped curve and the w-nullcline, g -- 0, is a monotone decreasing curve which intersects f = 0 at a unique point p0, as shown in Figure 7. We also assume that f > 0 ( f < 0) below (above) the v-nullcline and g > 0 (< 0) below (above) the w-nullcline. If P0 lies on the middle branch of f - 0, then (3.1) gives rise to a periodic solution for all e sufficiently small and we say that the system is oscillatory. In the limit e --+ 0, one can construct a singular solution as shown in Figure 7.
Geometric singular perturbation analysis of neuronal dynamics
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W
~
f=O
g =0
lP()
V
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If P0 lies on the left branch of f = 0, then the system is said to be excitable; Po is a stable fixed point and there are no periodic solutions for all e small. System (3.1) can be viewed as a simple model for a bursting neuron in which the active phase corresponds to the envelope of a burst's rapid spikes. (See also [43].) Of course, a two-dimensional model for a single cell cannot exhibit the more exotic dynamics described in the previous section for a bursting cell. However, by considering a simple relaxationtype oscillator for each cell, we will be able to discuss how network properties contribute to the emergent behavior of a population of cells. It is, of course, a very interesting issue to understand how this population behavior changes when one considers more detailed models for each cell. Some results for more detailed models are given in [63]. Networks of two coupled cells may display a variety of different rhythms. By a synchronous solution, we mean a rhythm in which both cells exhibit exactly the same behavior, oscillating in phase with each other. An antiphase solution is shown in Figure 8A. This is the simplest example of what we will call a clustered solution. In a larger population of cells, the network is said to exhibit clustering if the population breaks up into distinct subgroups such that all of the cells within each subgroup are synchronized with each other, but cells belonging to different subgroups are desynchronized. Figure 8B shows a suppressed solution. One of the cells oscillates periodically between the silent and active phases, while the other cell always remains in the silent phase. A more exotic solution is shown in Figure 8C. During each period of oscillation, one of the cells fires two action potentials, while the other cell fires just one. In the next section, we describe how we model the two mutually coupled cells. The form of coupling used is referred to as synaptic coupling and is meant to correspond to a simple model for chemical synapses, the primary means by which neurons in the central nervous
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Fig. 8. Nonsynchronous solutions for two mutually coupled cells [85]. Note that v represents a rescaled voltage. A. (top) Antiphase solution. B. (middle) Suppressed solution. C. (bottom) Two-to-one solution.
system communicate with each other. As we shall see, there are many different forms of synaptic coupling. For example, it may be excitatory or inhibitory and it may exhibit either fast or slow dynamics. We are particularly interested in how the nature of the synaptic coupling affects the emergent population rhythm. A natural question is whether excitatory or inhibitory coupling leads to either synchronous or desynchronous rhythms. There are four possible combinations and we will demonstrate that all four may be stably realized, depending on the details of the intrinsic and synaptic properties of the cells. Many of the results given here are complementary to those in [43] (Chapter 1 in this Handbook) which use the fast-slow structure in a somewhat different way.
3.2. Synaptic coupling We model a pair of mutually coupled neurons by the following system of differential equations !
vI -
f(vl, wl)
-
s2gsyn(l)l -
Ysyn),
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!
w I -- e g ( v l , w l ) , l)~ - - f ( v 2 ,
(3.2)
1/22) - - S l g s y n ( V 2
-- Vsyn),
l
W 2 = 8 g ( v 2 , W2).
Here (vl, wl) and (v2, 1/22) correspond to the two cells. The coupling term Sj gsyn(1)i -- l)syn) can be viewed as an additional current which may change a cell's m e m b r a n e potential vi. The parameter gsyn corresponds to the maximal conductance of the synapse and is positive, while the reversal potential Vsyn determines whether the synapse is excitatory or inhibitory. If v < Vsyn along each bounded singular solution, then the synapse is excitatory, while if v > Vsyn along each b o u n d e d singular solution, then the synapse is inhibitory. The terms si, i = 1, 2, in (3.2) encode how the postsynaptic conductance depends on the presynaptic potentials vi. There are several possible choices for the si. The simplest choice is to assume that si = H ( v i - 0 s y n ) , where H is the Heaviside step function and 0syn is a threshold above which one cell can influence the other. Note, for example, that if vl < 0syn, then sl = H (vl - 0syn) -- 0, so cell 1 has no influence on cell 2. If, on the other hand, vl > 0syn, then sl = 1 and cell 2 is affected by cell 1. Another choice for the si is to assume that they satisfy a first order equation of the form !
S i - - C ~ ( 1 -- s i ) H ( v i
- 0syn) - f l s i ,
(3.3)
where ot and/~ are positive constants and H and 0syn a r e as before. Note that ot and 13 are related to the rates at which the synapses turn on or turn off. For f a s t synapses, we assume that both of these constants are O(1) with respect to e. For a slow synapse, we assume that ot = O(1) and/~ = O(e); hence, a slow synapse activates on the fast time scale but turns off on the slow time scale. The synapses considered so far are referred to as direct synapses since they are activated as soon as a m e m b r a n e potential crosses the threshold 0syn. To more fully represent the range of synapse dynamics observed biologically, it is also necessary to consider more complicated connections. These are referred to as indirect synapses, and they are m o d e l e d by introducing new dependent variables x l and x2. Each (xi, si) satisfies the equations x~ - - eOtx (1 -- x i ) H ( v i s] -- or(1 - s i ) H ( x i
-- 0v) -- e f l x x i ,
(3.4)
- Ox) - f l s i .
The constants ax and fix are assumed to be independent of e. The effect of the indirect synapses is to introduce a delay from the time one oscillator j u m p s up until the time the other oscillator feels the synaptic input. For example, if the first oscillator j u m p s up, a secondary process is turned on when vl crosses the threshold 0v. The synapse sl does not turn on until x l crosses 0~; this takes a finite amount of (slow) time since x l evolves on the slow time scale, like the wi. Note that indirect synapses m a y be fast or slow. For fast, indirect synapses, the turn off of si, after the xi-induced delay, occurs on the fast time scale.
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3.3. G e o m e t r i c a p p r o a c h All of the networks in this paper are analyzed by treating e as a small, singular perturbation parameter. As in the previous section, the first step in the analysis is to identify the fast and slow variables. We then dissect the full system of equations into fast and slow subsystems. The fast subsystem is obtained by simply setting e = 0 in the original equations. This leads to a reduced set of equations for the fast variables with each of the slow variables held constant. The slow subsystems are obtained by first introducing the slow time scale r = et and then setting e - - 0 in the resulting equations. This leads to a reduced system of equations for just the slow variables, after solving for each fast variable in terms of the slow ones. The slow subsystems determine the evolution of the slow variables while the cells are in either the active or the silent phase. During this time, each cell lies on either the left or the right branch of some "cubic" nullcline determined by the total synaptic input which the cell receives. This continues until one of the cells reaches the left or right "knee" of its corresponding cubic. Upon reaching a knee, the cell may either jump up from the silent to the active phase or jump down from the active to the silent phase. The jumping up or down process is governed by the fast equations. For a concrete example, consider two mutually coupled cells with fast, direct synapses. The dependent variables (vi, wi, si), i -- 1,2, then satisfy (3.2) and (3.3). The slow equations are 0-
f (vi, Wi) -- Sjgsyn(Vi -- Vsyn),
(3.5)
lbi -- g ( v i , tOi), 0 -- Or(1 -- Si) H (Vi -- 0syn) -- flSi,
where differentiation is with respect to r and i ~ j. The first equation in (3.5) states that (vi, w i ) lies on a curve determined by s j . The third equation states that if cell i is silent (vi < 0syn), then si -- O, while if cell i is active, then si -- ~ / ( ~ + fl) = SA. We demonstrate that it is possible to reduce (3.5) to a single equation for each of the slow variables wi. Before doing this, it will be convenient to introduce some notation. Let ~ ( v , w, s) =-- f ( v , w ) - gsynS(V - Vsyn). If gsyn is not too large, then each Cs = {q~(v, w, s) --0} defines a cubic-shaped curve. We express the left and right branches of Cs by {v - q~L(W, s)} and {v -- q~R(W, s)}, respectively. Finally, let GL(w,s)
- g(~L(W,S),
w)
and
GR(w,s)
- g(cI)R(YO, s), 11o).
Now the first equation in (3.5) can be written as 0 - q~(Vi, tOi, S j ) with sj fixed. Hence, Vi -- Cl)ot(Vi, Sj) where ot - L if cell i is silent and ot - R if cell i is active. It then follows that each slow variable wi satisfies the single equation lbi - G ~ ( w i , s j ) .
(3.6)
By dissecting the full system into fast and slow subsystems, we are able to construct singular solutions of (3.2), (3.3). In particular, this leads to sufficient conditions for when there exists a singular synchronous solution and when this solution is (formally) asymptotically
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stable. The second step in the analysis is to rigorously prove that the formal analysis, in which e = 0, is justified for small e > 0. This raises some very subtle issues in the geometric theory of singular perturbations, some of which have not been completely addressed in the literature. For most of the results presented here, we only consider singular solutions. We note that the geometric approach used here is somewhat different from that used in many dynamical systems studies (see, for example, [59]). All of the networks considered here consist of many differential equations, especially for larger networks. Traditionally, one would interpret the solution of this system as a single trajectory evolving in a very large dimensional phase space. We consider several trajectories, one corresponding to a single cell, moving around in a much lower dimensional phase space (see also [87,86,71,85,63]). After reducing the full system to a system for just the slow variables, the dimension of the lower dimensional phase space equals the number of slow intrinsic variables and slow synaptic variables corresponding to each cell. In the worst case considered here, there is only one slow intrinsic variable for each cell and one slow synaptic variable; hence, we never have to consider phase spaces with dimension more than two. Of course, the particular phase space we need to consider may change, depending on whether the cells are active or silent and also depending on the synaptic input that a cell receives.
3.4. Synchrony with excitatory synapses Consider two mutually coupled cells with excitatory synapses. Our goal here is to give sufficient conditions for the existence of a synchronous solution and its stability. Note that if the synapses are excitatory, then the curve CA -- C~A lies 'above' Co --= {f -- 0} as shown in Figure 9. This is because for an excitatory synapse, v < Vsyn along the synchronous solution. Hence, on CA, f ( v , w) = gsynSA(V -- Vsyn) < 0, and we are assuming that f < 0 above Co. If gsyn is not too large, then both Co and CA will be cubic shaped. We assume that the threshold 0syn lies between the two knees of Co. In the statement of the following result, we denote the left knee of Co by (VLK, WLK). THEOREM 3.1. Assume that each cell, without any coupling, is oscillatory. Moreover, assume the synapses are fast, direct and excitatory. Then there exists a synchronous periodic solution of (3.2) and (3.3). This solution is asymptotically stable if one of the following two conditions is satisfied. (H1) Of /Ow < O, Og/Ov > O, and Og/Ow < 0 near the singular synchronous solution. (H2) [g(VLK, WLK)I is sufficiently small. REMARK 3.1. We note that the synchronous solution cannot exist if the cells are excitable and the other hypotheses, concerning the synapses, are satisfied. This is because along a synchronous solution, each (vi, wi) lies on the left branch of Co during the silent phase. If the cells are excitable, then each (vi, wi) will approach the point where the w-nullcline {g -- 0} intersects the left branch of Co. The cells, therefore, will not be able to jump up to the active phase.
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J.E. R u b i n a n d D. T e r m a n
lw CA
Vsyn
v
Fig. 9. Nullclines for an oscillatory relaxation oscillator with (CA) and without (Co) excitatory coupling. Note that cell 2 responds to cell 1 through Fast Threshold Modulation.
REMARK 3.2. The assumptions concerning the partial derivatives of f and g in (H1) are not very restrictive since we are already assuming that f > 0 (< 0) below (above) the v-nullcline and g > 0 (< 0) below (above) the w-nullcline. REMARK 3.3. A useful way to interpret (H2) is that the silent phases of the cells are much longer than their active phases. This is because g(vLK, WLK) gives the rate at which the slow variables wi evolve near the end of the silent phase. Note that g(vLK, wLK) will be small if the left knee of Co is very close to the w-nullcline. PROOF. We first consider the existence of the synchronous solution. This is straightforward because along a synchronous solution (vi, wl, Sl) = (v2, w2, s2) = (v, w, s) satisfy the reduced system
v' = f (v, w)
-
Sgsyn(V -
Vsyn),
w' = eg(v, w), s' = c~(1 - s) H (v - 0syn) - / ~ s . The singular solution consists of four pieces. During the silent phase, s = 0 and (v, w) lies on the left branch of Co. During the active phase s = SA and (v, w) lies on the right branch of CA. The jumps between these two phases occur at the left and right knees of the corresponding cubics. We next consider the stability of the synchronous solution to small perturbations. We begin with both cells close to each other in the silent phase on the left branch of Co, with cell 1 at the left knee ready to jump up. We follow the cells around in phase space by
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constructing the singular solution until one of the cells returns to the left knee of Co. As before, the singular solution consists of four pieces. We need to show that the cells are closer to each other after this complete cycle than they were initially. The first piece of the singular solution begins when cell 1 jumps up. When Vl (t) crosses 0syn, SI (t) --+ SA. This raises the cubic corresponding to cell 2 from Co to CA. If ]Wl (0) w2(0)l is sufficiently small, corresponding to a sufficiently small perturbation, then cell 2 lies below the left knee of CA. The fast equations then force cell 2 to also jump up to the active phase, as shown in Figure 9. Note that this piece takes place on the fast time scale. Hence, on the slow time scale, both cells jump up at precisely the same time. During the second piece of the singular solution, both oscillators lie in the active phase on the right branch of CA. Note that the ordering in which the oscillators track along the left and right branches has been reversed. While in the silent phase, cell 1 was ahead of cell 2. In the active phase, cell 2 leads the way. The oscillators remain on the right branch of CA until cell 2 reaches the right knee. The oscillators then jump down to the silent phase. Cell 2 is the first to jump down. When v2(t) crosses 0syn, s2 switches from SA to 0 on the fast time scale. This lowers the cubic corresponding to cell 1 from CA to Co. If, at this time, cell 1 lies above the right knee of CA, then cell 1 must jump down to the silent phase. This will certainly be the case if the cells are initially close enough to each other. During the final piece of the singular solution, both oscillators move down the left branch of Co until cell 1 reaches the left knee. This completes one full cycle. To prove that the synchronous solution is stable, we must show that the cells are closer to each other after this cycle; that is, there is compression in the distance between the cells. There are actually several ways to demonstrate this compression; these correspond to two different ways to define what is meant by the 'distance' between the cells. Here we consider a Euclidean metric, which is defined as follows: Suppose that both cells lie on the same branch of the same cubic and the coordinates of cell i are (vi, wi). Then the distance between the cells is defined as simply [wl - wz]. Note that during the jump up and the jump down, this metric remains invariant. This is because the jumps are horizontal so the values of wi do not change. If there is compression, therefore, it must take place as the cells evolve in the silent and active phases. We now show that this is indeed the case if (H 1) is satisfied. Suppose that when r = 0, both cells lie in the silent phase on Co. We assume, for convenience, that w2(0) > wl (0). We need to prove that wz(r) - Wl (r) decreases as long as the cells remain in the silent phase. Now each wi satisfies (3.6) with oe = L and sj = 0. Hence,
wi(r)=wi(O)+ f0 T GL(wi(~),O)d~ and, using the Mean Value Theorem,
~T
1/72(l") -- Wl ('t') -- 1/)2(0) -- 1/)1 (0) -Jr= w 2 ( 0 ) - w,(0) +
G L ( ~ 2 ( ~ ) , 0) -- GL(1/dl (~), 0) d~
fo r -0-~w OGL (w*, 0)(w2(~) - wl(~))des (3.7)
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J.E. Rubin and D. Terman !
for some w*. Now G L ( w , s ) = g(q~L(W), W). Hence, OGL/OW -- gv~L(W) + gu,. We asI sume in (H1) that gv > 0 and gw < 0 near the synchronous solution. Moreover, 4~L (w) < 0 because v = 4~L(w) defines the left branch of the cubic Co which has negative slope. It follows that OGL/OW < 0, and therefore, from (3.7), wz(r) - wl (r) < we(0) - wl (0). This gives the desired compression; a similar computation applies in the active phase. We note that if there exists g > 0 such that OGL/OW < --y along the left branch, then Gronwall's inequality shows that w2 (r) - w l (r) decreases at an exponential rate. We next consider (H2) and demonstrate why this leads to compression of trajectories. Suppose, for the moment, that g(VLK, WLK) = 0; that is, the left knee of Co touches the w-nullcline at some fixed point. Then both cells will approach this fixed point as they evolve along the left branch of Co in the silent phase. There will then be an infinite amount of compression, since both cells approach the same fixed point. It follows that we can assume that the compression is as large as we please by making g(VLK, WLK) sufficiently small. If the compression is sufficiently large, then it will easily dominate any possible expansion over the remainder of the cells' trajectories. This will, in turn, lead to stability of the synchronous solution. V3 REMARK 3.4. The mechanism by which one cell fires, and thereby raises the cubic of the other cell such that it also fires, was referred to as Fast Threshold Modulation (FTM) in [71 ]. There, a time metric was introduced to establish the compression of trajectories of excitatorily coupled cells, which implies the stability of the synchronous solution. A detailed discussion of the time metric can be found in [43]; see also [49]. REMARK 3.5. While the synchronous solution has been shown to be stable, it need not be globally stable. In [45], it is shown that this network may exhibit stable antiphase solutions if certain assumptions on the parameters and nonlinear functions are satisfied. We have so far considered a completely homogeneous network with just two cells. The analysis generalizes to larger inhomogeneous networks in a straightforward manner, if the degree of heterogeneity between the cells is not too large. The major difference in the analysis is that, with heterogeneity, the cells may lie on different branches of different cubics during the silent and active phases. The resulting solution cannot be perfectly synchronous; however, as demonstrated in [87], one can often expect synchrony in the jump-up, but not in the jump-down. Related work on heterogeneous networks include [72,55,8]. One may also consider, for example, an arbitrarily large network of identical oscillators with nearest neighbor coupling. We do not assume that the strength of coupling is homogeneous. Suppose that we begin the network with each cell in the silent phase. If the cells are identical, then they must all lie on the left branch of Co. Now if one cell jumps up it will excite its neighbors and raise their corresponding cubics. If the cells begin sufficiently close to each other, then these neighbors will jump up due to FTM. In a similar manner, the neighbor's neighbors will also jump due to FTM and so on until every cell jumps up. In this way, every cell jumps up at the same (slow) time. While in the active phase, the cells may receive different input and, therefore, lie on the right branches of different cubics. Once one of the cells jumps down, there is no guarantee that other cells will also jump down at this (slow) time, because the cells to which it is coupled may still receive input from other
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active cells. Hence, one cannot expect synchrony in the jumping down process. Eventually every cell must jump down. Note that there may be considerable expansion in the distance between the cells in the jumping down process. If ]g(VLK, WLK)] is sufficiently small, however, as in the previous result, then there will be enough compression in the silent phase so that the cells will still jump up together. Here we assumed that the cells are identical; however, the analysis easily follows if the heterogeneities among the cells are not too large.
3.5. Desynchrony with inhibitory synapses We now consider two mutually coupled cells with inhibitory synapses. Under this coupling, the curve CA now lies below Co. As before, we assume that gsyn is not too large, such that both Co and CA are cubic shaped. We also assume that the right knee of CA lies above the left knee of Co as shown in Figure 10. Some assumptions on the threshold 0syn are also required. For now, we assume that 0syn lies between the left knee of Co and right knee of CA. We will assume throughout this section that the synapses are fast, direct and inhibitory. The main results state that if a synchronous solution exists then it must be unstable. The network will typically exhibit either out-of-phase oscillations or a completely quiescent state and we give sufficient conditions for when either of these arises. We note that the network may exhibit bistability; both the out-of-phase and completely quiescent solutions may exist and be stable for the same parameter values. These results are all for singular solutions. Some rigorous results for e > 0 are given in [86]. The first result concerns the existence and stability of the synchronous solution.
W
g=0
CO
t
Vsyn
Fig. 10. Instability induced by mutual inhibition. Cell 2 jumps to CA when celll fires. The excitable case is shown, while the identical mechanism leads to the instability of the synchronous state in the oscillatory case.
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THEOREM 3.2. Assume that the synapses are fast, direct and inhibitory. If each cell without any coupling, is oscillatory and 0syn is sufficiently large, then there exists a singular synchronous solution. This solution is unstable. If each cell, without any coupling, is excitable, then there does not exist a singular synchronous solution. PROOF. The existence of a singular synchronous solution for oscillatory cells follows precisely as in the previous section. During the silent phase, the trajectory lies on the left branch of Co, while in the active phase it lies on the right branch of CA. Note that we require that the right knee of CA lies above the left knee of Co. Moreover, when the synchronous solution jumps up and crosses the threshold v = 0syn, it should lie to the right of the middle branch of CA; otherwise, it would fall down to the silent phase. This is why we assume that 0syn is sufficiently large. This solution is unstable for the following reason. Suppose both cells are initially very close to each other on Co. The cells then evolve on Co until one of the cells, say cell 1, reaches the left knee of Co. Cell 1 then jumps up to the active phase. When vl crosses the threshold 0syn, S1 switches from 0 to SA and cell 2 jumps from Co to CA, as shown in Figure 10. This demonstrates that the cells are uniformly separated for arbitrarily close initial data. The synchronous solution must, therefore, be unstable. The synchronous solution cannot exist if the cells are excitable for precisely the same reason discussed in the previous section. If such a solution did exist then each cell would lie on Co during its silent phase. Each cell would then approach the stable fixed point on this branch and would never be able to jump up to the active phase. D We next consider out-of-phase oscillatory behavior. One interesting feature of mutually coupled networks is that such oscillations can arise even if each cell is excitable for fixed levels of synaptic input. The following theorem gives sufficient conditions for when this occurs. We will require that the active phase of the oscillation is sufficiently long. To give precise conditions, we introduce the following notation. Assume that the left and right knees of Co are at (1)LK, tOLK) and (URK, WRK), respectively. If the w-nullcline intersects the left branch of CA, then we denote this point by (VA, WA) = PA- We assume that WA < WLK, as shown in Figure 10. Let "rL be the (slow) time it takes for the solution of (3.6) with ot = L and s = SA to go from w = WRK to W = WLK, and let "t'R be the time it takes for the solution of (3.6) with ot = R and s = 0 to g o f r o m w -- WLK to W ~ WRK. Note that "rL is related to the time a solution spends in the silent phase, while rR is related to the time a solution spends in the active phase. THEOREM 3.3. Assume that the cells are excitable for each fixed level of synaptic input and the synapses are fast, direct, and inhibitory. Moreover, assume that WA < WLK and rE < rR. Then the network exhibits stable out-of-phase oscillatory behavior. REMARK 3.6. We do not claim that the out-of-phase solution is uniquely determined or that it corresponds to antiphase behavior. These results may hold; however, their proofs require more analysis than that given here.
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REMARK 3.7. The rest state with each cell at the fixed point on Co also exists and is stable. Hence, if the hypotheses of Theorem 3.3 are satisfied, then the network exhibits bistability. PROOF. Suppose that we begin with cell 1 at the right knee of Co and cell 2 on the left branch of CA with WA < w2(0) < WLK. Then cell 1 jumps down and, when vl crosses the threshold 0syn, cell 2's cubic switches from CA to Co. Since w2(0) < wLK, cell 2 lies below the left knee of Co, so it must jump up to the active phase. After these jumps, cell 1 lies on the left branch of CA, while cell 2 lies on the right branch of Co. Cell 2 then moves up the right branch of Co while cell 1 moves down the left branch of CA, approaching PA. This continues until cell 2 reaches the right knee of Co and jumps down. We claim that at this time, cell 1 lies below the left knee of Co, so it must jump up. We can then keep repeating this argument to obtain the sustained out-of-phase oscillations. The reason why cell 1 lies below the left knee of Co when cell 2 jumps down is because it spends a sufficiently long amount of time in the silent phase. To estimate this time, note that because cell 2 was initially below the left knee of Co, the time it spends in the active phase before jumping down is greater than rR. Hence, the time cell 1 spends in the silent phase from the time it jumps down is greater than rR > rL. From the definitions, since cell 1 was initially at the right knee of Co, it follows that cell 1 must be below the left knee of Co when cell 2 jumps down, which is what we wished to show. [2 REMARK 3.8. To obtain sustained oscillations, it is not really necessary to assume that each cell is excitable for all levels of synaptic input. Suppose, for example, that the wnullcline intersects Co along its middle branch, but it intersects CA along its left branch. Then each cell, without any coupling, is oscillatory. The hypothesis rL < rR is no longer necessary for sustained out-of-phase oscillations. If rL > rR, then it is possible that both cells will lie in the silent phase on the left branch of Co at the same time. If there is no fixed point of this branch then the leading cell will be able to jump up. At this time, the trailing cell will approach the left branch of CA and remain there until the leading cell jumps down. The trailing cell may then either jump up, if it lies below the left knee of Co, or it may jump back to the left branch of Co. In this latter case, it will eventually jump up when it reaches the left knee of Co. REMARK 3.9. Wang and Rinzel [95] distinguish between "escape" and "release" in producing out-of-phase oscillations. In the proof of the preceding theorem, the silent cell can only jump up to the active phase once the active cell jumps down and releases the silent cell from inhibition. This is referred to as the release mechanism and is often referred to as postinhibitory rebound [24]. To describe the escape mechanism, suppose that each cell is oscillatory for fixed levels of synaptic input. Moreover, one cell is active and the other is inactive. The inactive cell will then be able to escape the silent phase from the left knee of its cubic, despite the inhibition it receives from the active cell. Note that when the silent cell jumps up, it inhibits the active cell. This lowers the cubic of the active cell, so it may be forced to jump down before reaching a right knee. These issues are also discussed in [43], where there is a detailed discussion of properties of the antiphase solutions, including the control of their frequencies.
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3.6. Synchrony with inhibitory synapses 3.6.1. Introduction. In the previous section, we showed that an inhibitory network cannot exhibit stable synchronous oscillations if the synapses are direct and fast. Synchronous oscillations in an inhibitory network have been observed experimentally, however [77,79, 98]. We now show that these oscillations are possible with slow and indirect synapses. Recall that the synapse is slow if ot = O(1) and fl - - O ( e ) with respect to e. We now assume that fl = e K where K does not depend on e. As before, we analyze the network by considering singular solutions. The first step in this analysis is to derive fast and slow equations. This is done in the next subsection. We then show that the synchronous solution may exist with either direct or indirect slow synapses and that this solution cannot be stable if the synapses are direct. In Section 3.6.5, we state the main result concerning the stability of the synchronous solution with indirect synapses. This result is proved in [85]. We then demonstrate that mutually coupled networks with slow inhibitory synapses may exhibit numerous other types of solutions besides the synchronous one. In fact, all of the solutions shown in Figure 8 are generated by this class of networks. 3.6.2. Fast and slow equations. We derive slow subsystems valid when the cells lie in either the silent or the active phase. There are several cases to consider and we only discuss two of these in detail. Here, we only consider direct synapses; the derivation of the slow equations for indirect synapses is very similar. If both cells are silent, then vi < 0syn and the first term in (3.3) is zero. Hence, after letting ~: = et and setting e = 0, (3.2) and (3.3) become 0--
f ( v i , W i ) -- S j g s y n ( V i --
lbi ~- g ( v i ,
l)syn),
tO/),
(3.8)
Si -- -- K s i ,
where j :~ i. This system can be simplified as follows. We write the left branch of C~ as v = q~L(W, s) and let GL(W, s) = g(q~L(W, s), s). Each (wi, sj) must then satisfy the system
&=GL(w,s),
(3.9)
~= -Ks.
These equations determine the evolution of the slow equations while in the silent phase. If both cells are active, then si is a fast variable. The only slow variables are the wi. Instead of (3.8), the slow equations are now 0 - ~ f (l)i, Wi) -- gsyn(1)i -1]Oi -- g(13i, Wi ), l=si.
Vsyn), (3.10)
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This can be reduced to a system for just the slow variables as before. Denote the right branch of C, by v = q~R(W, s) and let GR(w,s) = g(~R(W,S),S). Then each wi satisfies the scalar equation tb = GR(W, 1).
(3.1 1)
In a similar fashion, we can derive the slow subsystem for when one cell is active and the other is silent. For indirect synapses, there are further cases depending on whether the xi-variables have crossed their threshold 0~ or not. This is discussed in detail in [85]. 3.6.3. Existence of synchronous oscillations. The singular synchronous solution is easily constructed; here we consider the case of direct synapses. We begin with both cells at the right knee of the right branch of C j. From this point, the cells jump down to the silent phase. While in the silent phase, the slow variables evolve according to (3.9). The cells can only leave the silent phase once they reach a left knee of one of the left branches. If the cells are able to reach such a point, then they will jump up to the active phase and return to the starting point. Hence, the existence of the synchronous solution depends on whether the cells can reach one of the jump-up points while in the silent phase. If the cells are oscillatory, then the synchronous trajectory must reach one of the jumpup points. This is demonstrated in [85], where it is also shown that a synchronous solution can exist even though both cells, without any coupling, are excitable. This will be the case if the rate K of decay of inhibition is small enough and the cells are oscillatory for some fixed values of s E (0, 1); if the cells are excitable for all s c [0, 1], then the only stable solution is the quiescent resting state. Exit from the silent phase is not possible if K is too large, since then the inhibition decays quickly and the system behaves in the slow regime like the uncoupled excitable system with s -- 0. The construction of the synchronous solution for the case of indirect synapses is very similar. There are some additional complications due to the additional slow variables xi. The complete analysis is given in [85]. 3.6.4. Instability of the synchronous solution for direct synapses. The synchronous solution is not stable when the synapses are direct for precisely the same reasons described in Section 3.5. This holds because with direct synapses, when one cell jumps, the other cell begins to feel inhibition as soon as the first cell's membrane potential crosses threshold. This instantly moves the second cell away from its threshold by an amount that stays bounded away from zero no matter how close to the first cell the second cell starts. Thus, infinitesimally small perturbations are magnified, at this stage of the dynamics, to finite size, and the synchronous solution cannot be stable. 3.6.5. Statement of the main result. We now consider indirect synapses and show that the synchronous solution can be stable in some parameter ranges. We shall show that there are two combinations of parameters that govern the stability. Furthermore, only one of those two combinations controls stability in any one parameter regime.
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For this result, it is necessary to make some further assumptions on the nonlinearities and parameters in (3.2) and (3.4). It will be necessary to assume that fw < 0 ,
gv > O
and
gu, < O
(3.12)
near the v-nullcline. For Theorem 3.4 below, we also assume that f (v, w) is given by f (v, w) = f l (v) - gcw(v - VR),
where gc > 0 and VR ~< Vsyn represent a maximal conductance and reversal potential, respectively. This holds for the well-known Morris-Lecar equations [51]. The analytical framework we develop, however, also applies to more general nonlinearities which satisfy (3.12). Some technical assumptions are also required on the nonlinear function g(v, w). We need to assume that gv is not too large near the right branches of the cubics Cs, for example. We assume that the parameters O~x and fl~ are sufficiently large, and c~.r/(C~x +/3x) > Ox. This guarantees that each xi can cross its threshold in order to turn on the inhibition. Precise conditions on how large Oex and fix must be are given in [85]. We also need to introduce some notation. Let a_ be defined as the minimum of - O g / O w over the synchronous solution in the silent phase. Note from (3.12) that a_ > 0. Let (w*, s*) = (WE(S*), s*) be the point where the synchronous solution meets the jump-up ' (s*) be the reciprocal slope of the jump-up curve at this point in curve, and let )~ = w E (w, s)-space. Finally, let a+ denote the value of g(v, w) evaluated on the right hand branch of Co at the point where the synchronous solution jumps up. The main result is then the following. THEOREM 3.4. Assume that the nonlinear functions and parameters in (3.2) and (3.4) satisfy the assumptions stated above. I f fl = e K with K < a_ and Ks* < a+/I)~l, then the synchronous solution is asymptotically stable. REMARK 3.10. The first condition in Theorem 3.4 is consistent with the numerical simulations of [95], who obtained synchronized solutions when the synapses recovered at a rate slower than the rate at which the neurons recovered in their refractory period. REMARK 3.1 1. To interpret the second condition in Theorem 3.4, note that Ks* is the rate of change of s at the point at which the synchronous solution jumps, while a+ is the rate of change of w on the fight hand branch right after the jump. Since )~ = d w L / d s , multiplication by I)~l transforms changes in s to changes to w. Thus, the second condition is analogous to the compression condition that produces synchrony between relaxation oscillators coupled by fast excitation as described in Theorem 3.1 of Section 3.4. I~,l may be thought of as giving a relationship between the time constants of inhibitory decay and recovery of the individual cells; a larger I)~l (corresponding to a flatter jump-up curve) means that a fixed increment of decay of inhibition (As) has a larger effect on the amount of recovery that a cell must undergo before reaching its (inhibition-dependent) threshold for activation.
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REMARK 3.12. The two conditions given in the statement of Theorem 3.4 correspond to two separate cases considered in the proof of Theorem 3.4. These two cases correspond to whether the two cells preserve their orientation (case 1) or reverse their orientation (case 2) on the right branch of the s -- 1 cubic after one cycle. Theorem 3.4 says that, whatever case the synchronous trajectory falls into, if both conditions hold, then the synchronous solution is stable. Note, however, that the different cases require different conditions. Case 1 requires K < a_ and case 2 requires Ks* < a+/l,kl. Thus, by changing a parameter, such as gsyn, that switches the system between cases 1 and 2, one can change which combinations of time scales and other parameters control the stability of the synchronous solution. In particular, stability of synchronous solutions can be lost or gained without changing any time constants. See [85] for details. REMARK 3.13. In this section, we have considered rather simple models for each cell; in particular, each cell contains a single channel state variable w and there is only one intrinsic slow process. In this case, the synchronous solution can be stable only if the synaptic variable decays on the slow time scale. This is true even if the cells are oscillatory [85,64]. In fact, if the cells are excitable, then the slow synaptic variable is required even for the existence of a synchronous solution; it allows the cells to escape from the silent phase. In [63-65], more complicated models for the cells are considered. It is shown that mutually coupled networks with more complex cells can give rise to stable synchronous solutions even if the cells are excitable for all s E [0, 1] and the synapses are fast; the synapses must still be indirect, however. The main conclusion of the analysis in [63] is that what is needed for the existence of stable synchronous solutions is the presence of two slow variables; one of these slow variables may correspond to an intrinsic process and the other to a synaptic process, or both slow variables may correspond to intrinsic processes. REMARK 3.14. The models discussed here represent bursting neurons. Synchronization of spiking neurons, namely neurons with very short active phases, that are connected by inhibition is considered in [90,25,6,43]. We have also assumed that cells and coupling are homogeneous. The effect of heterogeneities on cells coupled with inhibition is analyzed in [26,97,12].
3.6.6. Nonsynchronous solutions. The network of two mutually inhibitory cells can display other behaviors. We will not give rigorous conditions for the existence and stability of these other solutions; instead, we give simulations of the other solutions and a general description of the parameter ranges in which they are expected. The heuristic explanations we give are based on the techniques developed in the previous section. For all of the examples, we consider direct synapses, although the analysis for indirect synapses is very similar. We start with the antiphase solution. Such a solution is shown in Figure 8A. The antiphase solution is the most well-known solution for a pair of mutually inhibitory oscillators, expected when the inhibition decays at a rate faster than the recovery of the oscillator (K/a_ large); see, for example, [94,68]. Though K/a_ small favors stability of the synchronous solution and K/a_ large favors the existence of a stable antiphase solution, there is a parameter range in which both solutions are stable.
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s = 1
C 2(0) = C 12 2 ) _..,,
~(~1 )
CELL 2 t CELL I
'
'
'
' ""lb~
. . . . . . . .
C l ( Z 1)
c1(0)
I
Fig. 11. The projection of an antiphase solution onto (w, s) [85]. One-half of a complete cycle is shown; during this time, cell 1 remains silent (solid curve). The dashed curve shows the evolution of cell 2 in the silent phase, while the dotted curve shows its evolution in the active phase; cell 2 jumps up at time rl and jumps down at time r2.
One can describe the evolution of the antiphase solution in phase space in a way that is similar to the description of the synchronous solution given earlier. In Figure 11, we illustrate the projection of an antiphase solution onto the (w, s) plane. We choose the initial (slow) time so that both cells lie in the silent phase after cell 1 has just jumped down from the active phase. This implies that the inhibition sl felt by cell 2 satisfies Sl (0) = 1. Both cells then evolve in the silent phase until cell 2 reaches the jump-up curve, say at time r -rl. At this time, the inhibition s2 felt by cell 1 jumps up to the line s -- 1. Cell 2 then evolves in the active phase; we illustrate the projection of cell 2's trajectory during the active phase with a dotted curve in Figure 11. Note that sl (r) still satisfies kl = --Ksl; hence, it keeps decreasing while cell 2 is active. During this time, cell 1 lies in the silent phase with s2 -- 1. This continues until cell 2 reaches the jump-down curve WR(s). We denote this time as r2. Cell 2 then jumps down and this completes one-half of a complete cycle. For this to be an antiphase solution, we must have that wl (r2) = w2(0) and sj (r2) = s2(0). Rigorous results related to the existence and stability of antiphase solutions for systems with slow inhibitory coupling are given in [86,62]. In the introduction to this section, we referred to another kind of nonsynchronous solution obtained in this system as a suppressed solution; an example is shown in Figure 8B. In such rhythms, one cell remains quiet while the other oscillates. They occur in the same parameter range as the stable synchronous solutions, i.e., K / a _ small. The behavior of these solutions is easy to understand: if the inhibition decays slowly enough, the leading cell can recover and burst again before the inhibition from its previous burst wears off enough to allow the other cell to fire. This type of solution cannot exist if the cells are excitable rather than oscillatory, since there is no input from the quiet cell to drive the active one. On the other hand, suppressed solutions only arise if the cells are excitable for some fixed levels of inhibition; i.e., some s ~ (0, 1]. If this is not the case, then the w-coordinate of the
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125
suppressed cell must keep decreasing until that cell eventually reaches the jump-up curve and fires. If the synaptic inhibition decays at a rate comparable to the recovery of the cell, complex hybrid solutions can occur, in which one cell is suppressed for several cycles, while the other fires, and then fires while the other is suppressed. An example is shown in Figure 8C. In this example, each cell is excitable when uncoupled but is oscillatory for some intermediate levels of inhibition. Hence, if K / a _ is sufficiently small, then a cell can fire a number of times while the other cell is suppressed. The inhibition of the firing cell must eventually wear off, such that that cell can no longer fire. This then allows the inhibition of the suppressed cell to wear off to the level from which it can fire. The roles of the two cells are then reversed. The synchronous solution exists stably in parameter regimes in which one or more of the above nonsynchronous solutions is also stable. Thus the choice of solution depends on the initial condition. The basin of attraction of the synchronous solution depends mainly on the delay of the onset of the inhibition. For a trajectory to be in the domain of attraction of the synchronous solution, the lagging cell must be activated before the inhibition from the leading cell suppresses it. As the onset time of inhibition decreases, the domain of stability of the synchronous state vanishes, but the nonsynchronous solutions remain.
3.7. Desynchrony with excitatory synapses We now briefly discuss results in [5] which demonstrate that excitatory coupling can lead to almost synchronous solutions if the active phase of a single cell is much shorter than the silent phase; that is, the cells here correspond to spiking neurons. This work is motivated by simulations in [55] which showed that there can be a stable state close to synchrony when the fully synchronized state is unstable. Here we write the equations for a single oscillator as
v'--f(v,w),
(3.13)
w' = eg(v, w)/r~c(V), where the function r ~ ( v ) is given by
vo, ifv>~v0.
1
r~c(v)-
if v <
e/y
The parameter vo is the threshold for entering the active phase and }, governs the rate of passage through the active phase. We assume that vo lies between the two knees of the cubic f = 0, which we again denote by Co. The equations for two mutually coupled oscillators in this model are v;
-- f (vi, wi)
-
gsynH(vj
to; -- 6g(vi, wi)/rvc(vi),
-
Vst)(vi
-
Vsyn),
(3.14)
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J.E. Rubin and D. Terman
where j =/=i. Here gsyn is the conductance of the synaptic current and vst is the synaptic threshold. The synaptic reversal potential Vsyn is chosen to be high so that the synapse is excitatory; that is, vi Vsyn < 0 along the singular solutions. The v-nullcline is the same as that of the uncoupled cell when vj < vst. For vj > Vst, the effect of the coupling term is to raise the nullcline and change its shape; if gsyn is not too large, then the v-nullcline is still qualitatively cubic. We denote this cubic by Cl. Note that there are two thresholds vo and Vst in the equations. To simplify the discussion, we assume here that Vst = vo. As in previous sections, we analyze the solutions by treating e as a small, singular perturbation parameter. We construct singular solutions after formally setting e -- 0. An interesting feature of the model considered here is that the structure of the corresponding flows is quite different depending on whether cells are silent or active. We first demonstrate how to construct a singular periodic orbit for a single uncoupled cell. When e is set equal to 0 in (3.13), we obtain the equations for the fast flow -
v' = f (v, w), ,
w --0
(3.15)
if v < vo and v' = f (v, w), w tm vg(v,w)
(3.16)
if v >~ vo. By introducing the rescaling r = et into (3.13) and then setting e -- 0, we obtain equations for the slow flow in the silent phase, 0 = f (v, w),
(3.17)
= g(VL(W), w)
if v < vo, where I)L(W) is obtained by solving 0 = f ( v , w) along the left branch of the cubic Co. Note that Equations (3.15) and (3.17) are simply scalar equations; in (3.15), the variable w serves as a parameter in the v' equation. The full two-dimensional system (3.13) has been reduced to two one-dimensional equations and solutions of these are easy to characterize. However, (3.16) is not reduced. In other fast-slow systems of the form (3.13), typically the entire v-nullcline consists of rest points for the fast flow. In this case, however, only the portion of Co with v < vo consists of rest points. The singular periodic orbit for a single uncoupled cell is constructed as follows. We begin the orbit with the cell at the left knee of Co, which we denote by (VLK, WLK). The first part of the singular orbit is a solution of (3.15) which connects (VLK, WLK) and (vo, WLK). The second part is a solution of (3.16) which connects (vo, WLK) to (vo, Wh), where Wh > WLK; this corresponds to the active phase. The third part is a solution to (3.15) which connects (vo, Wh) back to Co at some point (VL(Wh), Wh). The fourth and final part is a solution to (3.17) which connects (VL(Wh), Wh) to (I)LK,WLK). The construction of a synchronous solution for the coupled system is done in the same manner as that of the periodic solution for the uncoupled cell; the only difference is that the
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Geometric singular perturbation analysis o f neuronal dynamics
dynamics are changed as the voltage passes across vst -- vo. Thus, if each (3.16) is replaced by v; -- f (v, w ) - gsyn(V - Vsyn), !
Vi > ldst,
then
(3.18)
w --vg(v,w).
The synchronous solution is not the same as the uncoupled periodic solution, since they satisfy different equations while in the active phase. There is no difficulty in proving that these singular periodic solutions perturb, for s > 0 small, to actual solutions of either (3.13) or (3.14). See, for example, [50]. The actual periodic orbits for s small lie O(s) close to the singular orbits, except near the left knee of Co, where the distance is O(s 1/2). To construct the almost-synchronous solutions, we again work with singular solutions. There are now more cases to consider, depending on which of the cells is silent or active. If both cells are silent (that is, each vi < vo), then the fast flow corresponding to each cell is given by (3.15), while the slow flow is given by (3.17). If both cells are active, then the fast flow for each cell is (3.18); there is no slow flow. Finally, suppose that one of the cells, say cell 1, is active and cell 2 is silent. Then the fast flow for cell 1 is (3.16) and the fast flow for cell 2 is v' -
f (v, w )
w' -- 0.
-
gsyn(V - Vsyn),
(3.19)
There is no slow flow for this case since all cell 1 dynamics in the active phase occurs on the fast time scale. We now give a heuristic argument to explain why there may exist a stable almost synchronous solution. We describe the construction of a singular solution of (3.14) in which one of the cells, say cell 1, begins at the left knee of Co and cell 2 lies on the left branch of Co just above cell 1. We follow the cells around in phase space until one of the cells returns to the left knee. If the other cell returns to the initial position of cell 2, then this will correspond to an almost synchronous solution. As in the previous constructions, the singular solution consists of several pieces. The first piece starts as cell 1 leaves the left knee; it satisfies (3.15), moving horizontally in (v, w)-space, until it crosses vo. Suppose that this occurs when t - tl. At this time, H ( v l -- vo) switches from 0 to 1 and cell 2 then satisfies (3.19). If cell 2 initially lies below the left knee of Cl, then it will jump up, continuing to satisfy (3.19) until v2 crosses vo. Suppose that this occurs when t = t2. For tl < t < t2, cell 1 satisfies (3.16). Note that its trajectory is no longer horizontal. We assume in this heuristic argument that vl > vo for tl < t < t2. When t - t2, H ( v 2 - vo) switches from 0 to 1 and both cells then satisfy (3.18). Assume that cell 1 is the first cell to cross vo again and this occurs when t - Tl. Then cell 2 satisfies (3.16) until it crosses vo, say when t = T2. Once a cell crosses vo, it moves horizontally to a point on the left branch of Co. The cells then evolve along the left branch of Co until one of the cells returns to the left knee and this completes the cycle. We note that it is possible for either of the cells to be the first to reach the left knee.
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The singular flow naturally gives rise to a one-dimensional map, which we denote by 17. More precisely, if (v2, w2) is the initial position of cell 2 on the left branch of Co and @, ~) is the position of the trailing cell after the other cell has returned to the left knee, then 17(1/32) - - ~ . This map is well defined if tO2 - - tOLK is sufficiently small; in particular, this requires that cell 2 initially lies below the left knee of the excited cubic C l. A fixed point of H corresponds to a periodic singular almost-synchronous solution. The orbit may be orientation preserving or orientation reversing, depending on whether the first cell to reach the left knee of Co is cell 1 or cell 2, respectively. This periodic solution is referred to as an O(e)-synchronous solution for the following reason. The analysis demonstrates that if w, is the fixed point of 17, then this must lie below the left knee of the excited cubic C1. Hence, if cell 1 is the first to jump up, then cell 2 will jump up as soon as cell 1 crosses vo. The difference between the times the cells reach vo is therefore O(1) in t-time. Since the total period in t-time is O(1/e), the normalized time difference (Atime/period) is O(e). We prove the existence of an asymptotically stable fixed point of 17 by showing that H defines a uniform contraction on some interval. We find A l < A2 such that 17 maps the interval [Al, A2] into itself. In particular, H(A1) > Al so expansion between cells takes place if they are initially close to each other. This leads to instability of the synchronous solution. The reason why this expansion takes place is because the cells satisfy different equations during their initial times in the active phase. Cell 1 does not feel excitation from cell 2 when it first enters the active phase, so it satisfies (3.16) there. Cell 2, on the other hand, does feel excitation from cell 1 when it enters the active phase, so it satisfies (3.18). Note that in Section 3.4, we proved that the synchronous solution is stable for excitatory synapses if there is slow dynamics in both the silent and active phase. The main difference between the analysis for that system and the one considered here arises during the jumping up process. In Section 3.4, the jump-ups are horizontal and it does not matter that different cells satisfy different fast equations during portions of the jump-ups. Here the jump-ups are not horizontal, so expansion of trajectories can arise. We conclude by stating two theorems proved in [5]. We will need the following notation. Assume that the right knee of C1 is at (I)RK, tORK) and choose fl so that g(fl,//)RK) = 0. We say that constants 61 and 62, which depend on V, are O(v) apart if there exist K I and K2 such that K19/ < 161 -- 62 ] < K2)I for V small. THEOREM 3.5. Suppose that the constants vo and fl satisfy one of the following conditions: (R) Either vo > fl, or vo < ~ and fl - vo is sufficiently small. (P) vo < fl and gsyn is sufficiently small. Then, for g sufficiently small, there exist A1 < A2 such that H defines a uniform contraction from the interval [A1, A2] into itself If w, is the resulting asymptotically stable fixed point of 17, then w, and WLK are O(y) apart. If (P) is satisfied, then 17 is orientation preserving on (A1, A2), while if (R) is satisfied, then 17 is orientation reversing on (AI, A2).
THEOREM 3.6. The asymptotically stable singularperiodic solution given by Theorem 3.5 perturbs, for e > O, to an asymptotically stable O(e)-solution of (3.14).
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4. Globally inhibitory networks 4.1. Introduction We now consider the network illustrated in Figure 12. This is composed of a population of excitatory (E-)cells and a single inhibitory (J-)cell. Each E-cell sends excitation to the J-cell, while the J-cell sends inhibition back to every E-cell. We assume that all of the E-cells are identical, but they may differ from the J-cell. This network, analyzed in [63, 62] is motivated by models for thalamic oscillations involved in sleep rhythms. In those networks, there may be a population of J-cells with inhibitory coupling among the Jcells. This application is explored in Section 5. This type of network was also introduced in [87,91 ] as a model for scene segmentation. We distinguish two types of rhythms in which the network may engage. In a synchronous oscillation, the E-cells are completely synchronized. When the E-cells fire, or become active, they excite the J-cell to fire in response. However, the J-cell is not necessarily synchronized with the E-cells throughout the entire oscillation; this is because the J-cell need not have the same intrinsic properties as the E-cells. Alternately, in a clustered oscillation, the E-cells form subpopulations or clusters; the cells within a cluster fire synchronously but cells from distinct clusters act out of synchrony from each other. The J-cell will be induced to fire with each cluster. Two types of solutions for a network with just four E-cells are shown in Figure 13. Figure 13A shows a synchronous solution, while a 2-cluster oscillation is shown in Figure 13B. Note that in each of these figures, the E-cells within a cluster are perfectly synchronized; moreover, the J-cell jumps up together with each cluster. To obtain the different cluster states, as well as a 3-cluster state (not shown), we adjusted a parameter in the equations for the J-cell; this parameter controls the duration of the J-cell's active phase, which, in turn, controls the amount of inhibition sent back to the E-cells. The analysis that follows will clarify why the level of inhibition produced by the J-cell is an important factor in determining the network behavior of the E-cells. We model the globally inhibitory network as follows (see [63,62]). Here we assume that all of the synapses are direct. Each Ei satisfies equations of the form v;
11)i'
s~
f (vi, wi) -- eg(vi, Wi),
--
-
-
or(1
-
ginhSJ(Vi
si)H(vi
--
Vinh), (4.1)
-O)
- flsi,
Inhibition
0
0 0
Excitation
J-cell
0
0 E-cells
Fig. 12. Globally inhibitory network. The J-cell inhibits the E-cells, which excite the J-cell.
J.E. Rubin and D. Terman
130
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-20
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\
I
I
.......
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-30
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\
\
\
-50
\
\
\
_ \
-60
_
-70
_
-80
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20
40
60
80
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120
140
160
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19
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-70 -80
0
I
i
I
i
I
i
I
J
I
20
40
60
80
1O0 t
120
140
160
180
200
Fig. 13. Solutions for a globally inhibitory network of four E-cells and one J-cell. A. (top) A synchronous solution; the solid curve is the time course of E-cell voltage and the dashed curve is the time course of J-cell voltage. B. (bottom) A 2-cluster solution; the solid and dotted curves are the voltage time courses of two different E-cell clusters of two cells each. The dashed curve is the time course of J-cell voltage. Note that the J-cell fires less powerful, shorter bursts in B than in A.
while the J-cell satisfies the equations , Vj-
1 f J ( V J , Wj) -- ~
S i g e x c ( V J - Vexc),
Z i
(4.2)
tOj - - 6 g J ( l ) J , WJ), S~ - - OtJ(1 -- sJ)H(uJ - - O J ) -- 6 g j s j .
Unlike the previous section, we assume f > 0 ( f < 0) above (below) the v-nullcline for sj = 0 and g > 0 (g < 0) below (above) the w-nullcline. The nullclines are illustrated in Figure 14. Note that the nullclines are "upside-down" relative to the figures in Section 3. This orientation is motivated by the biological model in Section 5. The variable sj denotes the inhibitory synaptic input from the J-cell, while si denotes the excitatory synaptic input from cell Ei to the J-cell. The sum in (4.2) is taken over
Geometric singular perturbation analysis of neuronal dynamics
131
A) PL
/
/
sj=l
sj - 0
Ei
W
B) ~
~
Sto t -- 0 -..-..
/
J
./ Stot-- SA
Qo Q2 1
wj
Fig. 14. Nullclines for (A) E-cells and (B) J-cells in a globally inhibitory network [63]. The heavy lines and points Pi, Qi correspond to the singular synchronous solution discussed in the text. Note that sj decays on the slow time scale.
all N Ei-cells. Note that turn on of inhibition and excitation both occur on the fast time scale, while the inhibitory variable sj turns off on the slow time scale. This turn off may be 'fast' or 'slow', depending on whether Kj is large or small. We assume that/3 = O(1), representing fast turn off of excitation, although there is no problem extending the analysis if/3 = O(s). Note that if 1)i > 0, then Si ---> S A -m- Ol/(Ol -+- fl) on the fast time scale. Each synapse in (4.1) and (4.2) is direct. If the inhibition is indirect, then sj satisfies system (3.4), with the appropriate adjustment of subscripts and a new variable xj included, instead of the equation given in (4.2). Indirect synapses will be necessary to obtain stability of synchronous and clustered solutions to (4.1) and (4.2). As in previous section, we analyze this network by constructing singular solutions. The trajectory for each cell lies on the left (right) branch of a cubic nullcline during the silent (active) phase. Which cubic a cell inhabits depends on the total synaptic input that the cell receives. Nullclines for the Ei are shown in Figure 14A and those for J in Figure 14B. Note in Figure 14A that the sj -- 1 nullcline lies above the sj = 0 nullcline, while in Figure 14B, the Stot ~ ~1 Z Si -- SA nullcline lies below the Stot - - 0 nullcline These relations hold because the Ei receive inhibition from J while J receives excitation from the Ei. We
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assume that each cell is excitable for fixed levels of synaptic input. In the next section, we give sufficient conditions, first presented in [63], for when a globally inhibitory network exhibits a stable synchronous solution. Clustered oscillations are considered in Section 4.3, which follows [62]. Analyzing the stability of clustered solutions entails consideration of two issues, namely what prevents separate clusters from firing together and what maintains the synchrony of the cells belonging to the same cluster. Note that these issues may become especially subtle in cases for which a clustered solution is stable but the completely synchronous state is unstable.
4.2. Synchronous solution We now give sufficient conditions for the existence of a singular synchronous periodic solution and for its stability. For existence, we assume that each synapse is direct; there is no problem in extending the analysis to indirect synapses. Indirect synapses are needed for the stability of this solution. To state the main result, it is necessary to introduce some notation. Let Cs denote the cubic f ( v , w) - g i n h S ( V - - Vinh) = 0. Since the E-cells are excitable for fixed levels of the inhibitory input sj, there is a fixed point on the left branch of Cs of the first two equations in (4.1) with s = sj held constant. We denote this fixed point by (VF(Sj), WF(SJ)) and the left knee of Cs by (VL(Sj), WL(Sj)). THEOREM 4.1. A singular synchronous periodic solution exists if WF(1) > WE(0), Kj is sufficiently large, the active phase of the J-cell is sufficiently long, and the recovery of the J-cell in the silent phase is sufficiently fast. If the inhibitory synapse sj is indirect and the active phase of the J-cell is long enough, then the synchronous solution is stable. REMARK 4.1. The condition WF(1) > I/)L ( 0 ) simply states that the fixed point on the left branch of C l lies above the left knee of Co. This allows the possibility of the E-cells firing upon being released from inhibition from the J-cell. REMARK 4.2. Recall that K j corresponds to the rate of decay of the inhibition. We will see that synchronous oscillations cannot exist if this synaptic decay rate is too slow, given the fact that each E-cell is excitable for fixed levels of synaptic input. PROOF. We prove the existence by demonstrating how to construct the singular synchronous solution if the hypotheses of Theorem 4.1 are satisfied. We assume throughout this construction that the positions of the E-cells are identical. The singular trajectory is shown in Figure 14. We begin with each cell in the active phase just after it has jumped up. These are the points labeled P0 and Q0 in Figure 14. Then each Ei evolves down the right branch of the sj = 1 cubic, while J evolves down the right branch of the Stot - - SA cubic. We assume that the J-cell active phase is long, such that the Ei have a shorter active phase than J; thus, each Ei reaches the right knee PI and jumps down to the point P2 before J jumps down. The assumption of a long J-cell active phase implies that at this time, J lies above the right knee of the Stot = 0 cubic; otherwise, J would jump down as soon as the E i did. J
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must therefore jump from the point Q I to the point Q2 along the Stot - - 0 cubic when the Ei jump down. On the next piece of the solution, J moves down the right branch of the Stot = 0 cubic while the Ei m o v e up the left branch of the s~ = 1 cubic. When J reaches the right knee Q3 it jumps down to the point Q 4 o n the left branch of the Stot = 0 cubic. Now the inhibition sj to the Ei starts to turn off on the slow time scale; that is, k~ = - K j s j f o r ' = - d / d r , r -- et. Thus, the Ei do not jump immediately to another cubic. Instead, the trajectory for the Ei m o v e s upwards, with increasing wi, until it crosses the w nullcline. Then each wi starts to decrease. If the Ei are able to reach a left knee, then they jump up to the active phase and this completes one cycle of the synchronous solution. When the Ei jump up, J also jumps up if it lies above the left knee of the Stot = SA cubic; that is, the recovery of the J-cell in its silent phase must be sufficiently fast. Existence of the synchronous solution requires that the E-cells can reach the jump-up curve and escape from the silent phase. We demonstrate that this is indeed the case if the assumptions of Theorem 4.1 are satisfied. It will be convenient to first introduce some notation. This will allow us to obtain simple estimates for when a synchronous oscillation exists and what the period of the oscillation is. This notation will also be useful in the next section. As in earlier sections, we derive equations for the evolution of the E-cells' slow variables; these are (wi, sj). Let v = q~L(W, s) denote the left branch of the cubic f ( v , w) ginhS(V -- Vinh) = 0, and let GL(w, s) = g(q~L(W, s), s). Then each (wi, sj) satisfies the slow equations 6o =
GL(w,
kj -- - K j s j .
sj), (4.3)
Figure 15 illustrates the phase plane corresponding to this system. Recall that w = WE(sj) denotes the jump-up curve or curve of left knees, and the second curve, which is denoted by WF(Sj), consists of the fixed points of the first two equations in (4.1) with the input sj held constant. This corresponds to the w-nullcline of (4.3). We need to determine when a solution (w(r), sj(r)) of (4.3), beginning with sj(0) = 1 and w(0) < WF(1), can reach the jump-up curve WL(Sj). This is clearly impossible if WF(1) < WE(0). If WF(1) > WE(0) and w(0) > WE(0), with Kj sufficiently large, then the solution will certainly reach the jump-up curve; this holds because the solution will be nearly vertical, as shown in Figure 15. If, on the other hand, Kj is too small, then the solution will never be able to reach the jump-up curve. Instead, the solution will slowly approach the curve WF(Sj) and lie very close to this curve as sj approaches zero. This is also shown in Figure 15. We conclude that the cells are able to escape the silent phase if the inhibitory synapses turn off sufficiently quickly and the w-values of the cells are sufficiently large when this turn off begins. Escape is not possible for very slowly deactivating synapses (although it would be possible with slow deactivation if the cells were oscillatory for some levels of synaptic input). We assume that Kj is large enough so that escape is possible. Choose West so that the solution of (4.3) that begins with sj (0) = 1 will be able to reach the jump-up curve if and only if w(0) >Wesc. The existence of the singular synchronous solution now depends on whether the Ei lie in the region where wi > West when J jumps down to the silent phase.
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,
W
Wv(1)
+ ~,.~
9
./j/.. ~
Sj
l
;~i,"
~
-w, K j large
,,;.?" _.,*::>'" t
wv(O)
v
K j small
we(O)
W Fig. 15. The slow phase plane for an E-cell [63]. The curve WL(Sj)is the jump-up curve, which trajectories reach if Kj is large enough. The dotted curve WF(Sj)consists of zeros of GL(w, sj) in system (4.3); trajectories tend to WF(0) as sj ~ 0 for smallKj. Note that wt < 0 for w > WF.
This will be the case if the active phase of J is sufficiently long. One can give a simple estimate on how long this active phase must be as follows. Let rE (rj) denote the duration of the E-cell (J-cell) active phase. Further, let w + denote the value of w at the right knee of the sj = 1 cubic (see Figure 15). If gsyn is not too large, then w + < WL(0). Finally, let rest denote the time for wi to increase from w + to West under tb - GL(W, 1). Since the Ei spend time rj - rE in the silent phase before they are released from inhibition, the singular synchronous solution exists if resc < rj - rE. For the stability of the synchronous solution, we need to assume that sj corresponds to an indirect synapse, as in Section 3.6. Suppose we slightly perturb the synchronous solution. If sj is direct, then when one E-cell fires, it will excite the J-cell which will, in turn, inhibit the other E-cells on the fast time scale. This will prevent the other E-cells from firing and desynchrony will result. The compression of trajectories if sj is indirect and the J active phase is long enough follows because all the E-cells approach exponentially close to the point wF(1) in the silent phase while the J-cell is still active. This exponential compression easily dominates any possible expansion over the remainder of the cells' trajectories. The domain of attraction of the synchronous solution depends on the ability of the Ecells to pass through the 'window of opportunity' provided by the indirect synapse. The size of this domain grows as the J-cell's active phase increases: More powerful J-cell bursts provide increased inhibition to the E-cells and this further compresses the E-cells near the point wF(1) in the silent phase. We note that this source of compression in globally inhibitory networks is considerably more powerful than any compression mechanism in mutually coupled inhibitory networks. D
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REMARK 4.3. The analysis leads to simple formulas for the period of the synchronous solution. Let r~ be, as above, the time cell J spends in the active phase and let rs be the time for the E-cells to reach the jump up curve after the J-cell jumps down. Then the period of the synchronous solution is simply rj + rs. Now rj is determined by the dynamics of the J-cell, while rs is primarily controlled by the rate at which the synapses turn off; this is the parameter Kj in (4.3). Other parameters play a secondary role. Note, for example, that the parameter gsyn mildly influences the period by controlling the slope of the jump up curve. REMARK 4.4. The domain of attraction of the synchronous solution increases with Kj, since a large Kj yields rapid decay of sj. If Kj is too large, however, then the J-cells actually cannot recover in time to respond to the excitation from the firing E-cells. Hence, this analysis shows that the combination of fast J-cell recovery and a large Kj promotes stable synchronization. REMARK 4.5. An important difference between mutually coupled and globally inhibitory networks arises in the way they use inhibition to synchronize oscillations. In mutually coupled networks with a slow decay of inhibition, the slow decay allows the cells to escape from the silent phase and to come together as they evolve in phase space. In globally inhibitory networks, the J dynamics controls synchronization, and a decay of inhibition on the slow time scale is only needed to allow J-cell recovery.
4.3. Clustered solutions 4.3.1. Singular orbit. Here we describe the singular trajectory corresponding to a 2cluster solution. The number of cells in the network may be arbitrary, but we assume for ease of notation that the clusters have equal numbers of cells. The geometric construction will require certain assumptions on the equations and a precise theorem is stated and proved in the following subsections. As we shall see, the construction of a 2-cluster solution easily generalizes to an arbitrary number of clusters. For the geometric construction of a singular 2-cluster solution, it suffices to consider only one-half of a complete cycle. During this half-cycle, one cluster, call it El, fires, say at r = 0, and evolves to the initial position of the other cluster; the non-firing cluster, call it E2, evolves in the silent phase to the initial position of El. By symmetry, the solution then continues with the roles of the clusters reversed. When El jumps up, it forces J to jump up to the right branch of the Stot - - I S A cubic. Then El moves down the right branch of the sj = 1 cubic, while J moves down the right branch of the S t o t - I SA cubic and E 2 m o v e s up the left branch of the sj -- 1 cubic. We assume, as before, that the E-cells in E I have shorter active phases than the J-cell, so Ei jumps down before J does. The assumption that J has a longer active phase than El implies that it lies above the right knee of the Stot = 0 cubic at this time, so it moves down the right branch of the Stot = 0 cubic until it reaches the right knee and then jumps down. During the time that J remains active, both El and E2 move up the left branch of the sj = 1 cubic.
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After J jumps down, sj(r) slowly decreases. If E2 is able to reach the jump-up curve, then it fires and this completes the first half cycle of the singular solution. Suppose that r -- rF when this occurs. By abuse of notation, let wi denote the w-value of all cells in cluster El. For the trajectories described above to represent one-half of a 2-cluster solution, we need that w2(rF) = Wl (0), Wl (rF) = W2(0), and WJ(I'F) - - W J ( 0 ) . The analysis in Section 4.3.3 shows that a 2-cluster solution will exist, with stability between clusters, if the active phase of J is not too long or too short, compared with the active phase of the El. If J ' s active phase is too long, then the network exhibits synchronous behavior as described before. If J ' s active phase is too short, then the system approaches the stable quiescent state. To conclude that the 2-cluster state is stable, we must also consider stability within each cluster. The stability mechanism here is similar to that for a synchronous solution; as discussed further in Section 4.3.3. Since each cluster experiences a decay of inhibition and subsequent re-inhibition between firings, stability within clusters does not require as long a J-cell active phase as does stability of the synchronous solution. The singular trajectory for an n-cluster oscillation represents a natural generalization of that for the 2-cluster oscillation. In the singular n-cluster solution, if we start when the Jcell falls down, then inhibition to the E-cells decays until one E-cluster fires and causes the J to fire; while these are active, the other n - 1 E-clusters evolve in the silent phase. The active E-cluster falls down before the J-cell, and the E-clusters then evolve in the silent phase such that each cluster reaches the initial position of the cell ahead of it in the firing sequence at the moment that the J-cell falls down. Precise conditions for the existence and stability of such a solution are given in the next subsection. 4.3.2. Statement of the main result. In this subsection, we state our main result concerning the existence and stability of clustered solutions. To clarify the presentation and notation, we will make some simplifying assumptions. A more complete presentation is given in [62]. We begin by discussing the notation that is needed. While all cells in the network are silent, the E-cell slow variables (wi, sj) satisfy (4.3). Let WL(Sj) and wF(sj) be as in the previous section. We assume that fu, > 0 near the singular solutions. Since sj represents inhibitory input, implicit differentiation of the first l f equation in (4.1) then yields that WL(SJ) and WF(SJ) are both positive. As in the previous section, we assume that wF(1) > WE(0) and that Kj is sufficiently large to guarantee that escape from the silent phase is possible. Thus, there exists West such that the solution of (4.3) beginning at (w0, 1) will reach the jump-up curve if and only if w0 > West. We denote by rs the time it takes for this solution to reach the jump-up curve. Note that rs depends on the initial position w0; however, we ignore this dependence here to simplify notation. Otherwise, we could state our results in terms of minimum and maximum times "t'~nin and r~nax. Note also that West --+ tOL (0) and rs ~ 0 as Kj ~ cx). Let resc, rE and rj be as in the previous section. Recall that resc is the time for w to increase from w + to Wesc under tb - GL(W, 1). Since Kj is large and rs is thus small, it follows that the time for w to increase from w + to Wesc, with several excursions from sj -- 1 of duration rs, is still approximately resc; we use this below but again, a notational adjustment suffices if more precise conditions are desired (see [62]).
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Finally, we need to assume that the J cell jumps up if it receives excitation from sufficiently many E cells. Let C sJ denote the J cell cubic when the J cell receives excitation of strength S t o t - S. Note that (4.2) has a fixed point on the left branch of C0J, call it w J(0). The J can jump up upon receiving excitation of strength S if and only if it lies above the left knee of w~ (S) when it is excited. Thus, we assume that there exists M such that if m >~ M, then tOJ
(m) ~SA
<1/)3(0).
(4.4)
Since we assume that the J cell is active for longer than each E cluster, the J cell jumps down at the right knee of the Stot -- 0 cubic, call it w J (0). Let rR (M) be the time for the J cell to evolve on the left branch of the Stot- 0 cubic from w J(0) to w J ( ~ s A ) . With these notation and assumptions, we obtain the following theorem. THEOREM 4.2. Given N, fix the parameters in (4.1) and (4.2) such that the above assumptions are satisfied and assume there exits M such that (4.4) applies for all m ~ M. Let n be the unique positive integer such that both of the following hold: (i) n rj -- rE + (n -- 1) rs > rest, (ii) (n -- 1)rj -- rE + (n -- 2)rs < rest.
l f n <~ N / M , rR(M) < rs, and Kj is sufficiently large, then there exists an n-cluster periodic solution of (4.1) and (4.2) such that each cluster contains at least M cells. This solution is stable, for gj sufficiently large, if lGL(w, sj)l < IGR(w, 1)l for WE(0) < W < wF(1). REMARK 4.6. Theorem 4.2 does not rule out the existence of other stable solutions. In particular, there may be bistability between a clustered solution and the synchronous solution. REMARK 4.7. If n = 1, then condition (i) becomes rj > rE -r resc. This is exactly the condition derived before for the existence of a synchronous solution. Here, condition (ii) is not relevant. For the 2-cluster case, conditions (i) and (ii) reduce to rj - rE
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To show the stability of the type of clustered solution considered here, two issues must be addressed. One must show that E-cells within different clusters remain separated from each other; in particular, they can never lie in the active phase at the same time. This actually follows from the existence arguments in [62]. One must also prove that if the E-cells within a cluster are perturbed slightly, then they are compressed under the subsequent flow. This requires indirect inhibition, for the same reasons discussed in Section 3.6. Given that the inhibition is indirect and that IGL(w, sj)l < [GR(w, 1)1, stability within clusters follows from compression of E-cells in the jump up after they are released from inhibition. There is no compression or expansion in the jump down because all E-cells jump down from the right knee of the cubic C1 (see Figure 14). REMARK 4.8. Note that the compression mechanisms responsible for the stability of the synchronous solution and the stability of cells within a cluster for a clustered solution are very similar. In fact, this compression mechanism is also similar to the compression mechanism in Fast Threshold Modulation, discussed in Section 3.4. In each of these networks, the cells jump up (nearly) horizontally and the slow variable, w, evolves slower before the jump up than after. This produces compression in a time metric (see [43]). What distinguishes these networks is the release mechanism that allows them to jump up. In FTM, one cell reaches the jump-up point at a knee, so that it can escape from the silent phase. This, in turn, raises the other cells' cubics so that they are forced to jump up. For the synchronous solution in globally inhibitory networks, all the E-cells jump up after the J-cell falls down and releases them from inhibition. In a clustered solution, the cells within one cluster must wait until another cluster jumps down. The J-cell must then still jump down before a new cluster is released from inhibition.
5. Thalamic sleep rhythms Many of the results described in this paper were motivated by models for thalamic oscillations. In particular, recent models for the spindle sleep rhythm [21,28,93,22,84] resemble the globally inhibitory networks in Section 4. In this section, we describe some of the rhythms generated in the thalamus and a model for them. We then discuss how geometric analysis helps to explain the generation of these rhythms and transitions between them.
5.1. Description of the sleep rhythms In an awake mammal, the thalamus plays an important role in relaying sensory inputs to the appropriate cortical regions. In sleep, neurons in the thalamus engage in rhythms which are believed to interfere with attentiveness to incoming stimuli [78]. During different stages of sleep, different organized forms of thalamic activity arise. For example, drowsiness and shallow sleep are characterized by the spindle rhythm, whereas the delta rhythm occurs in deeper sleep. Moreover, the spindle rhythm may be transformed into spike-and-wave like epileptiform oscillations, which we may generally classify as paroxysmal discharges [79, 9,78].
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The network that displays the thalamic spindle rhythm falls into the framework of the globally inhibitory networks considered in the previous section. In this network, a population of thalamocortical relay (TC) cells serve as E-cells, sending fast excitation to a population of thalamic reticular (RE) cells, which act as the J-cells in the network. The RE cells send fast (GABAA) and slow (GABAB) inhibition to the TC cells; they also inhibit other RE cells with fast inhibition. The spindle rhythm corresponds to a clustered oscillation in this model. In contrast to the discussion in the previous section, this network has multiple J-cells; however, its clustered oscillations are qualitatively the same as the type of clustered oscillation discussed there, since the TC cells form separately synchronized clusters which take turns firing while all the RE cells synchronize and fire together in each oscillation. In experiments, RE-TC networks have been observed to display completely synchronized oscillations in addition to clustering. Synchrony arises, for example, when fast inhibition is removed from the entire network (see [22] and the references therein) or from between the RE cells only [80,19,17,36]. Recent experimental results have also shown that complete synchrony can occur if the RE population receives additional phasic excitation, from cortical input or another source [14,20,76,17]. Hence, the network can change from the clustering of the spindle rhythm to paroxysmal synchrony without any change in thalamic inhibitory synapses.
5.2. A model f o r the spindle sleep rhythm We present the model discussed in [63], which closely resembles that given in [28] (see also [84]). Individual cells are modeled using the Hodgkin-Huxley formalism [33]. Unlike the models in the previous section, two intrinsic slow variables appear in the equations for each cell. As shown in [63], however, the inclusion of additional slow variables in a globally inhibitory network has no significant effect on the synchronization and clustering mechanisms. In the case of synchronization, the strong compression of E-cell trajectories towards a fixed point in the silent phase, afforded by a long J-cell active phase, dominates all other effects with or without this added variable. This contrasts dramatically with the situation for mutually coupled networks, for which the inclusion of an additional slow intrinsic variable significantly impacts escape and synchronization mechanisms. The following equations include numerous parameters and nonlinearities. These are defined in more detail in [28,84,63]. The equations of each TC cell are: !
v i -- --IT(Vi, ri) -- l s a g ( v i , h i ) - I L ( V i ) - I A -
r;hI -
(v,), (h~(vi)-
IB,
(5.1)
hi)/rh(vi).
Note that the singular perturbation parameter s is absorbed in rh, rr rather than mentioned explicitly. The terms IT,/sag, and IL are intrinsic currents; they are given by: IT(V, r) = gcam~c(v)r(v -- Yea),/sag(V, h) -- gsagh(v - Vsag) and IL(v) -- gL(V -- VL). Here, gc~ and vc~, ot = Ca, sag, L, correspond to the maximal conductance and reversal potential of each
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current. Moreover, m ~ (v) and r correspond to the activation and inactivation variables of the current IT, while h corresponds to the activation variable of the c u r r e n t / s a g . We note that the cell fires primarily due to the Iv current, corresponding to positively charged Ca 2+ ions flowing into the cell through calcium channels. For these channels to be open, we need that the Iv current be both activated and deinactivated; this means that both moc and r need to be bounded away from zero. An interesting feature of this model is that deinactivation of the IT current requires the cell to be hyperpolarized; that is, the membrane potential v must decrease from its resting level. This can be seen from the explicit formula for the nonlinear function roc(v) given in [63]. Hence, a biological explanation for the initiation of TC cell firing is that the inhibition that the TC cells receive from the RE cells hyperpolarizes them which, in turn, deinactivates IT. This current is also activated at low membrane potentials, which can be seen from the formula for m ~ ( v ) . Once the calcium channels are open, calcium ions flow into the cell and an action potential is produced. The terms IA and IB represent the fast and slow inhibitory input from the RE cells. We 1
model the fast inhibition IA as in previous sections; that is, IA = gA(Vi -- V A ) ~w- Y~ s~ where gA and VA are the maximal conductance and the reversal potential of the synaptic current. The sum is over all RE cells which make synaptic connections with this TC cell and NTR represents the maximum number of RE cells which send inhibition to a single TC cell. Each synaptic variable s J is direct and satisfies the first order equation jt
9
9
9
(5.2)
SA -- OtR (1 - s~) H (v s - 0R) -- fiRS J ,
where v/~ is the membrane potential variable of the j th RE cell. Motivated by recent experiments [ 19,22], we model the slow inhibition IB somewhat differently from IA. We first discuss, however, the model for the RE cells. The equations of each RE cell are: i t VR -- --/RT(V~, h ~ ) '
-
(h
i 1AHP(VR, m i ) -- I R L ( V / R ) - I R A -
-
IE,
(5.3)
rni' = #l[Ca]i(1 -- m i ) -- # 2 m i , [Ca]i t -- - - V l R T - ),,[Ca]/.
The terms/RT, IAHP, and/RL represent intrinsic currents. These are given by IRT(1), h) = g R c a m 2 v c ( v ) h ( v -- VRCa), I A H p ( V , m ) -- g A H p m ( V -- VK) and IRL(V) -- gRL(V -- VRL). More details concerning the biophysical significance of each term are given in [28,84]. In (5.3), the term IRA denotes the inhibitory input from other RE cells. It is modeled as i IRA -- gRA (V R
-
VRA)
1 ~ SJA where the sum is over all RE cells which make synaptic
connections with the i th RE cell. Each synaptic variable S~A satisfies a first order equation similar to (5.2). The term IE represents excitatory (AMPA) input from the TC cells and is 1 Y~ S~ where the sum is over all TC cells which make expressed as IE -- gz(viR -- VE)-R-~RV
excitatory synaptic connections with the i th RE cell. The synaptic variables s~ are fast and also satisfy first order equations similar to (5.2).
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It remains to discuss how we model the slow inhibitory current lB. Similarly to [ 19], we assume that IB - -
s4 g B ~ ( 1bi )i
S~i - - k l H ( X b i
x'bi
=
k3[ NTR
--
VB) where
Sbi,
along with the variable
Xbi,
satisfies
- O x b ) ( 1 -- Sbi) -- k 2 s b i ,
EH(I)~--0Rb)
] (1--Xbi)--k4Xbi.
(5.4)
The parameters are such that Xbi can only become activated (i.e., exceed 0xb) if a suffii above the threshold ciently large number of RE cells have their membrane potentials vR 0Rb. The threshold is chosen rather large so the RE bursts must be sufficiently powerful to activate Xbi. Once Xbi becomes activated it turns on the synaptic variable Sbi; the expression s bi 4 in IB corresponds to synaptic receptor dynamics [ 19] and further delays the effect of the inhibition on the postsynaptic cell.
5.3. Insights from the geometric analysis 5.3.1. Removal of fast inhibition promotes synchronization. The spindle sleep rhythm corresponds to a clustered solution to (5.1)-(5.4). Biologically, RE cells oscillate at about 7-14 Hz in this rhythm; simulations of this model in [63], for example, yielded a frequency of about 12.5 Hz. TC cells, meanwhile, form clusters that each oscillate at a fraction of the spindle frequency, with different TC clusters firing with each RE burst. In this rhythm, fast inhibition from the RE cells to the TC cells and to other RE cells activates with each cycle. Slow inhibition does not activate at all, since the RE cells do not fire powerful enough bursts to keep vRi above the threshold ORblong enough to activate the variables Xbi Equations (5.1)-(5.4) also support a synchronous solution, in which all TC and RE cells fire in each cycle, with the RE cells firing almost instantly in response to TC bursts and a frequency of about half that of the clustered oscillation. Experiments and numerical studies have demonstrated that synchrony arises, for example, when fast inhibition is removed from the entire network or from between the RE cells only. The results in the previous section help to clarify why removal of fast inhibition promotes synchronization. Note that fast inhibition occurs in two places: the RE cells inhibit themselves as well as the TC cells. Removing fast inhibition has different consequences for each of these synaptic connections and both of these help to synchronize the TC cells (see also [28,22]). Removing the RE-TC fast inhibition is clearly helpful for synchronization among the TC cells. This holds because the fast inhibition has a very short rise time. This short rise time corresponds to a small window of opportunity for TC firing and a small domain of attraction of the synchronous solution. In fact, it is precisely this inhibition that is responsible for desynchronizing the TC cells during the clustered solution. Removing the RE-RE fast inhibition appears to be even more crucial for synchronizing the TC cells (see also [80,35,19,17,36]). This allows the RE cells to fire longer, more powerful bursts which, in turn, activate the slow inhibitory current lB. The analysis presented for globally inhibitory networks demonstrates that long, powerful bursting of the RE cells is needed for the TC cells to synchronize, unless the desynchronizing effect of inhibition is somehow removed as discussed in [84]. Numerical simulations in [63] show, in fact,
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that the TC cells will synchronize even if fast inhibition is removed from within the RE population but not from the RE-TC connections. Why removal of inhibition leads to stronger RE bursts can be easily understood by the analysis of trajectories in phase space. This removal forces the RE cells to lie on the right branch of a different cubic while in the active phase. The cubic of the disinhibited cells lies below the cubic of the cells with inhibition. The disinhibited cells therefore jump up to larger values of membrane potential; moreover, their jump-down point (right knee) lies below the jump down point of the inhibited cells. The disinhibited RE cells, therefore, have a longer active phase. Note that the slow inhibitory current IB, activated when RE-RE fast inhibition is removed, has slower rise and decay times than IA. The slow decay time can help to bring the cells closer together while in the silent phase, as indicated by the results, presented in Section 3.6, on mutually coupled cells with slow inhibition (see also [63]). The slow rise time enhances the domain of attraction of the synchronous solution by expanding the window of opportunity, if the RE-TC fast inhibition is also removed. Hence, both effects improve the ability of the TC cells to synchronize in the absence of fast inhibition, while the former effect may encourage synchrony even if RE-TC fast inhibition remains. 5.3.2. What determines the period of synchronous oscillations ? The analysis in Section 4 demonstrates that when TC cells oscillate synchronously, the period of their oscillation equals the time for which the RE cells stay active (rj) plus the time it takes for inhibition to decay sufficiently for the next TC firing to occur (rs). Thus, only those parameters that affect these times can influence period. Numerics in [63] verify that increasing the decay rate of inhibition, for example, sharply decreases the period (see also [ 17]). The rates of change of individual TC cell currents, however, have little effect on period. Geometric analysis of globally inhibitory networks explains this ineffectuality; in a synchronous oscillation, TC cells are compressed close to their silent phase rest state while the RE cells are active, and then the TC cells evolve with little change in their intrinsic currents as inhibition decays. The slow inhibitory conductance gB also has little effect on rj, rs and hence on period. The small influence that it does exert comes through its mild influence on the slope of the curve of knees for the TC silent phase, which actually causes a counterintuitive increase in the period as the strength of slow inhibitory coupling increases (see also [93]). 5.3.3. Cortical inputs can enhance synchronization. Recent papers have emphasized the importance of the cortex in the transformation of spindle oscillations into spike-and-wavelike (SW) epileptiform oscillations in the thalamus [74,75,14,20,17]. The experiments and modeling described in these works have suggested that this can arise without the removal of fast inhibition from the thalamus. In the network model, one can view the cortex as providing excitatory input to the RE cells. In the mechanism discussed in Section 5.3.1, disinhibition of the RE cells leads to more powerful RE bursts and this permits the TC cells to synchronize. The analysis in Section 4 supports the finding [14,20,76,17] that if one does not remove the fast inhibition among the RE cells, but instead induces sufficiently strong excitation from the cortex, then this will have the same effect: the RE cells will fire more powerful bursts, with a longer active phase, because of the additional excitation (their cubics are lowered).
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Due to these more powerful bursts, both fast and slow inhibition of TC cells ensue, and the resultant additional inhibition implies that TC cells are compressed towards a rest state with larger h, r values. Together with the extended RE active phase, this yields strong compression of TC cells, which combines with the fast decay of fast inhibition to promote TC synchrony, despite the fast rise time of the fast inhibition they receive. In particular, this explains the mechanism behind the results of Destexhe [ 17], in which sufficiently strong corticothalamic excitation (achieved by blocking only cortical fast inhibition) is found to be crucial for triggering powerful RE bursts. These bursts in turn lead to the activation of fast and slow inhibition in the thalamus and the generation of synchronized ~ 3 Hz oscillations in TC and RE cells.
Acknowledgments Research for this paper was supported in part by the NSF grants DMS-9802339 and DMS9804447.
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CHAPTER
4
Numerical Continuation, and Computation of Normal Forms Wolf-Jtirgen Beyn l, Alan Champneys 2, Eusebius Doedel 3, Willy Govaerts 4, Yuri A. Kuznetsov 5 and Bj6rn Sandstede 6 I Bielefeld University, Bielefeld, Germany 2 University of Bristol, Bristol, UK 3 Concordia University, Montreal, Canada 4 University of Gent, Gent, Belgium 5 Utrecht University, Utrecht, The Netherlands 6 Ohio State University, Columbus, OH, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Continuation of stationary and periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Parameter continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Pseudo-arclength continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Periodic solution continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Locating codimension- 1 bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Locating codimension- 1 equilibrium bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . .
151 152 152 153 154 155 157 157 158
3.3. Locating codimension- 1 bifurcations of periodic solutions . . . . . . . . . . . . . . . . . . . . . 3.4. Test functions defined by bordering techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Branch switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The algebraic branching equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Branch switching at simple branch points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Approximation of periodic solutions near a Hopf point . . . . . . . . . . . . . . . . . . . . . . .
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4.4. Approximation of double-period solutions near a flip point . . . . . . . . . . . . . . . . . . . . .
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5. Continuation of codimension- 1 bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Continuation of codimension- 1 equilibrium bifurcations . . . . . . . . . . . . . . . . . . . . . . 5.2. Standard augmented systems
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5.3. Continuation of codimension- 1 bifurcations of periodic solutions . . . . . . . . . . . . . . . . . 6. Continuation of codimension-1 homoclinic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. A truncated boundary-value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H A N D B O O K OF D Y N A M I C A L S Y S T E M S , VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 149
168 168 171 175 177 179
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6.2. Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Locating codimension-2 equilibrium bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Normal forms for codimension- 1 bifurcations
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7.2. Locating codimension-2 bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
181 182 182 185
8. Locating codimension-2 homoclinic bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.1. Test functions utilizing eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.2. Test functions for homoclinic flip bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8.3. Test functions detecting non-central homoclinic orbits
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9. Continuation of codimension-2 equilibrium bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. B ogdanov-Takens
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193 193
9.2. Fold-Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9.3. Double-Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
196 197
9.5. Generalized Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Normal forms for codimension-2 equilibrium bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. List of codimension-2 normal forms
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198 200 200
10.2. The normalization method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
203
10.3. The cusp bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
204
10.4. B ogdanov-Takens bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
205
10.5. Bautin (generalized Hopf) bifurcation
206
10.6. Fold-Hopf bifurcation
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10.7. Double-Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Branch switching at codimension-2 bifurcations
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207 208 209
11.1. Switching at a cusp point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Switching at a Bogdanov-Takens point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
210 212
11.3. Switching at Bautin (generalized Hopf) bifurcation
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11.4. Other codimension-2 cases References
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216 216
Numerical continuation, and computation of normal forms
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1. Introduction
In this chapter we give an overview of numerical methods for analyzing the solution behavior of the dynamical system
x'(t)-- f(x(t),o~),
x, f ( x , ~ ) EIR",
(1)
where, depending on the context, c~ denotes one or more parameters. Throughout we assume that f is as smooth as necessary. We do not consider Hamiltonian systems, reversible systems, and systems with symmetries in this chapter. The emphasis in this chapter is on numerical continuation methods, as opposed to simulation methods. Time-integration of a dynamical system gives much insight into its solution behavior. However, once a solution type has been computed, for example, a stationary solution (equilibrium) or a periodic solution (cycle), then continuation methods become very effective in determining the dependence of this solution on the parameter c~. Moreover, continuation techniques can also be used when the solutions are not asymptotically stable. Knowledge of unstable solutions is often critical in the understanding of the global dynamics of a system. We first review basic continuation techniques for following stationary and periodic solutions in one-parameter families of (1). We also describe algorithms for detecting codimension-1 bifurcations, namely folds and Hopf (or Andronov-Hopf) bifurcations, and methods for locating branch points. Branch switching techniques are also described. Once a codimension-1 bifurcation has been located, it can be followed in two parameters, that is, with oe ~ ~2 in Equation (1). We give details on this for the case of folds, Hopf bifurcations, period-doubling bifurcations and torus bifurcations. We also describe techniques for the detection and continuation of codimension-2 bifurcations, including the cusp, and the Bogdanov-Takens (BT), generalized Hopf, fold-Hopf, and double-Hopf bifurcations. Efficiency is an important issue in the design of algorithms for following such singularities. Particular attention is paid to connecting orbits, especially homoclinic orbits. Basic numerical techniques for the computation and continuation of such orbits are outlined. We also describe algorithms for the detection of higher codimension homoclinic bifurcations. In many cases, detection of higher codimension bifurcations requires computation of certain normal forms for equations restricted to center manifolds at the critical parameter values. We provide explicit formulas for such coefficients for codimension-1 and 2 equilibrium bifurcations. Finally, we discuss how to start continuation of local and global codimension-1 bifurcations from some codimension-2 equilibrium singularities. For an excellent introduction to the subject of numerical continuation and bifurcation methods we refer to the fundamental paper of Keller [42], which also appears, along with other important contributions, in Keller [44]. Comprehensive books on the subject include Kubi6ek and Marek [47], Rheinboldt [62], Keller [43], Seydel [73], Allgower and Georg [1], and Govaerts [32]; see also in Kuznetsov [51, Chapter 10]. For tutorial introductions, see Beyn [8], Doedel et al. [24,25] and Doedel [18].
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152 1.1. Notation
T h r o u g h o u t this chapter, t and ot denote the time and parameter variables, respectively. The derivative with respect to the time variable t is denoted by x' = d x / d t , where x(t) is a function of t. We write k = dx Ida for the derivative of a function x = x (or) with respect to the parameter c~. If A is a matrix (or a vector represented by a one-column matrix), then A* denotes the transposed matrix; if A has complex entries, then A* is the transpose and complexconjugated matrix. The scalar product of two vectors is then written as (v, w) = v* w, while the standard n o r m is defined by I]vll = ~/(v, v). The identity matrix in R n is denoted by In. We write A/'(A) and Tr for the nullspace and the range, respectively, of a matrix A. We also use the notation X = (x, c~) (or, more precisely, X = (x*, or*)*). Let f (x, c~) be a smooth function and (x0, c~0) a given point. We then use the notation
fOx v
-
f0x x V W
i x ( x o , ~ o ) v,
-
fxx(XO,
c~0)[v, w]
as a short hand for the derivatives of f with respect to x evaluated at (x0, or0) and, as a multi-linear form, applied to the vectors v and w. Analogous expressions are used for derivatives with respect to c~. Let A and B be n x n matrices with elements {aij } and {bij }, respectively, for 1 <~ i, j ~< n, and set m = 89 - 1). The bialternate product of A and B is an m x m matrix, denoted A 63 B, whose rows are labeled by the multi-index (p, q) (p = 2, 3 . . . . . n; q -- 1, 2 . . . . . p - 1), whose columns are labeled by the multi-index (r, s) (r = 2, 3 . . . . . n; s = 1, 2 . . . . . r - 1), and whose elements are given by
,{
(A 63 B)(p,q),(r,s) : -~
apr bqr
aps + bpr bqs aqr
bps aqs
o
2. Continuation of stationary and periodic solutions Here we consider the computation of one-parameter families of equilibria of (1), that is, solutions of
f(x,et) -0.
(2)
Such a continuum of solutions is often referred to as a solution branch. With ot 6 IK and X = (x, or) the above equation can be written as
f(X) -0, where f " R n+l --~ R n. A solution X0 of f ( X ) - - 0 is called regular if f o has m a x i m a l rank, that is, if R a n k ( f ~ -- n. Near a regular solution one has a unique solution branch, as we now make precise.
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THEOREM 2.1. Let Xo be a regular solution of f (X) = O. Then, near Xo, there exists a unique one-dimensional continuum of solutions X (s) with X (0) = Xo. PROOF. We have X ~ (x, or) and f o = ( f o i f o). If R a n k ( f ~ I f ~ = n then either f.o is nonsingular and by the Implicit Function T h e o r e m one has x = x(oe) near x0, or else one can interchange columns in f~9 to see that the solution can locally be parametrized by one of the components of x. Thus a unique solution branch passes through a regular solution. D REMARK. An example of the second case in the proof is the simple fold (or saddle-node bifurcation) described in Section 3.
2.1. Parameter continuation Here we take ot to be the continuation parameter. Suppose we have a solution x0 of (2) at c~o, as well as its derivatives :to with respect to the parameter or. We want to compute the solution x l at oel --= or0 + Aoe. See Figure 1 for a graphical interpretation. To find x l, we solve f (x l, Otl ) = 0 for x I by Newton's method,
f~(xl~),ot,)Ax(l~)----f(xl"),a,),
x(l ~+1) =x(l~')+ Axl v),
v - - O , 1,2 . . . . .
with x l~ = xo + Aot :to. If f~ (X l, 0t l) is nonsingular and Aot sufficiently small then the theory for Newton's method assures that this iteration will converge. After convergence, the new derivative vector :tl can be obtained by solving A ( X I , Ogl):tl -- - - f ~ ( X l , Ogl).
Xl
(zo, 1)
a?o
I OLO
~1
Fig. 1. Graphical interpretation of parameter continuation.
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This equation follows from differentiating f(x(c~), c~) = 0 with respect to c~ at c~ = c~l. In practice, the calculation of .fl can be done at negligible computational cost, since the numerical decomposition of the final Jacobian matrix fx (x l, a l ) in Newton's method can be reused.
2.2. Pseudo-arclength continuation This method, due to Keller [42], allows the continuation of any regular solution, including folds. Geometrically, it is the most natural continuation method. Suppose we have a solution (x0, c~0) of f (x, c~) = 0, as well as the normalized direction vector (ko, &o) of the solution branch at (x0, c~0). Keller's method consists of solving the following equations for x 1 and ~l :
f ( x l , otl) = O,
(xl -- xo)*.fo + (Otl -- otO)&O -- As = O.
See Figure 2 for a graphical interpretation. Newton's method for solving these equations can be written as
(~(,~)- ~o)*~o
-
(~(~) I
-a0)6t0-
(3)
Xl
271
X0
"
As
--
x0 -
X0
OL0
0/1
Fig. 2. Graphical interpretation of Keller's method.
(~0, ~0)
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155
Upon convergence, the next direction vector can be computed from
(o) x0"*
&o
&l
1
"
Note that, in practice, the decomposed Jacobian of the final Newton iteration can be reused to compute the new direction vector. Furthermore, the normalization x~)-fl + &0&l -- 1 ensures that the orientation of the branch is preserved if As is sufficiently small. The new direction vector must be rescaled to have unit length. In practice, the step size As is updated regularly during the computation of a solution branch. In the simplest case the choice of the new As is based on the convergence of the Newton iteration (3). More sophisticated methods rely on local error estimates; see, for example, Deuflhard et al. [ 16], Rheinboldt [62]. THEOREM 2.2. The Jacobian of Keller's equations is nonsingular at a regular solution. PROOF. With X = (x, or) 6 R ''+l , Keller's equations are
f(Xl) =0, where
(Xl - X o ) * X o - As = 0 ,
IIX011- 1. The matrix in Newton's
,cO
method at As - - 0 is (xJ.X). At a regular solution i /
./V(f o ) -- Span{Xo}. We must show that (x(~)is nonsingular. If, on the contrary, (xf~)is 13
singular, then f ~ 0 and . ~ z non-zero constant c. But then 0 contradiction.
O, for some vector z ~ O. Thus z - cX0, for some XoZ "* - c~'~Xo - cllX0112 - c, so z - - 0 , which is a D
REMARKS. The addition of Keller's continuation equation (Xl - X0)*X0 -- As is one of the simplest examples of an extended system that regularizes an otherwise singular system. In the current context, the singularity would arise at a fold (see Section 3.2.1), if we did not use Keller's method. We shall discuss many other extended systems in this chapter. The linear system (3) in Newton's method applied to Keller's equations has the form
(; If A is a sparse matrix, for example, if A is tridiagonal, then the bordered system can be solved efficiently using bordered LU-decomposition; see, for example, Doedel et al. [24].
2.3. Periodic solution continuation Suppose we want to compute a periodic solution of Equation (1), and follow it as a parameter c~ changes. Note that the period T of the solution is not known a priori, and moreover T
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typically varies with or. To deal with this, we fix the interval of periodicity by the transformation t --+ t~ T. Then (1) becomes
x' (t) = T f (x(t), ot),
(4)
and we now seek solutions of period l, that is, x (0) = x (1).
(5)
Note that the period T is one of the unknowns. In our continuation context, assume that we have computed (xk-1 (.), Tk-l, Otk-1) and that we want to compute (xk(.), Tk, c~k). For simplicity of notation, let (x(-), T, or) = (xk(.), Tk, o~k). Equations (4) and (5) do not uniquely specify x and T, since x(t) can be translated freely in time, that is, if x(t) is a periodic solution then so is x(t + or), for any o-. Thus, a "phase condition" is needed. An example is the Poincar6 orthogonality condition
(x(0~
xk _ ~(0))* x k' _ j ( 0 ) - 0 .
-
However, there is a numerically more suitable phase condition, that has the property of keeping sharp fronts or peaks in approximately the same position. To be more precise, let ~(t) be a solution. We want the phase-shifted solution that is closest to xk-l in phase, that is, we want the solution that minimizes
=
fo
II (, +
-
II dt.
The optimal solution ~(t + 6) satisfies the necessary condition dD/dcr = 0, that is,
fo I (~c(t -~ Cr) -- Xk-l (t))*X,t(t --~ cr) dt -- O. Writing x(t) -- Yc(t + ~) gives
fo I (x(t)
- x k _ l ( t ) ) * x ' ( t ) d t --0.
Integration by parts, using periodicity, gives
fo I x t, t ,), xk_ t 1(t) dt - O.
(6)
The phase condition (6) is very suitable for numerical computations. It is not difficult to establish the following version of Poincar~ continuation. THEOREM 2.3. Let (x0(.), To) define a solution of (4)-(6), when ot --oto. If the Floquet multiplier 1 is algebraically simple then (x0(.), To) can be continued locally as a function of or.
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157
REMARK. The Floquet multipliers are the eigenvalues of the monodromy matrix V (1), where V (t) is the fundamental solution matrix of the homogeneous linear equation, that is, V (t) satisfies V ' ( t ) - To f~ (xo(t), oeo) V (t),
v ( o ) = i.
Due to periodicity, V (1) always has an eigenvalue equal to 1, called the trivial multiplier. For the numerical computation of Floquet multipliers see Fairgrieve and Jepson [27] and Doedel et al. [25]. Above we have assumed oe to be fixed. However, in practice we use Keller's method (see Section 2.2) to trace out a branch of periodic solutions. In particular, this allows calculation past folds along such a branch. In the current function space application, Keller's continuation equation takes the form
fo
l ( x ( t ) - - X k _ l ( t ) ) * k k _ l ( t ) d t + ( T - - Tk_l)j~k_l -3t-(O/--Otk_l)O/k_l -- As.
(7)
The complete computational formulation then consists of Equations (4)-(7), which are to be solved for x(.), T, and oe. These equations correspond to a (generalized) boundary value problem (BVP), and are solved by a numerical boundary value technique. The most widely used discretization method for boundary value problems in ordinary differential equations is the method of orthogonal collocation with piecewise polynomials. In particular, orthogonal collocation is used in software such as COLSYS (Ascher et al. [3]), AUTO (Doedel [17], Doedel et al. [20]), COLDAE (Ascher and Spiteri [2]), and CONTENT (Kuznetsov and Levitin [52]). Its high accuracy (de Boor and Swartz [14]) and its known mesh-adaption techniques (Russell and Christiansen [67]) make this method particularly suitable for difficult problems.
3. Locating codimension-1 bifurcations An equilibrium without eigenvalues having zero real part, or a periodic solution without nontrivial Floquet multipliers of unit modulus, is called hyperbolic. When following an stationary or periodic solution branch, one can encounter bifurcation points, where the solution loses hyperbolicity. In the case of equilibria, the Jacobian matrix f~ has at least one eigenvalue with zero real part at such a point, while for the case of a periodic solution there is at least one nontrivial Floquet multiplier of unit modulus. In this section we introduce test functions (sometimes called bifurcation functions) to detect codimension-1 bifurcations of stationary and periodic solutions to (1).
3.1. Test functions Let X = X (s) be a smooth, local parametrization of the curve
f (X) - O,
f "]R''+l --+ IR",
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W.-J. Beyn et al.
such that s = 0 corresponds to a bifurcation point. DEFINITION 3.1. A smooth scalar function ~ :~,;+l __+ R l defined along the curve is called a test function for the corresponding bifurcation if g(0) = 0, where g(s) = 7t(X (s)). The test function 7t has a regular zero at the bifurcation point if g'(0) :/: 0. A bifurcation point is detected between two successive points X0 and X1 on the curve if a test function 7t = 7t (X) has opposite signs at these points r
< o.
In such case, one can attempt to locate the bifurcation point by applying Newton's method to the system
7~(x) - 0 ,
(8)
with initial point X0, for example. If the solution branch X (s) is regular near the bifurcation point, and if the test function is defined and differentiable in a neighborhood of the curve and has a regular zero at the bifurcation point, then Newton's method will converge, provided IIX0 - X111 is small. One-dimensional secant methods can be used to solve (8) under less restrictive conditions.
3.2. Locating codimension- 1 equilibrium bifurcations Consider the equilibrium curve (2) corresponding to a one-parameter system (1). There are two generic codimension-1 bifurcations that can be encountered along the equilibrium curve: the fold and the Hopf bifurcation. At a fold point, the Jacobian matrix fx has an algebraically simple zero eigenvalue and no other eigenvalues on the imaginary axis. At a Hopfpoint fx has a unique and algebraically simple pair of purely imaginary non-zero eigenvalues. In both cases, the topology of the phase portrait near the equilibrium changes when the parameter passes the bifurcation value (see Section 7). 3.2.1. Folds.
The function
l~f(X, 0e) = )~1 (X, 0e)~,2 (X, 0 / ) ' ' ' )~n (X, 0/), where ~,j (X, 13l) are the eigenvalues of the Jacobian matrix fx (x, ~), is a test function for the fold bifurcation. Since ~f(x, c~) -- det fx (x, or), this test function can be efficiently computed without computing eigenvalues.
(9)
159
Numerical continuation, and computation of normal forms
DEFINITION 3.2. A point (x0, c~0) in a one-parameter system (1) is called a simple fold if (S1) p. fO qq =/=O, and ($2)
p* f o r
0,
where f Oq _ f~0*p -- 0, q*q - p* q -- 1. Note that ($2) means f o ~t 7~(f,~
Simple folds are often called simple quadratic folds.
THEOREM 3.1. The test function (9) has a regular zero at a simple fold point. PROOF. Near a simple fold the equilibrium branch (2) has the parametrization
x(s) -- xo + 8q +
0(82),
a(s) = a o - -l P * f r ~ 2 p, fO
0(8 3) +
"
The Jacobian matrix f~(x(s), or(s)) has a simple eigenvalue ,k(s) with normalized eigenvector v(s), such that )~(0) = O, v(O) = q. Both )~(s) and v(s) are smooth. Differentiating the first equation of the eigenvalue problem
fx (x(s), ~(s))v(s) - ~(s)v(s) =0,
p* v(e) = 1,
with respect to s at s = 0 and multiplying the result from the left by p*, gives
X(O) = p* f,~ q =/=O.
D
The Jacobian matrix of (8) with X = (x, or) is nonsingular at a simple fold, so that Newton's method can be used to locate it. Another way to detect a fold point is to monitor extremum points with respect to the parameter ot on the equilibrium curve (2). At a simple fold point the a-component of the tangent vector to the curve (2) changes sign. 3.2.2. Hopfpoints.
Consider the real-valued function
~H(~,,~)- 1--I(~; (~, 0~) + ~; (~,,~)),
(10)
i>j
where, as before, the i~j(X, Ol) are the eigenvalues of the Jacobian matrix f~ (x, or). This function vanishes at a Hopf bifurcation point, where there is a pair of eigenvalues X 1,2 = +ico0. Clearly, 7rH is also zero if there is a pair of real eigenvalues ~,1
:
K,
~,2
:
--K.
We have to exclude such points when looking for Hopf bifurcations.
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DEFINITION 3.3.
A Hopf point (xo, oto) in a one-parameter system (1) is called simple if
/2(0) = Re[p*ii(c~o)q] r 0, where q, p 6 C" satisfy fOq _ icooq,
fO, p _ -icoop X
~
p*q -- 1
'
and d
A(OI) ---- ~
fx(Xe(Ol), Ol) - fxx(Xe(Ol), Ol).~e(Ol) nt- frot(Xe(Ol), Ol),
and where Xe(Ol) denotes the continuation of the equilibrium x0 for oe close to or0. It is easy to show that/2(0) is exactly the u-derivative of the real part of the critical pair of eigenvalues )~1,2(c~) = #(c~) 4- io)(c~), when it crosses the imaginary axis. The following theorem is then obvious, since
~(Ol)-- E1 (~. 1(or)
+ ~.2(ot))
THEOREM 3.2. The test function (10) has a regular zero at a simple Hopf point. The Jacobian of (8) is nonsingular at such a point, and Newton's method can be applied. As in the fold case, one can compute (10) without explicit computation of the eigenvalues of fx (Fuller [30], Guckenheimer and Myers [37]), by using the bialternate product defined in Section 1: THEOREM 3.3 (St6phanos [75]). Let A be an n x n matrixwith eigenvalues Jkl, ).2 . . . . . Then: (i) A (3 A has eigenvalues ~i ~.j, (ii) 2A (3 In has eigenvalues )~i nt- ~.j, w h e r e i = 2 , 3 . . . . . n; j = 1 , 2 . . . . . i - 1 .
~.n.
Therefore, the test function (10) can be expressed as
~H(X, Or)- det(2fx(x, or) (2) In).
(11)
The definition of the bialternate product (see Section 1) leads to the following formula for the elements of 2A @ I,:
(2A (3 In)(p,q),(r,s)=
--aps
ifr =q,
a pr
if r 7~ p and s = q,
app nt- aqq
if r = p and s = q,
aqs
if r = p and s r q,
--aqr
if s = p,
0
otherwise.
Numerical continuation, and computation of normal forms
16 1
Thus, given the matrix A, the computation of the elements of 2A 63/,7 can be efficiently programmed. Also, in Section 3.4, we give an efficient method for computing a function whose value is proportional to (1 1), by solving an extended linear system.
3.3. L o c a t i n g codimension-1 bifurcations o f p e r i o d i c solutions We now describe test functions for the location of codimension-1 bifurcations of periodic solutions in terms of the Jacobian matrix A = p~ of the Poincar6 map associated with the periodic solution: x w-~ P (x, ot ),
x ~ ]Rz'-I
~ ~ IR l
where x provides a smooth parametrization of a codimension-1 cross-section to the closed orbit. Let/z l, #2 . . . . . //,,7-l be the multipliers of the periodic solution, i.e., the eigenvalues of A evaluated at the fixed point corresponding to the periodic solution. Adding #,z -- 1 gives the set of Floquet multipliers introduced in Section 2.3. There are three generic codimension-1 bifurcations of periodic solutions: fold, flip (period-doubling), and Neimark-Sacker (torus) bifurcations. At a f o l d point, the matrix A has a simple unit multiplier #j = 1; at a flip point there is a simple multiplier #~ -- - 1 ; at a torus bifurcation A has a pair of non-real multipliers on the unit circle: #1,2
--exp(+i00),
0 < 00 < Jr.
In each of these cases, we assume that the critical eigenvalues are the only eigenvalues of A on the unit circle. The following test functions locate fold, flip, and Neimark-Sacker bifurcations, respectively: 11-- 1
Of -- U (#i - 11,
(12)
i=1
n-1
On- H(gi
+ 1),
(13)
i=1
7tNS -- 1-I (/zi/zj -- 1).
(14)
n>i>j
To detect a true Neimark-Sacker bifurcation, we must check that ~PNS -- 0 due to nonreal multipliers with unit product: ~i ~ j " - 1. As in the previous section, we can express the test functions (12)-(14) in terms of the Jacobian matrix, namely, Of -- det(A - L , - l ), O n - det(A + L,-l), O N S - det(A 63 A -
In,),
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162
where m -- 89 - 2)(n - 1). The last formula follows from statement (i) of St6phanos' theorem. Using the definition of the bialternate product we get
(A G A)(p,q),(r,s) -- apraqs - aqraps. Actually, a fold point can be more easily detected as an extremum point of the orcomponent of the (discretized) periodic solution branch. For the flip and torus bifurcations, the following approach is often applicable. Recall the BVP (4)-(6) for computing a periodic solution. After discretization, for example, using finite differences or collocation, these equations can be solved by Newton's method. Often the Newton matrix can be numerically decomposed such that the approximate monodromy matrix V (1) (see Section 2.3) is implicitly obtained as a by-product, namely in the form A1 V (1) = - A 0 , for certain nonsingular n x n-matrices A0 and A l; see Doedel et al. [25]. The test functions for flip and torus bifurcation can then be expressed as 1 0n = ~ d e t ( - A 0 + A l), detAl 1 ONs = det(A0 @ A0 - A 1 Q) A l). (detAl @ A1)
To obtain the last test function, we used the identities (AB) (7) (AB) = (A @ A)(B @ B) and (A @ A)-1 = A - 1 Q) A - I . These test functions are used in C O N T E N T (Kuznetsov and Levitin [52]).
3.4. Test functions defined by bordering techniques As we have seen, the detection of codimension-1 bifurcations of stationary and periodic solutions can be reduced to the detection of zeros of certain determinants along the corresponding branches. The numerical computation of a determinant of a square matrix is a simpler task than computing all its eigenvalues. However, scaling problems can arise when the system dimension is large. The following bordering technique avoids such scaling problems. The idea is to construct a scalar function g = g(s) that vanishes simultaneously with the determinant of a parameter-dependent (n x n)-matrix B(s). Suppose that the determinant vanishes at, say, s = 0, and also assume that the eigenvalue )~ = 0 of B0 = B(0) is geometrically simple. Such a g -- g(s) can be computed as the last component of the solution vector to the (n + 1)-dimensional bordered system:
. q0
.0)(.)(0; 0
g
--
1
,
where q0, p0 EI[{ n are certain fixed vectors.
(15)
Numerical continuation, and computation of normal forms LEMMA 3.1 (Keller [421).
163
lf qo q~~(B~) and Po q~~(Bo), then the matrix
M(s) _ ( B(S)q~ PO is nonsingular for all sufficiently small Isl. Therefore, generically, Equation (15) has a unique solution. In practical computations, the vectors q0, p0 should be adapted along the solution branch to make M(s) as wellconditioned as possible. Note that g(s) is proportional to det B(s). Indeed, by Cramer's rule
g(s) =
det B(s) det M(s) '
so g (0) -- 0. Moreover, if s = 0 is a regular zero of the determinant then it is also a regular zero of g. A more general result is given in Section 5. The function g(s) serves as a test function for the fold bifurcation if we let B(s) = f~(x(s), c~(s)), where (x(s), a(s)) is a regular parametrization of the equilibrium branch near the fold. For the detection of a Hopf bifurcation, one can use g(s) for
B(s) - 2 f , (x(s), oe(s)) 63 I,,. Taken literally, this method only applies when n is relatively small. To detect Hopf bifurcation in large systems, the method can be applied after reduction to an invariant subspace corresponding to the dominant eigenvalues; see, for example, the discussion in Govaerts et al. [34]. Alternatively, detection of Hopf bifurcation in such systems can be based on computing dominant eigenvalues using standard numerical linear algebra. If A is the Jacobian matrix of the Poincar6 map, then (15) with
B = A-L,, B = A+In, B = A63A-Ln, gives test functions for the fold, flip, and torus bifurcations, respectively. REMARK. The bordering approach can also be used to detect codimension-1 bifurcations of periodic solutions without explicit computation of the Jacobian matrix of the Poincar6 map. We illustrate this below for the case of the flip (period-doubling) bifurcation. Consider the BVP (4)-(6) for the continuation of a periodic solution. For given (x (-), T), introduce the following auxiliary BVP for (v(-), G) with fixed bordering functions ~oo,7t0:
I v'(t) = T f~(x(t), a)v(t) - Gqgo(t), v(1) -
-v(o),
fd v*(t)d/o(t)dt- 1
(16)
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(cf. Equation (15)). The functions r and 7r0 are selected to make (16) uniquely solvable, which is always possible. The solution component G of (16) defines a functional
which can serve as a test function for the flip bifurcation. Indeed, if G = 0 then the first equation in (16) reduces to the variational equation of the periodic solution x(t). The last equation in (16) normalizes the variational solution v(t), and the boundary condition v(1) = - v ( 0 ) corresponds to the multiplier # = - 1 of the Poincar6 map at the flip bifurcation.
4. Branch switching In this section we consider the computation of solution branches that emanate from certain bifurcation points. Specifically, we consider stationary solutions near simple branch points, and periodic solutions near Hopf and near period-doubling points.
4.1. The algebraic branching equation "]1~n + l ~ ~n be as in Equation (2). A solution Xo = X(so) of f ( X ) = 0 is called a simple singular point if fo _ f x (Xo) has rank n - 1. In the parameter formulation,
Let f
where f o _ ( f o I fo), we have that X0 - (x0, or0) is a simple singular point if, and only if, (i) dimA/'(f ~ - 1, f o 6 7~(fo), or (ii) dimA/'(f ~ = 2, f o ~ Tc(fo). Suppose we have a solution branch X (s) of f (X) = 0, where s is some parametrization. Let X0 ---- (x0, or0) be a simple singular point. Then we must have .A/'(f~ = Span{~b,, ~2},
A/(f ~
= Span{~r}.
From differentiating f ( X ( s ) ) = 0, we also have f o _ f ( X o ) - O, f ~ f o x XoXo + f o J(o -- 0. Thus 2(o = ot4~l + fl~2, for some or, fl 6 R l , and
O, and
lP*fOx (O/~bl -~- fl~b2)(ot~bl -~- fl~b2) -b ~*fOz~'O--0. Above, the second term is zero, and we can rewrite the equation as
Cl leg2 -+- 2Cl20tfl -+- C22fl 2 -- 0,
(17)
where l -
r
f~
,
f~
-
r
f~
Equation (17) is the algebraic branching equation (ABE); see Keller [42]. If the discriminant is positive, that is, if c22 - Cl lC22 > 0, then the ABE has two real nontrivial, linear
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independent solution pairs (or l, ill) and (ol2, f12), which are unique up to scaling. In such case we have a simple branch point, that is, two distinct branches pass through Xo. REMARK. Branch points are not a generic phenomenon in one-parameter solution families: they disappear under generic perturbations of the equation f (X) = 0, giving rise to nonintersecting branches. However, in systems with certain symmetries or invariant planes, branch points appear in a persistent manner.
4.2. Branch switching at simple branch points The direction vector of a bifurcating branch can be written Y0 = c~4~l + fl~b2, where (or, fl) is a root of the ABE (17). (Here we write "Yo" in order to distinguish this direction vector from the direction vector X0 of the "given" branch at the branch point.) We can take 4)1 = X0. Then (otl, fit ) -- (1,0) must be a root of the ABE and we must have cll -- 0. If A > 0 then also c12 # 0. The second root then satisfies 012/fl2 = -c22/2c12. To evaluate cl2 and C22 we need the nullvectors. Note that ~ is a simple nullvector of f o , . We already have
f~
chosen ~bl -- )~0. Choose 4)2 • ~bl. Then 4)2 is a nullvector of the matrix ()?~).
REMARK. Note that the above matrix is also the Jacobian of Keller's continuation system at X0. Its null space is indeed one-dimensional at a simple singular point. This implies that ~BP -- det
j~,
(18)
is a test function for singular points. This test function has a regular zero at simple branch points (for a proof, see Keller [43], or Kuznetsov [49]). Locating a branch point of the curve (2) using (18) might be difficult, since the domain of convergence for the Newton corrections (3) shrinks as one approaches such a point. This difficulty can be avoided by introducing (p, fl) E R 'l • •l and considering the extended system (see Moore [58] and Mei [56]): f (x, a) + tip - 0,
f,*(x,~)p - o , p* f~(x, a) --0, p ' p - 1--0.
(19)
A simple branch point (x0, c~0) corresponds to a regular solution (x, c~, fl, p) = (x0, or0, 0, p0) to (19). Therefore, the standard Newton method can be applied directly to (19) to locate a simple branch point. After determining the coefficients or2 and fi2, one must scale the direction vector Y0 = 0/2q91 + fl2~2 of the bifurcating branch so that IlY0ll- 1. The first solution X l on the bifurcating branch can now be computed from
f(Xl) =0,
(Xj - Xo)*I;'o- As = 0 .
(20)
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As initial approximation to X1 we can use X(l~ = X0 § As I;'0. For a graphical interpretation see Figure 3. This method is implemented in several codes, including STAFF (Borisyuk [10]), PITCON (Rheinboldt and Burkardt [63]), and C O N T E N T (Kuznetsov and Levitin [52]). Instead of Equations (20), one can also use f(Xl) =0,
(XI - Xo)*~b2 - As -- 0,
where (])2 is the second null vector of f o , with (])2 -[- q~l, t~l -- Xo. For a graphical interpretation see Figure 4. This method is implemented in AUTO and works very well in many applications, although there may be situations where it fails. Its advantage is that it does not require the computation of second order derivatives.
A s
Fig. 3. Switching branches using the correct branching direction.
~2
j m
_
As
Fig. 4. Switching branches using the orthogonal direction.
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4.3. Approximation of periodic solutions near a Hopf point Let (x0, or0) be a simple Hopf bifurcation point of Equation (1). By the Hopf Bifurcation Theorem, this ensures the existence of a bifurcating branch of periodic solutions. Moreover, one has the following asymptotic estimates for periodic solutions near the Hopf bifurcation: X(t" 8) -- XO 4- 8(/)(t) 4-
O(82),
T(e) -- TO 4- O(e2),
<,(E) - <*o + o ( . 2 ) .
Here e locally parametrizes the periodic solution branch. T (e) denotes the period, and To = 2re/co0. The function 4~(t) is the normalized nonzero periodic solution of the linearized, constant coefficient problem
4,'(t) - f.,~ To compute a first periodic solution (Xl, Tl, Otl)= (x, T, c~), near a Hopf bifurcation (x0, or0), we solve Equations (4), (5), (7), together with a modified version of the phase condition (6). An initial approximation for Newton's method is x(~ = xo 4- As4~(t), T (~ = To, ot (~ = o~0, where q$(t) is now a nonzero solution of the time-scaled, linearized equations
dp'(t) -- To f~~
4,(0) = 4,(1),
namely, 4~(t) = sin(2srt)ws + cos(2rct)wc, where (ws, Wc) is a null vector in
f,O
coo/,;
Wc
--
0
'
coO-- TO
(This nullspace is actually two-dimensional, with (-Wc, ws) being a second independent null vector.) For the phase equation we "align" x(t) with xo 4- ecb(t). Following the derivation that led to Equation (6), we obtain
fo
l x(t)* qb'(t) dt --0.
Finally, in the continuation equation (7) we have &0 - J"0 - 0.
4.4. Approximation of double-period solutions near a flip point Let xo(t) be a T0-periodic solution at ot = or0 with a simple Floquet multiplier # l = - 1 . Under certain genericity conditions, the Flip Bifurcation Theorem ensures the existence of a bifurcating branch of (approximately) double-period solutions to Equations (4)-(6) for
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nearby parameter values. Moreover, one has the following asymptotic estimates for these double-period solutions near the flip bifurcation:
x(t" 5`)- xo(2t) + e ~ ( 2 t ) + 0(5'2), =
T (5`) -- 2T0 + O(e2),
+
Here 5` locally parametrizes the branch of double-period orbits and r
_ ~ 4)(t), / -4)(t-
0~
1),
l~
where 4)(t) is the solution to
I ~'(t) - Tofx(Xo(t), o~)dp(t), 4~ ( ~ ) - -ep (o),
fo gr~)(t)~(t) dt - 1, that is, Equation (16) with G = 0. Since the flip point is a simple branch point for the second iterate of the Poincar6 map, one can also use the branch switching technique based on ABE described in Section 4.2.
5. C o n t i n u a t i o n o f c o d i m e n s i o n - 1 bifurcations
Suppose Equation (1) has a codimension- 1 equilibrium bifurcation at ot = a0. Generically, there is a curve ot -- or(s), with ot 6 ]R2 and s E IRl , along which the equation has an equilibrium with the given bifurcation. The bifurcation curve, say,/3, can be computed as a projection of a certain curve r in a space of larger dimension onto the c~-plane. Thus, we have to specify a continuation problem for F , that is, we shall define functions determining the curve in a certain higher-dimensional space.
5.1.
Continuation of codimension-1 equilibrium bifurcations
5.1.1. Minimally augmented systems. In this approach we simply append the relevant test function to the equilibrium equation, thus obtaining a system of n + 1 equations in the (n + 2)-dimensional vector of unknowns (x, a). More precisely, we have the following continuation problem
f (x, ~) - O,
(21)
det f~ (x, or) - 0, for the fold bifurcation, and the continuation problem { f ( x , o~) -- 0, det(2 fx (x, oe) 6) In) - 0,
(22)
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for the Hopf bifurcation, where fi) stands for the bialternate product defined in Section 1. Each system consists of n + 1 equations in n + 2 variables. It is called a minimally augmented system since it only has one extra equation, compared to the equilibrium equations. If a bifurcation point is detected while continuing an equilibrium and located as a zero of the corresponding test function ~pf or ~PH, then we have all the necessary initial data to start the continuation of the bifurcation curve defined by (21) or (22). The following theorem follows from the discussion in Section 3. THEOREM 5.1. The Jacobian matrices of the minimally augmented systems (21) and (22) have rank n + 1 at simple fold and simple Hopf points, respectively. Here "simple" means simplicity with respect to at least one parameter otl or or2. The defining system (21) has been implemented in LOCBIF (Knibnik et al. [46]). 5.1.2. Bordering techniques. As mentioned earlier, the defining systems (21) and (22) may have scaling problems. In addition it is generally not possible to express the Jacobian matrix explicitly in terms of the partial derivatives of f ( x , ~). Thus, we have to rely on numerical differentiation, even if the derivatives of f are known analytically. To overcome this difficulty, we replace the test function in (21) or (22) by a function g(x, ~) that vanishes together with the given test function but whose derivatives can be expressed analytically. We have already used this bordering technique in Section 3. The mathematical basis of the technique is provided by the following statement. THEOREM 5.2 (Govaerts and Pryce [33]). Let
M:
(A C*
D
be a nonsingular (n + m) x (n + m) block matr& with A ~ IKnX'', B, C E IK"xm, D ]Kmxm. Let the inverse M-I--(PR*
Q)S
be decomposed similarly. Let p <~min(n, m). Then A has rank deficiency p if and only if S has rank deficiency p. In the case where m << n and for small p this result is used to express that A has a desired rank deficiency. In the fold case, instead of (21), we introduce a modified minimally augmented system f ( x , o~) - O, g(x, a) = 0 ,
(23)
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where g = g(x, or) is computed as the last component of the solution vector to the (n + 1)dimensional bordered system:
(fx(x, Ot)qo, p o ) ( W g) o
= (0)1 '
(24)
for suitable vectors q0, P0 6 ]~n. If q0 is close to the nullvector of f~ (x, or) and p0 is close to the nullvector of fx* (x, or), then the matrix
M = ( f x (x~, Ot PO is nonsingular at (x, or) and (24) has a unique solution. In practical computations, q0 and P0 are the nullvectors of fx and fx*, respectively, at the previous point on the fold curve. For g = 0, system (24) implies
few--O,
qo w - 1.
Thus w is a scaled nullvector of fx (x, c~) and det fx (x, or) = 0 as in (21). The derivatives of g with respect to (x, or) can be computed by differentiating (24). Let z denote a component of x or c~. Then,
( and
00)t ) (0) w
L ( x , ~ )
q~
(wz, g=)T can (fx(x,
0
be found by solving the system
po ) ( Wz g= ) _ _ ( f,~:(x,0 W )ot
"
(25)
This system has the same matrix M as (24), while the right-hand side involves the known vector w and the derivative fxz of the Jacobian matrix fx. The derivative g: can be expressed explicitly if we introduce the solution (v, h)* to the transposed system
Multiplying (25) from the left by (v*, h) and taking into account that (v*, gives
h)M
= (0, 1),
gz = -v* fxz(X, Ot)w or
gx = -v* fxx(X, a)w,
ga = -v* La(x, ot)w.
(26)
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These equations can be used to independently verify that the Jacobian matrix of (23),
(.!; ..'-) is nonsingular at a simple fold point by applying Lemma 3.1. In the Hopf case, the modified minimally augmented system looks exactly like (23), with the function g = g(x, c~) now computed by solving the bordered system
This system is (m + 1)-dimensional, where 2m = n (n - 1), and is nonsingular if the vectors Qo, P0 6 R'" are the nullvectors of 2f~ Q L, and (2f~ E) LT)*, respectively, at a nearby generic point on the Hopf curve. The partial derivatives g: can be expressed in terms of f~-: as in the fold case. REMARK. Note that (27) is singular at a point where the equilibrium has two pairs of purely complex eigenvalues ("double-Hopf", see Section 7) regardless of the choice of vectors Q0 and P0. To overcome this difficulty, allowing the continuation to pass through such codimension-2 points, one can use double bordering. Instead of (27), double bordering uses the (m + 2)-dimensional system ( 2 f ~ . ( x ~ ) G I,,
0)(
Po
0
where Po, Qo are m x 2 matrices, selected to ensure regularity of the system (28), while W is a m x 2 matrix and G is a square 2 x 2 matrix. We then use g = det G as the defining function in (23). The system (23) with g computed via (24) and (28) is implemented in CONTENT.
5.2. Standard augmented systems 5.2.1. Folds. If one allows the dimension of the continuation space to be increased by more than one, then many more defining systems can be formulated for computing codimension- 1 bifurcation curves. For example, the following system of 2n + 1 scalar equations for the 2n + 2 components of (x, q, or), { f ( x , or) - 0, f~- (x, ol)q -- 0,
q*qo- 1--0,
(29)
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can be used to compute a fold bifurcation curve (Moore and Spence [60]). Here, q0 e ~;~n is a reference vector that is not orthogonal to the null-space of f t . In practical computations, q0 is usually the nullvector of f~ at the preceding point on the solution curve. The projection of the solution curve of (29) onto the parameter plane gives the fold curve/3. To start the continuation of a fold curve, we have to supply the nullvector q0 in addition to the critical equilibrium and the parameter values. THEOREM 5.3. Let (xo, oto) be a simple fold point of (1), and let qo denote a normal-
ized nullvector of f o = fx (xo, oto). Then the Jacobian matrix of the standard augmented system (29) has full rank, namely 2n + 1, at (xo, oto). PROOF. It is sufficient to show that the (2n + 1) • (2n + 1)-matrix
J
t :o o :o 1 fOrq o
fo
fOotqo
O*
qo*
0
is nonsingular, where ot now refers to the one-dimensional parameter that arises in Definition 3.2 of a simple fold. Suppose J is singular, i.e., (i) f ~ 1 7 6 (ii) f~
+ f~
+ wf~
= O,
(iii) q0 v = 0, for some nonzero u, v e R", w e R l . Multiplying (i) by the left nullvector P0 of f o , we get
:O - o . Using Property ($2) of a simple fold, it follows that w = 0 and hence
u = clqo,
cl E ~1.
From (ii) we get o Cl Po9 frxqoqo + Po, f Ox r _ 0
SO r
--
0 by Property (S 1) of the simple fold. Thus u = 0. From (ii) it now follows that f 0x v - - O ,
or v = r C2 E ]R 1 . But then, by (iii), r contradiction.
= 0. Thus, u = v = 0 and w = 0, which is a
[~
REMARK. The last equation in (29) may be replaced by the standard normalization condition q*q - 1 = 0.
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173
5.2.2. Hopfpoints. Next, consider the following system of 3n + 2 scalar equations for the 3n + 2 components consisting of (x, v, w, or) and co:
f (x,ol) = 0 , f ~ ( x , c~)v + cow = O,
(30)
f~ (x, a ) w - c o y =0,
v*vo + w * w o - 1 = 0 , v*wO -- v Ow -- O.
These equations are the real form of the complex system for (x, q, or, co) f ( x , oe) -- 0, (31)
fr (x, o~)q - icoq -- O, q*uo - 1 - - O,
that defines a necessary condition for Hopf bifurcation. Such systems were first introduced and analyzed by Griewank and Reddien [35]; see also Holodniok and Kubi~ek [41 ], Beyn [8], and Doedel et al. [24]. Here q = v + i w E C'; is the critical complex eigenvector and u0 = v0 + i w0 E C'; is a vector that is not orthogonal to the critical eigenvector corresponding to i co. As in the case of a fold, u0 is usually chosen as the critical eigenvector at the previously found solution point on the bifurcation curve. The projection of the solution curve F to (30) onto the (oel, otz)-plane defines the Hopf bifurcation boundary. To start the continuation of such a curve F from a Hopf point detected along an equilibrium curve, we also need to compute the Hopf frequency coo and the two real vectors v0 and w0. THEOREM 5.4. The Jacobian matrix of the augmented system (30) has full rank, namely
3n + 2, at a simple Hopf point (xo, do). PROOF. Let f o _ f~-(xo, ao) and
f~~
icooqo,
[f~~
-icoopo,
p ~ q o - 1.
It is sufficient to prove that the Jacobian matrix of (31) with respect to (x, q, or, co) evaluated at (x0, q0, c~0, coo), namely,
t
Zo
o
O*
u o*
f,Orqo f,o-icooI,,
I~
o t
fr~qo-iqo 0
,
0
has the trivial null-space. Here oe denotes the one-dimensional parameter in Definition 3.3 of a simple Hopf point. Note that the real form of J gives the square Jacobian matrix of (30). If J has a nontrivial null-space, then (i) f ~ + fifo __ O,
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(ii) fOxqoU + ( f o _ icooln)s + flfOaqo - i 6 q o = O, * = O, (iii) UoS for some u 6 IRn, s 6 C n, fl, 6 6 IR 1 with Ilull 2 + Ilsll 2 + t~2 + ~2 ~ 0. Denote by Xe(Ot) the continuation of x0 for small Iotl, Xe(~O) = xo. Then f ~ e -k- f g -~ O, and Equation (i) implies u = kxe(OtO),
fl = k,
for some k 6 IR l . Substituting these into (ii) and multiplying by p~ from the left, gives
k[pS~(oto)qo ] + p ~ ( f o _ icooln)s - i 6 p S q o -
O,
where, as in Definition 3.3, A(o~o) -
f O x x e ( O ) if- fOot.
It follows that k Re[p~i(ot0)q0] -- 0. Taking the Definition 3.3 of the simple H o p f point into account, we get k=0, so u = 0, fl = 0. Equation (ii) now takes the form
(fO _ icooIn)s - i6qo -- O, implying i6(pSqo ) - - 0 , which gives 6 - - 0 . From
(fO _ icooln)s = 0 we conclude that
s=zqo,
z ~ C I.
However, (iii) means z(uSqo ) - 0, and therefore z - 0, since u0 6 C n is not orthogonal to q0 by construction. We have u = 0, s = 0, t3 = 6 = 0, which is a contradiction. [2 REMARK. The second and third equations in (30) imply f 2 V .-}- O)2 v -- O.
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Thus w can be eliminated, thereby reducing the dimension of the augmented system (Roose and Hlava6ek [66]). This gives the following 2n + 2 equations in the 2n + 3 variables (x, v, a, K):
f (x, et) = 0 , [f,2(x,ot)+xl,;]v--O, v'v-
(32)
1 =0,
v*lo --0. Here the reference vector l0 6 R'; is chosen such that it is not orthogonal to the real twodimensional eigenspace of f~ corresponding to the eigenvalues )~l + )~2 = 0, ~1)~2 = x. A solution to (32) with x > 0 corresponds to a Hopf bifurcation point with 092 = K, while that with x < 0 correspond to a neutral saddle with two real eigenvalues ~ 1,2 - -+-~z--~. Unlike (30), the system (32) is also regular at a point where 0)2 = x = 0. Since [f~2(x, or) + xln] has rank defect 2 at points where f~,(x, or) has a pair of eigenvalues with )~l + )~2 = 0, )~1~2 = K, one can consider the following (n + 2)-dimensional system similar to (28): 0 0
At Hopf points, all elements of the (2 • 2)-matrix G vanish by Theorem 5.2. Therefore, the following defining system in the (x, or, x)-space specifies a Hopf curve in the c~-plane: { f(x,~)=0, gl l (x, or, K) --0,
(33)
g22 (x, or, x) -- 0, (cf. Werner [77]). Systems similar to (29) and (30) are used in AUTO, while the systems (29), (32), and (33) are implemented in CONTENT.
5.3. Continuation of codimension-1 bifurcations of periodic solutions Continuation of codimension- 1 bifurcations of periodic solutions to Equation (1), requires two problem parameters (or 6 R e) in addition to the period. This is a more delicate problem than the corresponding problem for equilibria. If the system is not very stiff then we can eliminate the period by computing the Poincar6 map and its Jacobian by numerical integration, and then apply continuation methods for fixed point bifurcations. This works satisfactorily in some cases; however, it fails if the periodic solution has very large or very small multipliers, which is often the case. In such situations, the BVP approach below is more reliable. 5.3.1. Folds. Recall Equations (4)-(6) for computing a periodic solution. We augment this BVP by the linearized, homogeneous equations
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Beyn et al.
I v'(t)- Tfx(X(t),~)v(t)--0-f(x(t),o~)--0, v(1) - v(O) = O, fo v* (t)~co(t) dt = O, where, as in Section 2.3, xo(t) denotes a reference solution. Normalize (v(.), 0-) by requiring
fo I v* (t)v(t) dt + 0-2 -- 1. This resulting augmented system can be used for the continuation of fold bifurcations for periodic solutions. It is to be solved for the functions x(t) and v(t) defined on [0, 1], scalar variables T and 0-, and two parameters otl and ot2. As in Section 2.3, the equations must be suitably discretized. 5.3.2. Period-doublings. tions (4)-(6) by
For the period-doubling (flip) bifurcation, we augment Equa-
I v ' ( t ) - Tfx(x(t),ot)v(t)=0, v(1) + v(0) = 0 ,
f) v* (t)v(t) dt - 1 - O,
dU
where the boundary condition v(1) = - v ( 0 ) expresses that the Jacobian matrix of the Poincar6 map has a multiplier # - - 1. After suitable discretization this augmented system can be used to compute flip bifurcation curves of (1). 5.3.3. Tori (Neimark-Sacker). For the continuation of the Neimark-Sacker bifurcation we introduce a complex eigenfunction w(t) and a scalar variable 0 parametrizing the critical multipliers # 1,2 = e +iO. The augmented system consists again of Equations (4)-(6), now augmented by
I w'(t) - T L (x(t), u)w(t) - O, w(1) -ei~ -0, fd w* (t)wo(t) dt - 1 - O, where wo(t) is a complex-valued reference function. Written in real variables, this augmented system can be discretized in order to continue generic Neimark-Sacker bifurcations. 5.3.4. Minimally augmented BVPs. The above standard augmented BVPs for continuing fold, flip, and Neimark-Sacker bifurcations of periodic solutions, involve double or triple the number of the differential equations in the underlying periodic solution continuation. It is also possible to derive minimally augmented BVPs to continue these bifurcations, using a bordering technique similar to that in the finite-dimensional case. We illustrate
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this approach for the flip bifurcation. The continuation of the flip bifurcation curve in two parameters can be reduced to the continuation of a solution of Equations (4)-(6) augmented by the single equation
G[u, T, ot] = 0,
(34)
where G is the test functional for the flip bifurcation defined in Section 3.4. The value of G is computed from the linear BVP for (v(.), G) with given bordering functions qgo, r and factor T
I v'(t) - Tf~(x(t),u)v(t) +Gqgo(t) --0, v(1) + v(O) = O, f l ~) (t)o(t) d t = 1.
(35)
The functions qg0 and ~0 are selected to make (35) uniquely solvable. Equation (34) is the flip bifurcation condition. To apply the standard continuation technique, we have to use suitable finite-dimensional approximations. It is also possible to efficiently compute the derivatives of G with respect to u, T, and or. A similar approach is applicable to the continuation of the fold and Neimark-Sacker bifurcations.
6. Continuation of codimension-1 homoclinic orbits
A heteroclinic solution x(t) connecting two equilibria x_ and x+ of an ODE system (1) satisfies lim x(t) = x_, t--+ --cx~
lim x(t) = x+.
(36)
t--+ + e ~
We shall concentrate on the special case where x+ = x_ - x0, called homoclinic solutions, as they have particular importance in global bifurcation theory. The approach is easily extended to the case where x+ =/=x_ provided a careful count is taken of the codimension of the connecting orbit, and the consequent number of free parameters ot required. For the case of a homoclinic orbit, their existence is of codimension- 1, given the following nondegeneracy condition. DEFINITION 6.1. A homoclinic orbit Xh (t) to a hyperbolic equilibrium of (1) at parameter value ot = or0, with ot 6 It~1, is called regular or nondegenerate if the linear variational equation
lim {y(t), y'(t)} exists,
/3 e R 1
t--->-+-2
has the unique solution (up to scalar multiples)/3 - 0 and y(t) - x'h (t).
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REMARK. The final condition of this definition can be expressed geometrically in terms of the stable and unstable manifolds W s (xo) and W" (x0) of x0 (see Section 8.2). Another possibility of codimension- 1 homoclinic connections is if A -- f x (x0, c~0) has a simple zero eigenvalue and all other eigenvalues are off the imaginary axis. Then one only requires the transverse intersection between the center-stable and center-unstable manifolds of x0 for a connection to occur. Since the sum of the dimensions of these manifolds is n + 1, such a connection will be of no extra codimension than that of a saddle-node equilibrium, i.e., codimension- 1. In order to define a regularity condition for such a connection, we need the notions of the stable and unstable sets B s (xo) and B u (xo) being the subsets of the center-stable and center-unstable manifolds composed of trajectories that tend to x0 as t --+ -+-oo, respectively. For precise definitions of these sets, see Deng [ 15] and references therein; Figure 5 illustrates the case n -- 2. DEFINITION 6.2. A homoclinic orbit Xh (t) to a saddle-node equilibrium x0 at parameter value c~o of (1), with c~ 6 ~ l, is called regular or n o n d e g e n e r a t e if A = f x (xo, or0) has nc -- 1 simple eigenvalues at zero, ns >~ 0 eigenvalues in the left-half plane and nu >~ 0 eigenvalues in the right-half plane (counting multiplicity), with nu + ns -- n - 1, and in addition the stable and unstable sets B s (xo) and B u (xo) intersect transversally along Xh (t), i.e., for each t 6 ]Rl , c o d i m ( T x h ( t ) B s (xo) + Txh(t)BU(xo)) = O.
(37)
REMARKS. (1) In (37), for both tangent spaces to be defined, it is implicit that x h ( t ) does not lie in the boundary of either B u (x0) or B s (xo), that is {Xh(t) I t ~ IR l } qs W s (xo) U W u (xo). Orbits which violate this condition we refer to as n o n - c e n t r a l s a d d l e - n o d e h o m o c l i n i c orbits (see Figure 5(b)). (2) Unlike Definition 6.1 above, we have not said anything about non-degeneracy with respect to the parameter, but this can be ensured by assuming that ot is well-chosen so that the equilibrium undergoes a simple fold bifurcation upon varying or.
Fig. 5. Illustrating (for n = 2, nc = 1, ns = 1, nu = 0), the differencebetween: (a) a codimension-1 central saddlenode homoclinic orbit, obeyingDefinition 6.2, and (b) a codimension-2 non-central saddle-node homoclinic orbit.
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Indirect methods for computing homoclinic orbits include numerical shooting (Hassard [39], Kuznetsov [48], Rodrfguez-Luis et al. [64]) and continuation of periodic solutions (as in Section 2) up to large period (Doedel and Kern6vez [19]). In this section we shall focus on a direct formulation as a two-point boundary-value problem which may then be solved using continuation methods, as outlined in Section 2 above, to trace out codimension-1 paths of homoclinic solutions in a two-parameter plane.
6.1. A truncated boundary-value problem Equation (1) subject to (36) defines a boundary-value problem on the real line, which cannot be solved directly for t e ( - c o , +co). There are two main approaches for defining problems on finite intervals. One is to use a different parametrization than time, say the arclength along the orbit, and the other is to truncate to t 9 ( - T _ , T+) for some suitably chosen T+ and approximate boundary conditions. We shall concentrate on the truncation method, but the interested reader is referred to Liu et al. [53,54], Moore [59], Bashir-Ali [5] for recent developments in arclength methods. Suppose now that xh (t) is a regular homoclinic orbit to the equilibrium x0 at parameter value or0. If x0 is hyperbolic, there exists a unique equilibrium Xe(O~) for all ot close to or0 such that Xe(Ot0) -- x0. If, on the other hand, x0 is a saddle-node equilibrium, we set Xe(OI) ~ XO.
Consider the following boundary-value problem on an infinite interval x' (t) = f (x(t), o~),
(38)
lim x ( t ) =Xe(Ot).
(39)
t---+-+-oo
Note that any homoclinic solution to the equilibrium Xe is a solution of (38)-(39). Since any time shift of the solution x ( t ) is still a solution, a condition is required to fix the phase. Suppose that some initial guess Z(t) for the solution is known. Then the following integral phase condition
f ~ (x(t) - $(t)) *$' (t) dt - 0
(40)
oo
is a necessary condition for a minimum of the L2-distance between x and s over time shifts (cf. Equation (6)). The boundary-value problem (38)-(40) defined on an infinite time-interval can be approximated by truncation to a finite interval [ - T _ , T+], with suitable boundary conditions as follows (Beyn [6,7]). Suppose that A(ot) = f~-(Xe(~), or) has ns eigenvalues (counting multiplicities) with negative real part, nc eigenvalues with zero real part, and n, eigenvalues with positive real part, so that ns + nc + n, = n. In the hyperbolic case, nc = 0, while if the equilibrium x0 is a saddle-node, one has nc = 1.
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Note that (39) can be cast as
x(-T_)
E BU(xe(ot)),
x(T+) E BS(xe(Ol)).
Linearizing this condition about the equilibrium Xe(Ot), w e obtain the projection boundary conditions
Ls(ol)(x(-T_)
- Xe(Ot)) =0,
Lu(Ol)(x(T_I_ ) - Xe(Ol)) --0,
(41) (42)
which replace (39). Here Ls (or) is a ns x n matrix whose rows form a basis for the stable eigenspace of A*(ot). Accordingly, Lu(ot) is a n , x n matrix such that its rows form a basis for the unstable eigenspace of A*(ot). The boundary conditions (41) and (42) place the solution at the two end points in the center-unstable and center-stable eigenspaces of A (or), respectively. Finally, take the phase condition of the truncated problem to be
f T~(x(t) - Yc(t))*Yc'(t)dt = 0 .
(43)
It is not difficult to see that the truncated problem (38), (41)-(43) is a formally wellposed codimension-1 problem, when x0 is hyperbolic, since one has n boundary conditions (41) and (42) plus one integral constraint (43). More precisely, we have the following result. THEOREM 6.1 (Beyn [7], Schecter [72], Sandstede [70]). Let Xh(t) be a regular homoclinic orbit o f (38) to the equilibrium xo at ot = oto. Suppose 2 ( t ) is such that (40) is satisfied with x ( t ) = xh(t), where
f ~ x'h*(t)'2' (t) dt r O. oo
Then there exist constants p, C, T, > 0 such that, f o r any T_, T+ > T,, there exists a solution (2, 6t) to the truncated boundary-value problem (38), (41)-(43) that is unique in the ball
{(x,~) ~ C~([-T-, T+],R ") x R ~" IIx --xhlt-T_.T+]II~ + Iot--OtOI ~
Ce
(44)
[~ -- ot0[ <~ C e -min{(2x"+lUI)T-'(21xsl+~")T+} Here ]]. Ill denotes the usual Cl-norm, and Re{U} < )s < 0 and 0 < )J~ < Re{~U}, where )~, i = 1 . . . . . ns, and I. bl i , i - 1 . . . . . nu, are the stable and unstable eigenvalues
of L (xo, ~o).
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If x0 is a saddle-node, then (41) and (42) give only (n - 1) boundary conditions. We then have the following theorem. THEOREM 6.2 (Schecter [71], Sandstede [70]). Let xh(t) be a regular homoclinic orbit o f
(38) to the saddle-node equilibrium xo at ot - o~o. Suppose ~(t) is such that (40) is satisfied with x ( t ) - xh(t), where
f ~ x'h*(t)~' (t) dz # o. OG
Then there exist constants p, C, T, > 0 such that, f o r any T_, T+ > T,, there exists a solution s to the truncated boundary-value problem (38), (41)-(43) with o~ - o~ofixed that is unique in the ball
{x ~ c~ (L-T_, T+], IRn)" IIx --xhl[-~,T§
~ P(T -~ +
T~)},
and satisfies the error estimate
I,
+
PROOF. The statements follow readily from the proofs of Schecter [72, Theorem 2.1 ] and Sandstede [70, Theorem 3.1 ], where non-central homoclinic orbits to saddle-node equilibria have been addressed. D In the case that the saddle-node homoclinic solution is continued in two parameters, we simultaneously need to compute the curve of saddle-node equilibria. This issue has been addressed in Section 5. For numerical experiments, we refer to Friedman [29] and Bai and Champneys [4].
6.2. Implementation details The above codimension-1 truncated boundary-value problems for regular homoclinic orbits or central saddle-node homoclinic orbits can be solved using standard boundary-value codes. For example, a particular implementation is available in AUTO (Doedel et al. [20]), using a suite of routines for homoclinic continuation called HOMCONT (Champneys et al. [12]). Here one can compute codimension-1 curves of homoclinic orbits with two free parameters, detect various codimension-2 points along such branches (see Section 8 below) and switch to continuation of higher codimension homoclinic bifurcations in three or more parameters. Some of the numerical issues involved in the continuation of solutions to such boundary-value problems include: efficient computation, ensuring smoothness with respect to or, of the boundary conditions (41), (42); the choice of T_ and T+; and the accurate determination of starting solutions for two-parameter continuation. The evaluation of (41), (42) requires a method for obtaining robust bases for the stable and unstable subspaces of A(a)*. A good choice is to use the Schur factorization of A*,
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as in Champneys et al. [ 12] and Doedel et al. [22]. In the latter reference, smoothness with respect to c~ has been achieved by adding the coefficients of this factorization, subject to various defining equations, to the list of unknowns to be solved for by the continuation algorithm. This is likely to lead to little extra computational work provided n 2 is small compared to the size of the linear systems to be solved by the BVP-solver. A simpler but less robust method is to perform the Schur decomposition exactly at each continuation step using blackbox linear algebra methods; see Champneys et al. [12]. An additional step is then made to normalize the stable and unstable subspaces in order to make them approximately smooth with respect to ot (Beyn [7, Appendix C]). The choice of T_ and T+ can be made adaptively during the continuation process using the error estimate (44) to achieve some desired tolerance, assuming the BVP to be solved exactly (Beyn [7, Section 4]). Starting points for homoclinic orbit continuation may be periodic solutions computed to large period (Doedel and Kern6vez [19]) or small amplitude solutions constructed near certain local codimension-2 bifurcations such as Bogdanov-Takens points (see Section 11.2.2). If neither of the above is available, a careful homotopy technique may be used to successively continue a small piece of the unstable manifold of x0 at a parameter value away from the true homoclinic orbit, into a full solution of the truncated boundary-value problem. An account of this latter method is beyond the scope of this Handbook, but the interested reader is referred to Doedel et al. [22] for the theory and Doedel et al. [21,23] for some applications.
7. Locating codimension-2 equilibrium bifurcations When investigating a two-parameter problem, one usually encounters higher-order degeneracies along codimension-1 bifurcation curves. Some of these degeneracies are determined by the Jacobian matrix, while others can only be detected taking into account nonlinear terms. For this reason we start this section with the nonlinear normal forms for codimension-1 equilibrium bifurcations, namely the fold and Hopf. Appropriate coefficients in these normal forms play the role of test functions for detecting codimension-2 bifurcations. Codimension-2 equilibrium bifurcations are important, as they serve as organizing centers, from which several curves of codimension-1 bifurcations can emanate. For example, the Bogdanov-Takens (BT) point, discussed in this section, gives rise to curves of Hopf points, folds, and homoclinic orbits. The switching to such codimension-1 curves from a codimension-2 point is discussed in Section 11.
7.1. N o r m a l f o r m s f o r c o d i m e n s i o n - 1 b i f u r c a t i o n s Suppose (1) has an equilibrium x = x0 at ot = or0, where ot e R. Let F ( x ) = f ( x , oto) represent the multivariate Taylor series 1 F(x)
1
-- A ( x - xo) + -~ B ( x - xo, x - xo) + -~ C ( x - xo, x - xo, x - xo)
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1 + ~ D ( x - xo, x - xo, x - xo, x - xo) 1
-Jr-~
E(x-xo,
x-xo, x-xo, x-xo, x-xo)+O(llx-xoll6),
(45)
where A = f ox ,
B ( p , q) -- f ox x P q ,
C ( p , q, z) - f,,~
To analyze codimension-1 bifurcations we need to take into account the linear, quadratic, and cubic terms. We also introduce the following multilinear terms of order 4 and 5 here, as they will be needed in Section 10. D ( p , q, z, v)
-
-
f~)(4)pqzv,
E ( p , q, z, v, w) -- fA~
where p, q, z, v, w 6 R". The dependence of A, B, C, D and E on (x0, c~0) is not indicated to simplify notation. Assume further that x0 = 0, or0 = 0. If the solution x -- 0, ot -- 0 of (1) corresponds to a f o l d bifurcation, then the Jacobian matrix A has a simple zero eigenvalue )~l - 0 and no other critical eigenvalues. Let Aq =0,
A*p --0,
be the nullvectors, normalized according to (p,q) = (q,q)=
1.
LEMMA 7.1. The restriction o f (1) at ~ = 0 to the o n e - d i m e n s i o n a l center m a n i f o l d W c has the f o r m w ' - a w e + O( Iw 13),
w~R,
where the coefficient a can be c o m p u t e d by the f o r m u l a
l(p
a = -~
,
B(q q))= 1 , o , -~ P f ~ x q q .
(46)
If we have a simple fold with respect to the parameter ot then the restriction of (1) to the (parameter-dependent) center manifold is locally topologically equivalent to the normal form
W' = fl -+-aw 2, where/~ is the unfolding parameter. This normal form predicts the collision of two equilibria when the parameter/~ passes through zero. Note that the unfolding parameter/~ can be expressed in terms of the original parameter ot as
= (p, f (0, (see Section 10).
+
:)
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If the equilibrium x -- 0 of (1) has a H o p f bifurcation at ot = 0, then the Jacobian matrix A -- f~ (0, 0) has a simple pair of purely imaginary eigenvalues )~1,2 -- +i coo, coo > 0, and no other critical eigenvalues. Introduce two complex vectors Aq = icooq,
A* p -- -icoop,
and normalize them according to (p,q) = 1. LEMMA 7.2. The restriction of (1) at ot = 0 to the two-dimensional center manifold is locally smoothly orbitally equivalent to the complex normal form w' -- icoow -+-II wlwl 2 -+- O(Iwl4),
w 9 C 1,
where the normal form coefficient I i can be computed by the formula
11 =~1 Re(p, C(q . q ~ ) + .B(-~, (2icooln . - A) . - 1 B ( q . q)) - 2 B ( q A -I B(q -~))). (47) If the Hopf point is simple and its first Lyapunov coefficient 11 7~ 0, then the restriction of (1) to the (parameter-dependent) center manifold is locally topologically equivalent to the normal form w' -- (~ + icoo)w + It wlwl 2.
This normal form describes a bifurcation of a unique periodic solution branch from the equilibrium w = 0, when the parameter fl passes through the bifurcation value 13 = 0. The direction of the bifurcation is determined by the sign of l l. The unfolding parameter fl has the following asymptotic representation: 13 -- [Re(p, ii (0)q)]or + O(ot2), where, as in Definition 3.3, d
f~ ( Ol) -- - ~ f x (Xe(Of), Or),
and Xe(Ol ) is the continuation of the equilibrium for small Ic~l, Xe(O) -- O. The formulas (46) and (47) were derived using the center-manifold reduction and subsequent normalization on the center manifold in Kuznetsov [49]. Originally, an expression equivalent to (47) had been obtained by Lyapunov-Schmidt reduction and asymptotic expansions for the bifurcating periodic solution by Kopell and Howard in Marsden and McCracken [55] and by van Gils [76]. These formulas will be rederived in Section 10 below. The first algorithms to determine numerically the direction of Hopf bifurcation were developed by Hassard et al. [40] and implemented into the code B IFOR2.
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7.2. Locating codimension-2 bifurcations 7.2.1. Codimension-2 points on the fold curve. While tracking a fold curve one can encounter the following singularities. (1) An additional real eigenvalue )~2 meets the imaginary axis, with their geometric multiplicity remaining one, while the center manifold W c becomes two-dimensional: )~1,2 = 0 . These are the conditions for the Bogdanov-Takens (or double-zero) bifurcation. A test function to detect this bifurcation is given by OBT = r ' q ,
(48)
where
A q = A*r=O,
q*q = r * r = l.
Indeed, if grBT # 0, then the zero eigenvalue is simple. If the standard augmented system (29) or the modified minimally augmented system (23), (24) is used to compute the fold branch, then the normalized nullvector q is known from the continuation. The function (48) has a regular zero at a generic Bogdanov-Takens point. (2) Two extra non-real eigenvalues )~2,3 meet the imaginary axis, and W C becomes three-dimensional: )~1 = 0,
~2.3 = -+-i~o0,
for o)0 > 0. These conditions correspond to the fold-Hopfbifurcation, also known as the Gavrilov-Guckenheimer bifurcation. The Hopf-bifurcation test function ~H based on the bialternate product (see Equation (11)) can be used to detect this singularity. It is regular at a generic fold-Hopf point. However, 7rH will also vanish at Bogdanov-Takens points. (3) The eigenvalue )~l - 0 remains simple and the only one on the imaginary axis (dim W C = 1), but the normal form coefficient a in Equation (46) vanishes: )~l = 0 ,
a =0.
These are the conditions for a cusp bifurcation. One cannot detect this bifurcation by looking at eigenvalues of the equilibrium, because quadratic terms of f (x, 0) are needed to compute a. The coefficient a can be used as a test function to detect this bifurcation: ~cp = a .
(49)
Again, if the standard or the modified minimally augmented system is used to compute the fold branch, the nullvector q is known from the continuation. The function (49) has a regular zero at a generic cusp point.
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7.2.2. Codimension-2 points on the H o p f curve. While following a Hopf bifurcation curve for Equation (1), one can encounter the following new singular points: (4) Two extra non-real eigenvalues )~3,4 m e e t the imaginary axis and W c becomes fourdimensional: XI,2 - -4-icol,
X3,4 - - -4-ico2,
with o91,2 > 0. These conditions define the two-pair or double-Hopfbifurcation. This bifurcation is most easily detectable if the double-bordered bialternate-product system (28) is used for the Hopf continuation. At this bifurcation, 2A (3 In has rank defect 2, so that all elements of the matrix G vanish. For example, the test function ~ D H = g22
will have a regular zero at a generic double-Hopf point. (5) Finally, the first Lyapunov coefficient l l (Equation (47)) may vanish, while X 1.2 = +ico0 remain simple and therefore dim W e = 2: )~1,2 ~-~ +ioa0,
11 = 0.
At this point, a "soft" Hopf bifurcation turns into a "hard" one (or vice versa). It is often called a generalized Hopfbifurcation (or Bautin bifurcation). The test function is given by lpG H "-- ll.
The Bogdanov-Takens bifurcation can also be located along a Hopf bifurcation curve defined by the augmented system (32) (or (33)) as x passes zero. At this point, two purely imaginary eigenvalues collide and we have a double zero eigenvalue. Following the curve further, we will continue a neutral saddle equilibrium with real eigenvalues X l = -X2. The value of tc can be calculated if the defining system (27) (or (28)) is used to continue the Hopf curve. One has in that case (v*Av)(w*Aw) - (w*Av)(v*Aw) K
-(V*V)(W*tO)
--
(V'W) 2
where v, w 6 ~;~n are two real vectors such that Q = v A w, where A denotes the wedge product, and Q is a right nullvector of 2A (3 In. We recall that the wedge product v A w of two vectors in R n is a vector in R n(n-l~/2 indexed by pairs (i, j ) where 1 ~< j < i ~< n such that (v A w ) ( i , j ) = l ) j w i -- l ) i w j . In the present case v, and w span the two-dimensional eigenspace that corresponds to the pair of eigenvalues with zero sum. This is an invariant subspace of A. An easy computation shows that ~ 2 ( = / r is the product of the two zerosum eigenvalues. A fold-Hopf bifurcation can also occur while tracing a Hopf bifurcation curve. In this case it can be detected as a regular zero of Of = det A.
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REMARK. In order to be able to use the above test functions to detect codimension-2 points, the underlying defining systems for the codimension-1 bifurcations must remain regular at the codimension-2 points. Otherwise, the continuation algorithms may not be able to pass such a point. The following two lemmas provide necessary information. LEMMA 7.3. If (x, c~) is a point corresponding to any generic codimension-2 equilibrium bifurcation of (1), except the double-Hopf singularity, then rank J = n § 1, where J is the Jacobian matrix of the corresponding minimally augmented system (21) or (22). A generic double-Hopf point is a simple branch point for (22). LEMMA 7.4. The Jacobian matrix of the augmented system (29) has rank 2n + 1 at generic Bogdanov-Takens and cusp bifurcation points of (1), while the Jacobian matrix of the augmented system (30) has rank 3n + 2 at generic Bautin, fold-Hopf and doubleHopf bifurcation points of (1). The first interactive software to detect all codimension-2 points was LOCBIF (Khibnik [45], Khibnik et al. [46]). Detection of all codimension-2 points as described above is implemented in CONTENT (Kuznetsov and Levitin [52]).
8. Locating codimension-2 homoclinic bifurcations In Section 6 we considered numerical methods for the two-parameter continuation of homoclinic orbits to equilibria. Suppose that we continue a branch of regular codimension-1 homoclinic orbits to (1) in two parameters, i.e., ot E R 2, so that a homoclinic loop to the equilibrium xe(s) exist whenever ot = or(s) with s 6 ~1; see Section 2.2 and Section 6. Here the one-dimensional parameter s is typically Keller's pseudo-arclength. We refer to these homoclinic solutions as the primary homoclinic orbits. Along this primary branch, codimension-2 homoclinic bifurcation points may arise. Such bifurcations may, for instance, lead to more complicated homoclinic connections such as so-called n-homoclinic orbits which follow the primary homoclinic loop n times. Another possibility is that the stability of the periodic solutions which accompany the primary homoclinic orbit changes. The issue addressed in this section is the detection of such codimension-2 homoclinic bifurcation points. We shall focus only on those known codimension-2 bifurcations that, at the critical parameter value, involve a unique finite-amplitude homoclinic orbit. Also, we confine ourselves to numerics. Details of the dynamics near each codimension-2 point are given elsewhere in the Handbook; see also the review papers: Fiedler [28], Champneys and Kuznetsov [ 11 ]. Codimension-2 homoclinic bifurcation points are detected along the primary branch ot = or(s) by locating zeroes of certain test functions; see Section 3 for the concept of test functions. The issue of defining these test functions is actually two-fold. First, consider the primary codimension-1 homoclinic orbits to the original problem (38)-(40) on the infinite time interval. A test function for a certain codimension-2 homoclinic bifurcation is a smooth function defined along the primary branch such that its regular zeroes correspond to the occurrence of the bifurcation. Afterwards, we need to define test functions for the
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truncated boundary-value problem (38), (41)-(43) on the finite time interval ( - T _ , T+). We require that each such test function is a smooth function along the branch of primary homoclinic orbits to the truncated problem such that the limit of the test function exists as T_, T+ --+ oo, and is equal to the test function of the original problem on the infinite time interval. We call such test functions well-defined. In the simplest cases, test functions are computable via eigenvalues of the equilibrium. In other cases, the homoclinic solution at the endpoints or solutions to the adjoint variational equation with appropriate boundary conditions are utilized. We address these two different types of test functions in the following two sections. Test functions along branches of central saddle-node homoclinic orbits are considered in the last section. The stable and unstable eigenvalues of A - f x ( X e ( S ) , a ( s ) ) are denoted by ~.~, i -ll 1 . . . . . ns, and )~i, i = 1. . . . . nu. In addition, if a branch of central saddle-node homoclinic orbits is computed, we have nc = 1, i.e., there is a simple eigenvalue ,kC1 -- 0 at zero. We assume that the eigenvalues are ordered so that Rel1.,s } ~<... ~< Re{)~i } < 0 < Rel)v71 ~<... ~< Rel~.,u, }.
(50)
The stable (unstable) eigenvalues with real part closest to zero are termed the leading stable (unstable) eigenvalues.
8.1. Test functions utilizing eigenvalues Let nc - - 0 so that Xe(S) is hyperbolic. The following test functions have been shown in Champneys and Kuznetsov [ 11 ] to be well-defined: Resonant saddle: S
ll
7t - ~ j + )v1 .
Neutral saddle, saddle-focus or bi-focus:
7* -- Re{,k{ } + Re{X]' ]. Double real stable leading eigenvalue:
{(Re{)v]}-Re{)v{})2, Im{~.~}--0, gr --
(Im{~.~} - Im{~.~ })2,
(51)
Im{)~)} --/=0.
A test function for double real unstable leading eigenvalues is obtained by replacing ~s. J with 1.j in (51). We remark that zeroes of (51) are regular at generic double eigenvalues since the expressions in (51) represent the discriminant of the quadratic factor of the characteristic polynomial corresponding to this pair of eigenvalues. Transitions to non-hyperbolic equilibria are detected in the following fashion. First, the truncated problem should be formulated in such a way that the branch of homoclinic orbits
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s-O
189
s>O
Fig. 6. Continuation through a Shilnikov-Hopf bifurcation arising at s = 0. The homoclinic orbit exists for s ~<0. For s > 0, the solution of the truncated problem is a connection between the equilibrium and a periodic solution with small amplitude. can be continued through parameter values at which eigenvalues cross the imaginary axis. Secondly, the labeling (50) has to be modified: the ns leftmost and the nu rightmost eigenvalues are labeled )v~. and )~." respectively, without regard to the sign of their real part. The l l ' test functions detecting fold and Hopf bifurcations are then defined as follows: Non-hyperbolic equilibria (crossing of stable or unstable eigenvalues): gr -- Re{)~{ },
gr -- Re{~./i' }.
We comment further on the first condition that the branch of homoclinic solutions to the truncated problem can be continued through the bifurcation point. For the Shilnikov-Hopf bifurcation, where the equilibrium undergoes a Hopf bifurcation at, say, s = 0, there exists a solution to the truncated boundary-value problem (38), (41)-(43) beyond the codimension-2 bifurcation point. This solution corresponds to a heteroclinic connection between the equilibrium Xe and the bifurcating periodic solution of small-amplitude; see Figure 6 for an illustration. We refer to Champneys and Kuznetsov [ 11 ] for more details. A similar situation arises if we continue along a primary branch of homoclinic solutions towards a fold bifurcation. This scenario is explained in more detail in Section 8.3.
8.2. Test functions for homoclinic flip bifurcations Homoclinic flip bifurcations are related to a discontinuous change of the exponential rate of convergence of either the homoclinic orbit itself or a certain solution to the adjoint variational equation along the homoclinic loop; see below. Let nc -- 0 so that Xe(S) is hyperbolic and assume that the leading eigenvalues are simple and real, i.e., }
< <
< o <
<
}.
Hence, the matrix A - f r ( x e ( s ) , or(s)) has normalized eigenvectors v~ and v'[ associated with the eigenvalues )c~l and )~'(, respectively. Analogously, the adjoint matrix A* has nor-
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190
(b)
WS(xe)
Fig. 7. Illustrations of a homoclinic orbit at (a) an orbit-flip and (b) an inclination-flip bifurcation both occurring in the stable manifold. The dotted vectors shown in the right-hand picture are the solution Yh (t) _1_ W s (Xe) of the adjoint variational equation.
malized eigenvectors w~ and w/{ belonging to X~ and X/[. These quantities depend smoothly upon s. In this situation, a codimension-1 homoclinic orbit is expected to converge to zero exponentially with rates Xsl and Xul as t ~ 4-oo respectively. An orbit-flip bifurcation occurs if the homoclinic orbit picks up a faster exponential rate, i.e., if it is contained in either the strong stable or the strong unstable manifold of the equilibrium; see Figure 7(a). If the homoclinic solution is contained in the strong stable manifold, we have ,~e~lim e
_X s It (x(t)
- X e ) * 1131s _ 0,
see Sandstede [69]. The following test function detects this codimension-2 bifurcation for the truncated problem. Orbit-flip (in the stable manifold):
~ -- e-X~ T+ (x(T+) -- Xe)* WlS. The exponential factor guarantees that the test function converges to the test function of the original problem; without it, this limit would be identically equal to zero. Analogously, the test function Orbit-flip (in the unstable manifold):
~P -- eX'~T- ( x ( - - T - ) -- Xe)*W~ detects homoclinic orbits in the strong unstable manifold. Next, we focus on inclination-flip bifurcations. For any regular homoclinic orbit xh (t) to a hyperbolic equilibrium Xe, the adjoint variational equation
y' (t) -- -- f * (Xh (t), oto) y(t)
(52)
has a unique bounded solution yh(t), which in fact converges to zero exponentially. This solution has the following geometric interpretation. Since the homoclinic orbit Xh(t) is
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191
regular, we have that T~hr W s (Xe) 0 Zxh(t ) W u (Xe) is one-dimensional by Definition 6.1. In particular, the codimension of Txh(t) WS (Xe) + Txh(t) W u (Xe) is one. The aforementioned solution Yh (t) to (52) satisfies
yh(t) _L
(T~h(t)WS(Xe) Jr- Txh(t ) W u (Xe))
for all t. In analogy to the situation for homoclinic orbits, yh(t) is expected to decay exponentially with rates U{ and )~ as t --+ -+-cx~. An inclination-flip bifurcation occurs if the solution Yh (t) converges to zero with a higher exponential rate. We refer to Figure 7(b) for an illustration. In order to detect inclination flips, we have to calculate the solution yh(t). Note that yh(t) can be regarded as a homoclinic orbit to (52). We can then follow the procedure outlined in Section 6 in order to derive an appropriate truncated boundary-value problem. Since (52) is non-autonomous, we do not need a phase condition. It is replaced by a condition which fixes the amplitude of YB(t); note that (52) is linear. For the boundary conditions, let L* (or) be the ns • n matrix whose rows form a basis for the stable eigenspace of fr(Xe(Ot),Ol). Similarly, L*(ot) is a n,, • n matrix such that its rows form a basis for the unstable eigenspace of f~(xe(O~), ~). The truncated boundary-value problem for the computation of Yh (t) is then given by
y'(t) + f~*(x(t),ot)y(t) + ef(x(t),c~) =0, L* (ot)y(T+) --0, Lt* (ot)y(-T_) = 0 , f_~+ (y(t) - y(t))* y(t) dt -- O, where x =
x(t)
denotes the homoclinic solution, and ~(t) is such that
f ~+ y~(t)~(t)dt ~0. The truncated system is then solved with respect to (y, e) for a given function x(t) at the parameter value or. Here, E e R 1 is an artificial parameter, which makes the boundary-value problem well-posed. We refer to Champneys et al. [12] for more details. Analogously to the case of an orbit-flip bifurcation for the homoclinic orbit, we then obtain the following test function for inclination-flip bifurcations. Inclination-flip (of the stable manifold): s
S
7r--e-Z~T- y*(_T_)vi. Inclination-flip 7r
-
-
(of the unstable manifold):
e) ll' T+y ~< (T+)vill "
It can be shown that these test functions are well defined.
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192
8.3. Test functions detecting non-central homoclinic orbits Suppose that a branch of central homoclinic orbits is continued in two parameters. Recall that in this case the truncated boundary-value problem is c o m p o s e d of (38), (41)-(43) plus conditions for the continuation of the saddle-node equilibrium; see Section 5 for the latter. One is then interested in detecting a transition to non-central homoclinic orbits. Note that, by assumption, the equilibrium Xe(S) has precisely one central eigenvalue )~lC = 0, i.e., nc = 1. Let w 1c be a normalized eigenvector corresponding to ~1C = 0 of the adjoint matrix A* = f*(xe(s),~(s)). The following test functions will then detect noncentral saddle-node homoclinic orbits. Non-central saddle-node homoclinic orbit (in center-stable and center-unstable manifold): o =
1 (x(T+) - Xe)*wCn
~ = T-~-_I( x ( - T _ ) - Xe)*~Cl
These functions measure the spectral projection of the end points of the approximate h o m o clinic orbit onto the one-dimensional center space. Note that, for the original p r o b l e m on the infinite time interval, the branch SNH of central homoclinic orbits cannot be continued b e y o n d the point c~ -- 0 since no homoclinic loop exists along the branch SN; see Figure 8. For ( - T _ , T+) = R1, the above test functions are defined on account of Schecter [71, L e m m a 3.1 ], are smooth along SNH including the point ot -- 0, and have a smooth (artificial) extension along the curve SN. For finite time intervals, the test functions are also smooth and they converge to the test functions of the original problem as T_, T+ --+ c~
C . l<
~
o
H
l~
YT
Oll
SN',
A I
a=0 SNH
SN
Fig. 8. Continuation through a fold bifurcation arising at ot = 0. If a branch of central homoclinic orbits is continued, we have ot ~ SNH. The algorithm will then continue through the fold bifurcation by computing solutions along the curve SN. On the other hand, if the branch H of homoclinic orbits to hyperbolic equilibria is computed, the algorithm continues along the branch C.
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along the branch SNH including ot = 0. Note, however, that the artificial solution of the truncated problem along the curve SN shown in Figure 8 has no limit as T_ --+ cxz. Finally, suppose that we continue along a primary branch H of homoclinic solutions to hyperbolic equilibria towards a fold bifurcation. At the fold bifurcation, another equilibrium is created. To be definite, we suppose that an unstable eigenvalue approaches zero; see Figure 8. Beyond the bifurcation point, the solution to the truncated boundary-value problem is then a heteroclinic orbit connecting the newly created equilibrium to the original equilibrium. In other words, the algorithms continues on the branch C shown in Figure 8; see Champneys et al. [ 12] for more details.
9. Continuation of codimension-2 equilibrium bifurcations In this section we give regular defining systems based on bordering techniques for continuing codimension-2 equilibrium bifurcations of (1) in three parameters. Detailed proofs of regularity can be found in Govaerts [32]. Test functions to detect codimension-3 bifurcations due to linear terms are also given. All defining and test functions are implemented in CONTENT (Kuznetsov and Levitin [52]). Earlier methods based on minimally augmented systems with determinants were proposed by Khibnik [45] and implemented in LOCBIF (Khibnik et al. [46]).
9.1. Bogdanov-Takens For computational purposes a Bogdanov-Takens point is characterized by the fact that the Jacobian matrix A -- f~(x, ~) has a double eigenvalue zero with geometric multiplicity one and no other eigenvalues on the imaginary axis (the dependence on (x, or) will be suppressed in the following). In particular, the characteristic polynomial p0~) = det(A )~I,,) satisfies
p(0) = 0, pz(0) = 0 ,
(53)
and A has rank defect 1. Hence there exist vectors Vl, Wl 6 IR'~ such that
M(X) _ ( A - XLv,l,
Wl)0
is nonsingular in a neighborhood of)~ -- 0. If we define v0~) ~ ITS"and g 0 0 6 R by solving
(:) (0;,) then
g(Z) =
p(,b det MOO
(54)
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194
Differentiating (54) with respect to )~ we obtain
g)~
=
0
g~x
(55)
'
(56)
o
0
The defining equations for the Bogdanov-Takens (BT) curve are the equilibrium equation (2) supplemented by g(0) -- 0, gz (0) - 0. These two equations are mathematically equivalent to (53), but they are better scaled and their derivatives can be computed more easily. In practical computations Vl and wl are chosen as approximations to the normalized right and left nullvectors of the local Jacobian matrix A. On a BT-curve, one can encounter the following singularities of certain linear terms: (1) Triple zero eigenvalue: 7el = 0. (2) Hopf-BT: lp2 -- 0. Here Tel -- gzz, ~2 = g22, where gzz is obtained from (56). The construction of g22 is more complicated and requires auxiliary data vlb, vzb, wlb, wzb E ~m, 2m -- n(n -- 1), and scalars d12, d2! such that the matrix 2A @ L7
wlb
v*lb
0
* 132b
d21
mb =
tO2b) d12 0
is nonsingular. Then g22 is computed by solving V
,2),
where
G_(gll g21
g12). g22
Numericalcontinuation,and computationof normalforms The vectors Vlb, l/)lb are chosen as normalized right and left nullvectors of 2A (3 spectively, initialized and updated essentially like Vl and Wl. Finally, the vectors
195
In, re-
(d;bl) ' (dl2ff) are updated jointly to make Mb as well-conditioned as possible.
9.2.
Fold-Hopf
A fold-Hopf point is characterized by the fact that the Jacobian matrix A = f~ (x, or) has an algebraically simple eigenvalue zero, a pair of pure imaginary algebraically simple eigenvalues -+-ico0, coo > 0, and no other eigenvalues on the imaginary axis. This implies that there exist vectors vl, 1/)1 E •n, Ulb, U2b, llOlb, l102b E I[4~m, 2m = n(n - 1), and scalars d12, d2! such that the matrices
f and
Mb
t 2 A Q L, Wlb web t * lJlb
0
d12
* 132b
dzl
0
are nonsingular. The defining equations for the fold-Hopf curve are the equilibrium equation (2) and
det G -- 0, where the scalar g results from solving
(o,) and the 2 • 2 matrix
G - ( gllg21 g22g12)
(57)
is obtained by solving the system
v
/2 )"
(58)
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The bordering rows and columns in M and Mb are initialized and updated essentially as in the BT case. Along the fold-Hopf curve, the following test functions can be computed:
{
~1 - - g22, ~ 2 - - ga ( 0 ) ,
is defined by (57), (58), and gz is obtained from (55). The following linear singularities can be detected and located as regular zeros of the aforementioned test functions: (1) Fold + double-Hopf: ~1 = 0, lP2 5~ 0. (2) Hopf + BT: 7tl = 0, 7r2 = 0. (3) Triple zero eigenvalue: 7rl r 0, 7t2 = 0.
w h e r e g22
9.3.
Double-Hopf
A double-Hopf point is characterized by the fact that the Jacobian matrix A -- fx has two pairs of purely imaginary algebraically simple eigenvalues -+-icol, q-ico2, COl, 092 > 0 and no other eigenvalues on the imaginary axis. This implies that 2A (3 L,, where A = fx (x, or), has rank defect 2 and there exist vectors rib, 132b,1/)lb, W2b E ~ n ( n - l ) / 2 such that the matrix
mb m
2 A G L, Vl*b
tOlb
0
//32b 1
0
* l)2b
0
0
is nonsingular. The defining equations for the double-Hopf curve are the equilibrium equation (2) and g i l j l - - O, gi2j2 --" O,
where gij = gij (u,
or) are the components of the matrix
G_(gll g21
gl2) g22
obtained by solving the system
G
(0m
I2)"
The vectors 131b, l)2b, //31b, tO2b and the indices i l, jl, i2, j2 are updated along the curve in the following fashion. The vectors Vlb, VZb are chosen to form an orthogonal basis of the right null space of 2A Q L,, and wlb, wzb are chosen similarly for the left null space.
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197
The choice of (il, j l ) , (i2, j2) is such that the space spanned by the gradient vector giljjz has the largest component orthogonal to the equilibrium surface and gi2j2z has the largest component orthogonal to the space spanned by both the tangent space to the equilibrium surface and gitj, z; here z ranges over the state variables and free parameters. Along the double-Hopf curve, the following test functions can be computed:
{
~,-detG
1,
gr2 - det A,
r -- P'q,
where the matrix G 1 is obtained by solving
0),
V 1
and p E R n and q 6 R" are obtained by solving
A e,.,) (.) 0 s
)
and
(q * s) ( A * epl
ePe ) -- (O,* 1) 0
Above, epi is the (pi) th unit vector and s denotes a dummy real number. The values for pl and p2 are obtained by looking for the smallest pivot elements in the decomposition with complete pivoting of the (Jacobian) matrix A = f~-(x, or). The following linear singularities can be detected and located as regular zeros of the above defined test functions: (1) Resonant double-Hopf: grl = 0. (2) Fold + double-Hopf: gr2 = 0, ~P3 r 0. (3) Hopf + BT: ~P2 = 0, lP3 -- 0.
9.4. Cusp Since a cusp is a fold point where A = f r has rank defect 1, there exist vectors vl, wl 6 IRn such that
.
(A v*1
.) 0
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is nonsingular. In addition, there has to be a degeneracy in the second derivatives. The defining system of the cusp curve then consists of the equilibrium equation (2) together with
g(x, or) - O, gl (x, ot) - 0 ,
(59)
where g is obtained by solving the bordered (n + 1)-dimensional system
g(x, ot) and g l is obtained by solving
M ( v l (x,ot) ) g l(x,~)
_ (-B(v, v) ) 0
'
where B(v, v) -- f,~x(X, ot)vv. 9.5. Generalized Hopf 9.5.1. Minimally augmented system. This method is suitable when the Jacobian matrix of the defining equations is computed numerically. The idea is, as in the simple Hopf case, that at a generalized Hopf point the bialternate-product matrix 2A | In is singular. Here A = fx(x, or). The auxiliary data are vectors rib, V2b, Wlb, W2b E JR" and scalars d12, d21 such that the matrix
mb
2A 0 In
Wlb
V*lb * 132b
0
d12
d21
0
m
tO2b 1
is nonsingular. The defining equations for the generalized Hopf (Bautin) curve are f(x, or) -- 0, det G(x, or) - O, ll (x, or) - - 0 ,
where the matrix G_ (gll g21
g12) g22
is obtained by solving the system V
( 0m
,2)
(60)
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199
and where 11 denotes the first Lyapunov coefficient defined as 11-
~lRe(p . C(q . q -~) . - . 2 B ( q. A - I B ( q
-~))
+ B(~-, (2io9oi,, - A) - 1 B ( q , q))) (see formula (47)). The complex vectors p, q 6 C" satisfy A q = )~q,
A* p = ~.p,
( R e q , R e q ) = ( p , q ) = 1,
(Req, Imq) = 0 , where ~. = i co0 along the curve (60) and the last condition is added to ensure smoothness of the vector q. The multilinear functions B ( p , q) and C ( p , q, r) are defined by (45) and thought of as functions of (x, or). 9.5.2. Standard augmented system. We now describe another computational scheme for the generalized Hopf bifurcation. When available, symbolic derivatives up to order 4 can be used in this scheme. The continuation data consist of 8n + 5 real numbers. As before, it is convenient to express some of the equations in complex form. The idea is to express first that A = f~ (x, or) has an imaginary eigenvalue ico0 with right eigenvector q E C" and left eigenvector p 6 C n, and then to add the condition that the first Lyapunov value vanishes. To fix the right and left eigenvectors we add the normalization conditions (q0, q) = (P, q) = 1, where q0 E C n is the normalized right eigenvector q at a previously computed point on the curve. To simplify the expression for l l, we introduce v E R 'l and w E C n as additional unknowns, where v -- A - I B(q,-~),
w = (2icooL, - A) -I B(q, q).
Thus, the continuation space consists of the state variables, the parameter variables, and the components of (q, p, v, w, coo,)O.
The defining equations for the generalized Hopf (Bautin) curve are then given by (2) supplemented by the complex system A q - icooq = 0 , A T p + )~p=O, (qo, q) - 1 = O, (p,q)-
1 =0,
A v - B(q, -~) = O, (2icool,7 - A ) w - B(q, q) = O,
Re(p, C(q, q,-~) - 2 B ( q , v) + B(-~, w ) ) - O.
(61)
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The complex variable i. is introduced artificially to regularize the system; formally, along the generalized Hopf curve, 1. = iw0. When written in real variables, (2) and (61) form a system of 8n + 5 equations with 8n + 6 variables including the three free parameters.
10. Normal forms for codimension-2 equilibrium bifurcations Bifurcations of phase portraits of (1) are determined by the normal form coefficients at critical parameter values. For example, depending on the values of certain coefficients for the fold-Hopf and double-Hopf bifurcation, the system may exhibit quasi-periodic and "chaotic" behavior. In this section we show how such coefficients can be computed numerically, while the final section deals with the nondegeneracy of parametric unfolding. If the critical parameter values and the equilibrium position are known exactly then symbolic manipulation software such as MAPLE or Mathematica can be applied; see, for example, Sanders [68].
10.1. List o f codimension-2 normal f o r m s As we have seen in Section 7, there are five codimension-2 equilibrium bifurcations: 1. The cusp (1.1 = O, a = 0).
Equation (1) restricted to the center manifold at the critical
parameter values is given by tO , - - btO 3
()
-+- 0 tO4 ,
tO E R 1.
(62)
If b -~ 0 and if the system (1) depends generically on the two parameters (0/1,0/2), then its restriction to the center manifold is locally topologically equivalent to the normal form wt=fll
nt-fl2tOn t-ctO 3,
where (ill, f12) are unfolding parameters that are related to (0/1,0/2) via an invertible smooth transformation. This normal form predicts a hysteresis phenomenon near the bifurcation. The restriction of (1) to the center manifold at the critical parameter values is locally smoothly equivalent to the normal form
2. Bogdanov-Takens (1.1,2 = 0).
!
toO -- tol, W!I - -
aw2 -t- b w o w l
-Jr- O(lltoll3),
(63)
where w = (w0, Wl)* E R 2. If ab 7~ 0, and if the parameters (0/1,012) enter (1) generically, then the restricted system is locally topologically equivalent to the normal form 1180 - - t o 1 ,
i W l!1 - -
f l l -nt- f l 2 t o 0 +
aw 2 +
bwowl.
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201
An analysis of this normal form reveals a parameter plane curve of saddle homoclinic bifurcations emanating from the codimension-2 point. The unique periodic solution family born in the Hopf bifurcation approaches the homoclinic orbit and terminates there as its period T --+ oc. 3. Generalized Hopf (Z 1,2 - - +ico0, 11 -- 0). The restriction of (1) to the center manifold at the critical parameter values is locally smoothly orbitally equivalent to the normal form w' -- icoow 4-12wlwl 4 4-0(Iw16),
(64)
w ~ C l,
where the second Lyapunov coefficient 12 is real. If 12 7~ 0 then, generically, the restricted system (1) is locally topologically equivalent to the normal form ! 1/3 - -
(/~1 4 -
icoo)w 4- fl2wlwl 2 4-12wlwl 4.
(65)
This normal form predicts the existence of a curve originating at the codimension-2 point in the parameter plane, where two periodic orbits collide and disappear through a nonhyperbolic periodic solution with a nontrivial multiplier #l -- 1. 4. Fold-Hopf (ZI --O, Z2,3 --+icoo). The normalized restriction of (1) to the center manifold at the critical parameter values has the form
C2oo g + ao,,
I +
+ a l l l woIwl 12 4- O(ll(w0, wl, Wl)ll4),
-
+
+ 89a2,0 o
(66)
, + ' c 0 2 , m, Ira, I
4- O(ll (wo, Wl, Wl) 114). Here wo 6 R 1 and wl 6 C 1 , the coefficients Gklm in the first equation are real, while those in the second equation are complex. If G2ooGoll =/=0, then, generically, the restriction of (1) to the center manifold is locally smoothly orbitally equivalent to the system U' -- fll 4- bu2 4- clzl 2 4- O(ll(u, z, z)]14),
(67)
z' - (fl2 4- ico)z 4- duz 4- eu2z 4- O(ll(u, z, z) ll4), where co, b, c, e are real functions of 13, while d is a complex function of 13 with 1
co(0)
- - coo,
b(0)
--
~G200,
c(0) - Goll,
G3oo d(0) - G11o - ioJo 3G2oo' and 1 [ (ReG021 . G30.0 . Jr- G . ill) e(0) -- ~ Re G21o 4- Gl lo Goll G200 G011
Go21G200]. 2Goll
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202
In general, the O-terms in (67) cannot be truncated, since they affect the topology of the bifurcation diagram of the system near the bifurcation. Depending on the coefficients b, c, d, e the system can have two-dimensional invariant tori, chaotic dynamics, NeimarkSacker bifurcations, and Shil'nikov homoclinic bifurcations.
5. Double-Hopf()~l,2 = -[-io)l, ~.3,4 -- +ico2). kcol =/=1o)2,
k, I > 0,
Assume that
k + l ~< 5,
(68)
where k, l are integer numbers. The normalized restriction of system (1) to the center manifold has then the form WZl = icOlWl + IG2100WlIWlI2 +GlollWlIW2I 2 -1- 163200tOl ]tO114 -t- 1621111/31 ]tO112]w212 + 1Glo22wllw214
+ O(ll(wl, ~1, W2, ~2) ll6),
(69)
f
W2 -- ic02W2 -'F alllOW2lWll 2 + 89
2
-t- I a z z l o w z l w l [4 + 1allZltOZltollZlwzl 2 --1-laoo32wz]wz14
+ O(ll(wl, wl, W2, ~2) ll6), where (Wl, 1/)2)* E C 2, and Gjklm E C 1. Moreover, if
(ReG2100)(ReGloll)(ReGlllo)(ReGo021) 7~ 0 and the critical eigenpairs cross the imaginary axis with nonzero velocities, then the system (1) is locally smoothly orbitally equivalent near the bifurcation to the system V'1 -- (ill + iO)l )v 1 + l p l l V 1Iv 112 -+- P12Vl ]v212
+ i Rl v| ]Vl [4 + 1 SI Vl ]1)2]4 -+- O(]] (Ul, ~1,1)2, v2) []6),
(70)
1); - (f12 -4- i(-o2)1)2 -+- P211)2[1)1 ]2 + 1 P221)2]1)2[2
+ 1 $21)2]1)1 ]4 _+_iR21)2[1)2]4 _+_O(]] (1)1, Vl, 1)2, v2) ]]6), where (Vl, 1)2)* E C 2, and the coefficients Pj~ and Sk are complex, while the numbers Rk are real. Moreover, the real parts of the critical values are given by the expressions Re PII -- Re G2100,
Re P12 = Re G 1011,
Re Pzl = Re Gill0,
Re P22 -- Re G0021,
and 1
Re SI -- Re G 1022 -+- g Re G 1011
•
Re GII21 _ 4 Re G0032 6 ReGlllO Re Goo21
(Re G3200) (Re Go021) ] (Re G2100) (Re G lllO) J '
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203
1
Re $2 -- Re G2210 -+- ~ Re Glllo x
ReGzlll _ 4 R e G 3 2 o o 6 Re Glol~ Re G21o---------~
(Re G2100) (Re G0032) ]. (Re Glol 1)(Re Goo21) J
As in the fold-Hopf case, the O-terms in (70) cannot be truncated, since they affect the topology of the bifurcation diagram of the system. Depending on the values of the normal form coefficients, the system can have invariant tori and chaotic dynamics, as well as Neimark-Sacker bifurcations and Shil' nikov homoclinic bifurcations. Proofs of the results formulated above can be found in Kuznetsov [49] with relevant bifurcation diagrams and bibliographical references.
10.2. The normalization method Our aim is to derive efficient formulas to numerically compute the coefficients of the normal forms (62), (63), (64), (66), and (69). The following normalization technique is due to Coullet and Spiegel [13] (see also E1phick et al. [26]). Suppose that the system (1) has the equilibrium x = 0 at ot -- 0, and suppose that the Jacobian matrix A = f , (0, 0) has nc eigenvalues with zero real part (counting multiplicities). Let T c be the corresponding generalized critical eigenspace of A. Write the system at ot = 0 as x ' = F(x),
(71)
x 6 IR",
with F given by (45), and restrict it to its nc-dimensional invariant center manifold parametrized by w E R"': x = H(w),
H : IR"' --->N".
(72)
The restricted equation can be written as w' = G(w),
G : R"~ -+ R ''c.
(73)
Substitution of (72) and (73) into (71) gives the following homological equation: Hw(w)G(w)-
(74)
F(H(w)).
We expand the functions G, H in (74) into multivariate Taylor series, 1
Ivl~l
1
Ivl~l
and assume that the restricted equation (73) is put into the normal form up to a certain order. The coefficients gv of the normal form (73) and the coefficients hv of the Taylor
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204
expansion for H(w) are unknown, but can be found from (74) by a recursive procedure, from lower to higher order terms. (Obviously, one has Y~'~lvl=lhvwV ~ Tc') Collecting the coefficients of the wV-terms in (74) gives a linear system for the coefficient hv
Lhv=Rv.
(75)
Here the matrix L is determined by the Jacobian matrix A and its critical eigenvalues. The right-hand side Rv depends on the coefficients of G and H of order less than or equal to Ivl as well as on the terms of order less than or equal to Ivl of the Taylor expansion (45) of F. When Rv involves only known quantities, the system (75) has a solution because either L is nonsingular or Rv satisfies Fredholm's solvability condition (p, Rv) = 0 , where p is a nullvector of the adjoint matrix L*, and (p, q) = p*q. When Rv depends on the unknown coefficient g, of the normal form, L is singular and the above solvability condition gives the expression for gv. For all codimension-2 bifurcations, except Bogdanov-Takens, the invariant subspace of L(L*) corresponding to the zero eigenvalue is one-dimensional in C n, i.e., there are unique (up to scaling) nullvectors q and p,
Lq=O,
L'p=0,
(p,q}=l,
and no generalized nullvectors. Then the unique solution h v to (75) satisfying (p, h v) = 0 can be obtained by solving the following nonsingular (n + 1)-dimensional bordered system:
("o)
L p*
We write h v = LINV Rv. The Taylor expansion of H (w) simultaneously defines the expansions of the center manifold, the normalizing transformation on it, and the normal form itself. Since we know a priori which terms are present in the normal form, the described procedure is a powerful tool to compute the normal form coefficients at the bifurcation parameter values. In the following sections, we summarize results obtained by this method with the help of the symbolic computation software MAPLE. Details can be found in Kuznetsov [50].
10.3. The cusp bifurcation At such a bifurcation point, the system (1) has an equilibrium with a simple zero eigenvalue )~1 -- 0 and no other critical eigenvalues. Let q, p 6 R" satisfy
Aq = 0 ,
A*p = 0 ,
(p,q} = 1.
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205
The restriction of (1) to the corresponding center manifold has the form t O ' - - a w 2 if- b to 3 -+- O ( w 4 ) ,
with unknown coefficients a and b. Applying the normalization technique described above, one gets an~
l (p , B(q , q)},
which already appears in (46) for the fold bifurcation. For the coefficient b we obtain
b--~l(p . C(q. q. q)+ . 3B(q . h2)) where h2 = -AINV[B(q, q) - (p, B(q, q))q], which can be computed by solving the nonsingular (n + 1)-dimensional bordered system A p*
qo)(h2)_(-B(q'q)q-(P'B(q'q)) s 0
q)
"
Recall that a = 0 at the cusp bifurcation. Thus the coefficient b in the normal form (62) can be expressed more compactly as
1 C(q, q, q) - 3B(q, b - -~(p,
10.4.
A INV B(q,
q))).
Bogdanov-Takens bifurcation
Here the equilibrium of system (1) has a double zero eigenvalue )~1.2 = 0, and there exist two real, linearly independent, (generalized) eigenvectors, q0, q l 6 R", such that
Aqo = 0 ,
Aql = q o .
Moreover, there exist corresponding vectors Po, Pg 6 R n of the transposed matrix A*: A*pl = 0,
A ' p 0 = Pl.
One can choose these vectors so that they satisfy (q0, p0) = (ql, pl) = 1,
(ql, p0) = (q0, pl) = 0.
Then one obtains
1 a -- ~(p,, B(qo, qo)) and
b-{p0
, B(qo, qo))+ (Pl, B(qo, ql)),
for the coefficients a, b of the normal form (63).
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10.5. Bautin (generalized Hopf) bifurcation At such a bifurcation point the system (1) has an equilibrium with a simple pair of purely imaginary eigenvalues, k 1,2 - - -+-i O)0, COO > 0, and no other critical eigenvalues. As in the simple Hopf case, we introduce two complex eigenvectors q, p 6 C nA q - i cooq ,
A *p -- - i coop ,
and normalize them according to (p,q) = p ' q - - 1.
The normalized restriction of (1) to the two-dimensional center manifold can be written as 1 G32wlw 14 + O( Iw 16), w , -- icoow + ~1 G21wlwl 2 + -~ where Gjk E C I. An application of the normal form algorithm gives the cubic normal form coefficient G21 - - ( p , C ( q , q,-q) + B(-~, (2icOoIn -
A) -1B(q, q)) - 2B(q, A -1B(q,-~))), (76)
while I I - - 1 Re G21 is given by (47). The second Lyapunov coefficient is given by
1
12 -- ~ Re G32, with G32 - (p, E(q, q, q, q, q ) + D(q, q, q, h 2 0 ) + 3D(q,-~,-q, h20) m
+ 6D(q, q,-~, hll) + C(-~,-~, h3o) + 3C(q, q, hzl) + 6C(q,-~, hzl) + 3C(q, hzo, h2o) + 6 C ( q , h l l , h l l ) +6C(-q, hzo, hll) + 2B(q-, h31) m
+ 3B(q, h22) + B(hzo, h3o) + 3B(hzl,h2o) + 6 B ( h l l , h 2 1 ) ) ,
where h20 = (2icooln - A) -1B(q, q), hll -- - A - 1 B ( q , - 4 ) .
The complex vector h21 is found by solving the nonsingular (n + 1)-dimensional complex system iwoIn - A p*
0
s
=
0
'
Numerical continuation, and computation of normal forms
207
while h3o - (3icooi,, - A) -l [C(q, q, q) + 3B(q, h20)], h31 -- (2icooln - A ) - l [ D ( q , q , q , - q ) + 3 C ( q , q , h l l ) + 3C(q,-q, h20)
+ 3B(h2o, hll) + B(q-, h3o) + 3B(q,h21) - 3G21h2o], h22 -- - A - I [ D ( q , q, q, q ) + 4C(q,-~, hll) + C(-~,-~, h2o) + C(q, q, h20)
+ 2 B ( h l l , h l t ) + 2B(q, h21) + 2B(q-, h21) + B(h20, h20) - 2hll(G21 +G21)].
10.6. Fold-Hopf bifurcation At a fold-Hopf bifurcation point of (1) the Jacobian matrix A -- fx (0, 0) has a simple zero eigenvalue ,kl -- 0 and a pair of purely imaginary simple eigenvalues: X1 -- 0,
~k2.3 -- -+-icoo,
with wo > 0, and no other critical eigenvalues. Introduce two eigenvectors, qo 6 R" and ql E C n,
Aqo - 0,
Aql -- icooql ,
and two adjoint eigenvectors, Po 6 R 'z and pl E C '7, with A*po -- 0,
A'p1 -- -icoopl.
Normalize these vectors such that (po, qo) -
(pl,qt)
- - 1.
The following orthogonality properties hold" (pl, qo) -- (po, ql) - 0. One obtains the following expressions for the quadratic coefficients in (66)" G2oo - ( p o , B(qo, qo)),
Gl 1o - - ( P l , B(qo, ql)),
G o l l - - ( p o , B(ql,-ql)), and the following formulas for the cubic coefficients in (66): G3oo - ( p o , C(qo, qo, qo) + 3B(qo, h2oo)),
Glll --(PO, C(qo, qt,-ql) + B(qo, holl) + B(ql, hllO) + B ( q l , hi 1o)), G21o--(pl, C(qo, qo, ql) + 2B(qo, hllo) + B(ql,h2oo)), Go21 --(Pl, C(ql,ql,-ql) + 2B(ql,holl) + B(-qi,ho2o)),
208
W.-J. Beyn et al.
where h2oo = - A I N V [ B ( q o , qo) - ( P o , B(qo, qo))qo], h020 = (2icooIn - A) - | B(q|, ql), hll0 -- (icoOIn -- A)INV[B(qo, ql) - ( P l , B(qo, ql))ql], h01l -- - a INV [B(ql,-ql) - (PO, B(ql, q-l))q0]. Here the vectors h200 and h01| can be computed by solving the nonsingular (n + 1)dimensional real systems A
Po
s
0
s ) (
0
and
(Ao.
while the vector h 110 can be found by solving the nonsingular (n + 1)-dimensional complex system
icooln -- A
q.) 0)(.+q..+.+oq.>q.) 0
s
=
0
"
10.7. Double-Hopf bifurcation At the double-Hopf bifurcation the system (1) has an equilibrium for which the Jacob|an matrix A -- fr (0, 0) has two pairs of purely imaginary simple eigenvalues" )~1,4 = -+-|col,
)~2,3 = -+-|co2,
with col > 0)2 > 0, and no other critical eigenvalues. Assume that condition (68) holds. Since the eigenvalues are simple, there are two complex eigenvectors, ql, q2 E C n, corresponding to these eigenvalues"
Aql -- icolql,
Aq2 -- ico2q2.
Introduce the adjoint eigenvectors pl, P2 E C n by
A* pi = -|col Pl ,
A ' p 2 -- -ico2P2.
These eigenvectors can be normalized using the standard scalar product in C n, (pl, ql) -- (p2, q2) -- 1.
Numerical continuation, and computation of normal forms
209
The resonant cubic coefficients in the normal form (69) are given by G2100 - (pl, C ( q l , q i , - q l ) + B(h2000, q-l) + 2B(hlloo, ql)), GI011 - - ( p l , C(ql, q2,-q2) + B(h 1010, q-2) -+- B(hlool, q2) -+- B(hool 1, ql)), G i l l 0 - - (P2, C ( q l , - q l , q 2 ) + B(hlloo, q2) + B(hlolo, q-l) + B(h-lool, ql)), Go02! --(P2, C(q2, q2, q-2) + B(ho020, q-2) + 2B(hool l, q2)), where hi loo -- - - A - l B(ql, -ql), h2000
--
(2icol L,
-
A) -1B(ql,
h 1010 - - [i (c01 + co2)In
--
ql),
A] - 1 B ( q l , q2),
hl001 -- [/(col - co2)I,, - A] - 1 B ( q l , q2), hoo2o = (2ico2l, - A) - l B(q2, q2), hool l -- - A -1B(q2,-q2)" All matrices involved in the above formulas are nonsingular, due to the conditions (68) on the critical eigenvalues. Expressions for the fifth-order coefficients in (69) are given in (Kuznetsov [50]).
11. B r a n c h s w i t c h i n g a t c o d i m e n s i o n - 2
bifurcations
Suppose that the system (1) has a codimension-2 equilibrium x --- 0 at ot = 0. Generically, one expects curves of codimension- 1 bifurcations to emerge from the origin in the a-plane. In this section we describe methods to start the continuation of such curves based on the information available at the singularity. As we mentioned in Section 10, there are not only codimension-1 equilibria that emanate from codimension-2 points but, depending on the type, there may also be codimension-1 families of periodic and homoclinic orbits. The analogous problem of switching from codimension- 1 bifurcations to codimension-0 equilibria or periodic solutions was treated in Section 4. Here we will not consider switching between bifurcations of the same codimension, which is typical for nongeneric parameter situations (e.g., in systems with symmetries). One way to set up a computational procedure is to consider the normal form of the codimension-2 singularity including the parameters/3 E R 2 w' - G(w, fl),
G" R no+2 --+ R n'.
(77)
For all five codimension-2 bifurcations these normal forms are listed in Section 10. Suppose that an exact or approximate formula is available that gives the emanating codimension-1 bifurcations for the normal form (77). In order to transfer this to the original equation (1) we need a relation a -- K (fl) ,
K" IR2 -+ ]R2
(78)
210
W.-J. Beyn et al.
between the unfolding parameters/3 and the given parameters c~ and, moreover, we need to extend the center manifold parametrization (72) with respect to 13 x = H(w, fl),
H:IR n~+2 ---+IR".
(79)
Taking (78) and (79) together as (x, or) = (H(w, fl), K(fl)) yields the center manifold for the suspended system x' = f (x, c~), or' = 0. The homological equation (74) now turns into
(80)
Hw(w, fl)O(w, fl) -- f (H(w, fl), K(fl))
and the method of Section 10.2 extends readily to this case. We assume the Taylor series of G to be known as 1
G(w, fl) -
v!#v
wV
flu,
Ivl+lul~>l and the Taylor series of H and K to be unknown
H(w, ~) --
y~
1 hvuw vfl/~, v!#!
Ivl+l~l~>l
1
;5 lul~l
t~ ~
We insert these expansions into Equation (80) and apply a recursive procedure as in Section 10.2. For # = 0 this reproduces the coefficients from Section 10 while the coefficients with [/z[ ~> 1 yield the necessary data on the parameter dependence. Again, as in Section 10, we emphasize that this approach requires the knowledge of the normal form (77) and derives necessary conditions for the transformations. We also notice that the Taylor terms are not always determined uniquely in this way and that the number of terms needed depends on the type of codimension-1 singularity we want to follow. For example, folds and Hopf points usually need fewer Taylor terms than periodic and homoclinic orbits. In the following sections we give some results for the cusp, Bogdanov-Takens, and Bautin bifurcations and comment on the remaining cases.
11.1. Switching at a cusp point The cusp is the simplest case for branch switching because there is a smooth and regular curve of folds passing through it, and no other codimension-1 points are nearby. To be more specific, suppose that a cusp of (1) has been located at some (x0, or0) by solving the augmented system which consists of (2) and (59). The tangent vector to the curve of folds passing through it is then given by (q, 0, 0) where fOq = 0. Note that according to (26) we have gx (xo, c~o)q = - p* fxx (xo, oeo)qq = 0
Numerical continuation, and computation of normal forms
211
and hence (q, 0, 0) spans the null space of the Jacobian of (f, g). We may then continue the fold branch with the predictor (xo, do) + s(q, O, O)
for some small s r O.
To obtain more information on the location of the fold curve in the parameter plane we use the normal form
G ( w , fl )
= fl l -+- f1211o + c w 3 - + - . . . .
For simplicity, let x0 = 0, o~0 = 0 and consider the expansions K(r
gl/~ -4-0(11r
1
1
H(w, fi) -- Holfl + qw + -~ H20w 2 + H, lwfl + -6 H30w 3 +
1
2
o(11/~11 + w4),
1
f (x, or) - a x + alo~ + -~ B(x, x) + B1 (x, or) + -~ C(x, x, x) + O(llc~ II2 + Ilx 114). Using the method described above one finds as in Section 10.3 that H20 = - A I N V B ( q , q ) ,
C ~
l p*(3B(H2o , q ) + C ( q
,
q , q))
and Hol, Hi l, H20, H30, Kl are in addition determined through the following bordered systems
( A p*Bq p*
A1 )(O01)_ (~ 0)l p*Blq 0
Kl 0
Hll -- A INv ([H20,q] -
B(q,
0
H01)-- B1(q, K1)),
/-/3o -- A INv (6cq - 3B(H2o, q) - C(q, q, q)). It turns out that the first bordered system is nonsingular for generic cusps and can be reduced to one left and two right solves with AINV and to a small 2 x 2 system. Since the curve of folds for the truncated normal form is given by 112=/3,
fll = 2ce 3,
f12 = - 3 c s 2
we obtain an O(6 4) approximation of the fold curve in the original system as follows 2
._
(..o1 ( o ) 2
1 H2o) 1
W.-J. Beyn et al.
212
11.2. Switching at a Bogdanov-Takens point 11.2.1. Switching to folds and Hopfpoints. According to Section 7.2, any generic BTpoint (x0, c~0) in a two-parameter system is a regular solution of the minimally augmented fold equations f = 0, gF = 0 (see (23), (24)) as well as of the corresponding Hopf equations f = 0, gH = 0 (see (23), (27)). In this sense there is no problem of starting either of the two branches by solving the appropriate system. However, if one replaces the large system (27) by the Hopf system (30), then the BTpoint becomes a simple (actually symmetry-breaking) branch point for (30) and the branch switching methods of Section 4 apply, cf. Spence et al. [74]. Starting Hopf curves at BTpoints was suggested by Roose [65] and the use of bordered systems in Griewank and Reddien [36]. We provide here some higher-order approximation for the functions in (78), (79) which will be used in the next subsection for the fold and Hopf branches as well as for the homoclinic branch. Let x = 0, ot = 0 be a BT-point; then choose vectors q0, ql, p0, pl and compute the coefficients a, b in the normal form
G(w, fl) --
(Wl
)
/31 § fl2tO0 § aw 2 + b w o w l §
'
as in Section 10.4. We use the homological equation (80) with the expansions 1
K ( / 3 ) - K1/3 § ~K2/3 2 + 0 ( / 3 2 § 1/3t/321 § 1/3213), 1
H(w,
fl) --
1
H01/~ § [qo, ql]W + -~ H20,OW2 4- H20, lWOWl 4- -~ Ho2,1fl 2 + O(w 2 + Iwow~l + Iwol3 §
§ I/~0r I + 1/3213),
1
1
f (x, ~ ) - Ax + Alc~ + -~ B ( x , x ) + B l ( x , ~ ) + -~ B2c~2 + O(llxll 3 + Ilotl13). (81) The linear terms K l, H01 are easily determined through
KI--(Y?+Y2)-I(Yly2-Y2),yl
where (Yl, Y2) - Pl A l,
H01 - - A INv ([ql, 0] - A ! K1).
(82)
Here we have defined A INV by solving the bordered nonsingular matrix [A ~,q~Pl0) as in Section 10.2. Note that Pl*([ql, 0] - A I K 1 ) - - ( 1 , 0) - (p~A1)Kl = O. This equation does not define K1 uniquely and the above choice of Kl was made for convenience.
Numerical continuation, and computation of normal forms
213
Letus introduce columns in K1 -- [KI,0, Kl,l] and in H01 = [H01.0, H01, l], respectively. Then the fold curve for the normal form ll) 0 = / 3 ,
1101 "~ 0 ,
f l l --- a /3 2 ,
132 --- - - 2 a / 3
transforms into ot -- -2a/3 KI. 1 + O(/32),
x --/3(q0 -- 2all01,1) + 0(/32)
and the Hopf curve w0 = wl = 0, fll = 0, f12 = --/3 < 0 into -
x = --/3Hol.1 with/3 > 0.
+
With AINV as defined above, one finds for the quadratic terms H20,0 -- A INV (2aql - B(qo, qo)), -
-(pTz)K,
o,
H20,1 = AINV(bql + H 2 0 , 0 -
B(qo, ql)),
H02,1 =--AINV(z -+- A1K2),
(83)
where
z = B(HoI,1, Hol,l) -+- 2Bl (H01,1, Kl.l) -+- B2(Kl,1, Kl,1). Using the expressions for a and b it is easily verified that the right-hand sides in the first two equations are in the range of A. With the formulae above and (81) one can write down an 0(/3 3) approximation of the fold and the Hopf curve. 11.2.2. Switching to homoclinic orbits. The key to the construction of the homoclinic orbits in the normal-form system is a blowup transformation which anticipates the cuspoidal shape of the phase curves in the (wo, Wl)-plane. Introduce new parameters r,/3 and phase variables ~, r/via
f12 -- Z'62,
1
fll -- ~aa (T2 -- 1)/34 , /33
(t)
-
Ua
,
/3 t) .
This transforms w' = G (w, fl) into ~' /7'
r/ ~2 _ 4
+ ~a
0 b~r/- 2br
~).
At e = 0 this system is Hamiltonian and has a homoclinic orbit given explicitly by (~0, r/0) (t) -- 2(1 - 3 sech 2 (t), 6 sech 2 (t) tanh(t)).
(84)
214
W.-J. Beyn et al.
One can now apply well-known techniques due to Pontryagin [61] and Melnikov [57] to obtain periodic and homoclinic orbits for the perturbed Hamiltonian system. Alternatively, according to Hale [38], one may treat this problem as a bifurcation problem in suitable function spaces. For this purpose take r as a parameter and write (84) as an operator equation
F(z, r) -- O,
where
1 Z -- (~,/7, 8) 9 Cbounde d X ]l~1 .
Setting z0 = (~0, r/0, 0) we have F (z0, r) = 0 for all r 9 ]1~ 1 and a computation shows that a simple (in fact, a pitchfork) branch point occurs at r0 - - 5 . This proves the existence of a branch of nontrivial homoclinic orbits for the system (84) and leads to a first order approximation of the bifurcating branch as z = (~0, 00, s),
r = r0.
For the normal form system we obtain an approximate homoclinic at
6 64 _[_0(66)
fll --
- 49----a
82((2) Wo(t)-- ~aa ~0
'
f12 --
5 82 + 0@4)
7
10) t -t--- -t-0(83),
'
83 ( 8 ) Wl(t)-- 8aa 170 2t
-F0(84).
Finally, with the data collected in (82), (83) and using (78), (79), (81) we arrive at a homoclinic predictor for the original system
or----if5 e2K,, 1 + ~
+ 0(8 6)
--6Kl,o + -~- K2
--ffHol,l+~a a
~o
t
lO
))
+--7-+O(8)qo
+~aar/O
~t
ql
--FO(64). This predictor can also be used to set up a phase fixing condition and thus start the continuation of a branch of codimension-1 homoclinics with the methods from Section 6. Starting homoclinic orbits at Bogdanov-Takens bifurcations was first considered in Rodr/guez-Luis et al. [64] and Beyn [9].
11.3. Switching at Bautin (generalized Hopf) bifurcation Since for this bifurcation there exists a continuation of the critical equilibrium xo(a) for all sufficiently small I1~ II, with x0(0) = 0, we can write (1) in a coordinate system with the origin at x0(ot) as
x' = A(ot)x + F(x, or),
F :N n+2 --+ R n,
Numerical continuation, and computation of normal forms
215
where F = O(llx 112). This allows one to avoid expanding the center manifold and the normal form into Taylor series in the parameters. We have 1
1
F(x, or) -- -~ B(x, x, or) -t- -~ C(x, x, x, or) + O(llx 114), 1
1
H(w,-~,ot) -- wq(a) + ~ ( o t ) + ~h20(ot)w 2 -F-h,,(a)w-~ + ~h02(c~)w 2
(85)
+ O(Iw13), and w' -- )~(c~)w + cl (ot)wlwl 2 + c2(ol)wlwl 4 -~- O(Iw16), where )~(ot) = #(or) + / c o ( a ) , #(0) = 0, co(0) = coo > 0,
A(ot)q(c~) = X(ot)q (or),
A* (ot)p(ot) = ~.(c~)p(ot),
and p, q are normalized in the standard way for all IlotII small. From the homological equation (80) we now get 1
c, (or) -- -~{p(ot)C(q(ot), q(a), ~(a), a) + B(~(c~), h2o(a), (~) + 2B(q(ot),h,,(ot),ot)), where h2o(u) -- [2)~(u)I,, - A ( u ) ] - ' B(q(ot), q(ot), or), h,, (or) -- [(k(ot) q- k(ot)) In -- A(c~)] -I B(q(ot),-~(ot), or). Note that cl (or) is not uniquely defined and the above choice gives a C1 (0) that coincides with (76) from Section 10.5. To arrive at the normal form with real coefficients of the nonlinear terms
w' -- (#(o~) + ico(ot))w + ll (oe)wlwl 2 -I- 12(ot)wlwl 4 + O(Iwl6),
(86)
one has to reparametrize time (Kuznetsov [49]) according to
dt-
1-
Imcl (Or) 2 3) co(a) Iwl +O(Iwl ) dr,
which gives 11(or)= Recl(ot) -
~o(~)
Imcl (or).
(87)
216
W.-J. Beyn et al.
The unfolding parameters of the normal form (65) can then be expressed as ill -- #(Or), f12 -- 11 (0/),
(88)
where 11 (or) is defined by (87). Equations (88) can be written as fl = K I ~ "1- 0(11c~112),
ce, r ~ R 2,
where K1 is the 2 x 2 Jacobian matrix of (88) that is nonsingular at a generic Bautin point. Thus, o t - g l l / 3 --t-0(11/3112). This allows relating the equation for the curve of fold periodic solutions in the truncated normal form (86), namely
Iwl : ~ ,
~ =12e 4,
f12 : --21262,
to the original parameters. We also obtain an asymptotic estimate for the periodic solution using (85). Recall that an expression for 12(0) is given in Section 10.5. Relating the Hopf curve in the truncated normal form with that in the original system is easy, since it is defined by fll = 0, f12 : 6, w -- 0.
11.4. Other codimension-2 cases A more delicate situation appears at a fold-Hopf bifurcation point (Section 10 and [49]). Assuming an approximate normal form as in Section 10.1, asymptotic formulae for a curve of Shilnikov homoclinic orbits and for a curve of torus bifurcations have been derived in Gaspard [31 ] and applied to a specific example, the R6ssler model. Note that the Shilnikov orbits appear in an exponentially narrow wedge in the parameter plane and their precise form depends on the higher-order terms of the approximate normal form. In addition there may be curves of heteroclinic tangencies of periodic solutions and torus bifurcations. Similar phenomena arise at double-Hopf bifurcations (cf. [49]). A complete set of formulae suitable for switching to homoclinic and heteroclinic orbits in the fold-Hopf and the double-Hopf case seems not to be available.
References
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Numerical continuation, and computation of normal forms
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[28] B. Fiedler, Global pathfollowing of homoclinic orbits in two-parameter flows, Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, G. Dangelmayr, B. Fiedler, K. Kirchg~issner and A. Mielke, eds, Pitman Res. Notes Math. Ser., Vol. 352, Longman, Harlow (1996). [29] M.J. Friedman, Numerical analysis and accurate computation of heteroclinic orbits in the case of centre manifolds., J. Dynamics Differential Equations 5 (1993), 59-87. [30] A.T. Fuller, Condition for a matrix to have only characteristic roots with negative real parts, J. Math. Anal. Appl. 23 (1968), 71-98. [31] E Gaspard, Local birth ofhomoclinic chaos, Phys. D 62 (1993), 94-122. [32] W. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, PA (2000). [33] W. Govaerts and J.D. Pryce,, Mixed block elimination for linear systems with wider borders, IMA J. Numer. Anal. 13 (1993), 161-180. [34] W. Govaerts, J. Guckenheimer and A. Khibnik, Defining functions for multiple Hopf bifurcations, SIAM J. Numer. Anal. 34 (1997), 1269-1288. [35] A. Griewank and G.W. Reddien, The calculation of Hopfpoints by a direct method, IMA J. Numer. Anal. 3 (1983), 295-303. [36] A. Griewank and G.W. Reddien, Computation of cusp singularities for operator equations and their discretizations, J. Comput. Appl. Math. 26 (1989), 133-153. [37] J. Guckenheimer and M. Myers, Computing Hopf bifurcations. II. Three examples from neurophysiology, SIAM J. Sci. Comput. 17 (1996), 1275-1301. [38] J.K. Hale, Introduction to dynamic bifurcation, Bifurcation Theory and Applications, L. Salvadori, ed., Lecture Notes in Math., Vol. 1057, Springer, Berlin (1983), 106-151. [39] B.D. Hassard, Computation of invariant manifolds, New Approaches to Nonlinear Problems in Dynamics, P.J. Holmes, ed., SIAM, Philadelphia, PA (1980), 27-42. [40] B.D. Hassard, N.D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge Univ. Press, London (1981). [41] M. Holodniok and M. Kubi6ek, DERPER - An algorithm for the continuation of periodic solutions in ordinary differential equations, J. Comput. Phys. 55 (1984), 254-267. [42] H.B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, Applications of Bifurcation Theory, E Rabinowitz, ed., Academic Press, New York (1977), 359-384. [43] H.B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Springer, Berlin, Notes by A.K. Nandakumaran and M. Ramaswamy, Indian Institute of Science, Bangalore (1987). [44] H.B. Keller, Numerical Methods for Two-Point Boundary-Value Problems, Dover, New York (1992). [45] A.I. Khibnik, LINLBF: A program for continuation and bifurcation analysis of equilibria up to codimension three, Continuation and Bifurcations: Numerical Techniques and Applications, D. Roose, B. De Dier and A. Spence, eds, Kluwer, Dordrecht (1990), 283-296. [46] AT Khibnik, Y. Kuznetsov, V.V. Levitin and E.V. Nikolaev, Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps, Phys. D 62 (1993), 360-371. [47] M. Kubi6ek and M. Marek, Computational Methods in Bifurcation Theory and Dissipative Structures, Springer, Berlin (1983). [48] Y.A. Kuznetsov, One-dimensional invariant manifolds in ordinary differential equations depending upon parameters, FORTRAN Software Series, Vol. 8, Research Computing Centre, USSR Academy of Sciences, Pushchino (1983) (in Russian). [49] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, Berlin (1995). [50] Y.A. Kuznetsov, Explicit normal form coefficients for all codim 2 bifurcations of equilibria in ODEs, Centrum voor Wiskunde en Informatica, Amsterdam, Report MAS-R9730 (1997). [51] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edn., Springer, New York (1998). [52] Y.A. Kuznetsov and V.V. Levitin, CONTENT." A multiplaOCorm environment for analyzing dynamical systems, Dynamical Systems Laboratory, Centrum voor Wiskunde en Informatica, Amsterdam (1997), ftp.cwi. nl/pub/CONTENT. [53] L. Liu, G. Moore and R.D. Russell, Computation and continuation ofhomoclinic and heteroclinic orbits with arclength parametrizations, SIAM J. Sci. Comput. 18 (1997), 69-93. [54] Y. Liu, L. Liu and T. Tang, The numerical computation of connecting orbits in dynamical systems: a rational spectral approach, J. Comput. Phys. 111 (1994), 373-380. [55] J. Marsden and M. McCracken, HopfBifurcation and its Applications, Springer, Berlin (1976).
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[56] Z. Mei, A numerical approximation for the simple bifurcation points, Numer. Funct. Anal. Optimiz. 10 (1989), 383-400. [57] V.K. Melnikov, On the stabili~ of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12 (1963), 1-57. [58] G. Moore, The numerical treatment of non-trivial bifurcation points, Numer. Funct. Anal. Optimiz. 2 (1980), 441-472. [59] G. Moore, Computation and parametrisation of periodic and connecting orbits, IMA J. Numer. Anal. 15 (1995), 319-331. [60] G. Moore and A. Spence, The calculation of turning points of nonlinear equations, SIAM J. Numer. Anal. 17 (1980), 567-576. [61] L.S. Pontryagin, On dynamical systems close to Hamiltonian systems, J. Exptl. Theoret. Phys. 4 (1934), 234-238 (in Russian). [62] W.C. Rheinboldt, Numerical Analysis of Parametrized Nonlinear Equations, Wiley, New York (1986). [63] W.C. Rheinboldt and J.V. Burkardt, Algorithm 596: A program for a locally-parametrized continuation process, ACM Trans. Math. Software 9 (1983), 236-241. [64] A.J. Rodrfguez-Luis, E. Freire and E. Ponce, A method for homoclinic and heteroclinic continuation in two and three dimensions, Continuation and Bifurcations: Numerical Techniques and Applications, D. Roose, A. Spence and B. De Dier, eds, Kluwer, Dordrecht (1990), 197-210. [65] D. Roose, Numerical computation of origins for Hopf bifurcations in a two-parameter problem, Bifurcation: Analysis, Algorithms, Applications, T. Kfipper, R. Seydel and H. Troger, eds, Birkhfiuser, Basel (1987), 268-273. [66] D. Roose and V. Hlava6ek, A direct method for the computation of Hopf bifurcation points, SIAM J. Appl. Math. 45 (1985), 897-894. [67] R.D. Russell and J. Christiansen, Adaptive mesh selection strategies for solving boundary value problems, SIAM J. Numer. Anal. 15 (1978), 59-80. [68] J.A. Sanders, Versal normal form computation and representation theory, Computer Algebra and Differential Equations, E. Tournier, ed., Cambridge Univ. Press, Cambridge (1994), 185-210. [69] B. Sandstede, Verzweigungstheorie homokliner Verdopplungen, Ph.D. thesis, University of Stuttgart, Germany (1993). [70] B. Sandstede, Convergence estimates for the numerical approximation of homoclinic solutions, IMA J. Numer. Anal. 17 (1997), 437-462. [71] S. Schecter, Numerical computation of saddle-node homoclinic bifurcation points, SIAM J. Numer. Anal. 30 (1993), 1155-1178. [72] S. Schecter, Rate of convergence of numerical approximations to homoclinic bifurcation points, IMA J. Numer. Anal. 15 (1995), 23-60. [73] R. Seydel, From Equilibrium to Chaos. Practical Bifurcation and Stability Analysis, 2nd edn., Springer, New York (1994). [74] A. Spence, K.A. Cliffe and A.D. Jepson, A note on the calculation ofpaths of Hopf points, J. Comput. Appl. Math. 26 (1989), 125-131. [75] C. St6phanos, Sur une extension du calcul des substitutions lin~ares, J. Math. Pures Appl. 6 (1900), 73-128. [76] S.A. van Gils, On a formula for the direction of Hopf bifurcation, Centre for Mathematics and Computer Science, Report TW/225 (1982). [77] B. Werner, Computation of Hopf bifurcations with bordered matrices, SIAM J. Numer. Anal. 33 (1996), 435-455.
CHAPTER
5
Set Oriented Numerical Methods for Dynamical Systems Michael Dellnitz and Oliver Junge Department of Mathematics and Computer Science, University of Paderborn, D-33095 Paderborn, Germany E-mail: dellnitz @uni-paderborn, de http ://math- www. uni-pade rbo rn. de/-a g de l ln itz
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The computation of invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Brief review on invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The computation of relative global attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Convergence behavior and error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. 2.6. 3. The 3.1.
The computation of chain recurrent sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . computation of invariant manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Convergence behavior and error estimate 3.3. 4. The 4.1. 4.2.
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Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . computation of SRB-measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief review on SRB-measures and small random perturbations . . . . . . . . . . . . . . . . . . . . Spectral approximation for the Perron-Frobenius operator . . . . . . . . . . . . . . . . . . . . . . .
223 223 223 224 227 228 230 232 233 233 234 235 238 239 241
4.3. Convergence result for SRB-measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
244
4.4. Numerical examples
245
5. The 5.1. 5.2. 6. The
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identification of cyclic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extraction of cyclic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . computation of almost invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1. Almost invariant sets
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6.2. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Adaptive subdivision strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Adaptive subdivision algorithm 7.2. Numerical examples
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8. Implementational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Realization of the collections and the subdivision step . . . . . . . . . . . . . . . . . . . . . . . . .
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8.2. Realization of the intersection test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. I m p l e m e n t a t i o n of the m e a s u r e c o m p u t a t i o n Acknowledgments References
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Set oriented numerical methods for dynamical systems
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1. Introduction Over the past years so-called set oriented numerical methods have been developed for the study of complicated temporal behavior of dynamical systems. These numerical tools can be used to approximate different types of invariant sets or invariant manifolds but they also allow to extract statistical information on the dynamical behavior via the computation of natural invariant measures or almost invariant sets. In contrast to other numerical techniques these methods do not rely on the computation of single long term trajectories but rather use the information obtained from several short term trajectories. All the methods which are described in this chapter are based on multilevel subdivision procedures for the computation of certain invariant sets. This multilevel approach allows to cover the object of interest - e.g., an invariant manifold or the support of an invariant meas u r e - by several small subsets of state space. Since outer approximations are produced and long term simulations are avoided these methods are typically quite robust. Recently also adaptive subdivision strategies have been developed and moreover concrete realizations have been proposed which allow to make the computations rigorous. The numerical methods presented here are similar in spirit to the so-called cell mapping approach, see, e.g., Kreuzer [31], Hsu [24]. However, a significant difference lies in the fact that in the cell mapping case the numerical effort depends crucially on the dimension of state space whereas for the multilevel subdivision procedures the efficiency essentially depends on the complexity of the underlying dynamics. We would also like to mention that by now there exist several relevant extensions and adaptations of the set oriented approach as described here. For instance, in [42,15] the authors develop and analyze set oriented algorithms which can be used for the identification of so-called conformations for molecules. Roughly speaking, these are almost invariant sets for a specific type of Hamiltonian systems. Another direction has been considered in Keller and Ochs [29]. There the set oriented approach has been successfully adapted to the context of random dynamical systems. In this chapter we give an overview about the developments in the area of set oriented methods for general deterministic dynamical systems. We report on both theoretical properties of the numerical methods and details concerning the implementation.
2. The computation of invariant sets In this section we present set oriented multilevel algorithms for the approximation of two different types of invariant sets, namely attracting sets and chain recurrent sets. We demonstrate the usefulness of this multilevel approach by several numerical examples.
2.1. Brief review on invariant sets We consider discrete dynamical systems
Xj+l = f (xj),
j -- O, 1, 2 . . . . .
(2.1)
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where f:]l~ n ---> ]t~n is a diffeomorphism and begin by recalling some types of invariant sets of such dynamical systems.
Attracting sets.
A subset A C ~n is called invariant if
f(A)=A. Moreover, an invariant set A is an attracting set with fundamental neighborhood U if for every open set V D A there is an N 6 N such that f J (U) Q V for all j ~> N. Observe that if A is invariant then the closure of A is invariant as well. Hence we restrict our attention to closed invariant sets A, and in this case we obtain
A -- A f j (U). jcN
By definition all the points in the fundamental neighborhood U are attracted by A. For this reason the open set UjcN f - J (U) is called the basin of attraction of A. If the basin of attraction of A is the entire ]1~n then A is called the global attractor. REMARKS 2.1. (a) Although the global attractor may not be compact, it typically happens in applications that all the orbits of the underlying dynamical system eventually lie inside a bounded domain, and in that case the compactness of A immediately follows. (b) The global attractor contains all the invariant sets of the dynamical system. This can easily be verified using the definitions.
Chain recurrent sets. Sometimes it is of interest to analyze the fine-structure of the dynamics on the global attractor. This is, e.g., accomplished by extracting recurrent subsets. A notion of recurrence which proved to be particularly useful is that of chain recurrence (see Conley [5]): DEFINITION 2.2. A point x E U C IR" belongs to the chain recurrent set of f in U if for every s > 0 there is an s-pseudoperiodic orbit in U containing x, that is, there exists { x - - x0, x I . . . . . xe_ l } C U such that
]lf(xi)-xi+,model]
<~e
fori-O
..... g-1.
It is easy to see that the chain recurrent set is closed and invariant.
2.2. The computation of relative global attractors We now present an algorithm for the computation of parts of the global attractor of a dynamical system.
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Relative global attractors. DEFINITION 2.3. Let Q c IRn be a compact set. We define the global attractor relative to Q by
AQ - ~ f J (Q). j>~o
(2.2)
In the following remark we summarize some basic properties of A Q. REMARKS 2.4.
(a) The definition of AQ in (2.2) implies that AQ C Q and that f - 1 (AQ) C AQ, but not necessarily that f (A 0) C A Q. (b) A 0 is compact since Q is compact. (c) A Q is a subset of the global attractor A. In fact,
AQ -- {x 6 A" f - J (x) ~ Q for all j >~O}. (d) Denote by A the global attractor of f . Then in general
AQ~AAQ. Subdivision algorithm. The following algorithm provides a method for the approximation of relative global attractors. It generates a sequence B0,/31 . . . . of finite collections of compact subsets of IR" such that the diameter diam(/3k) = max diam(B) converges to zero for k --+ oo. Given an initial collection/30, we inductively obtain/3k from ~k-1 for k - 1,2 . . . . in two steps" (i) Subdivision: Construct a new collection/3k such that U
B-
U
B
(2.3)
and (2.4)
diam(~'k) ~< Ok diam(Bk_ l), where 0 < 0min ~ Ok ~ 0max < 1. (ii) Selection" Define the new collection/3k by 13, -- {B ~ &" 3B" E & such
that f - l (B) (q B"~k fJ}.
(2.5)
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M. Dellnitz and O. Junge
By construction k xdiam(Bo)~0 diam(Bk) ~< 0ma
fork~cx).
EXAMPLE 2.5. We consider f : R --+ R, f (x) -- otx,
where ot 6 (0, 89 is a constant. Then the global attractor A = {0} of f is a stable fixed point. We begin the subdivision procedure with/3o - {[ - 1 , 1]} and construct Bk by bisection. In the first subdivision step we obtain A
z,-
B5 - {[-1, o], [o, ~]}.
No interval is removed in the selection step, since each of them is mapped into itself. Now subdivision leads to
Applying the selection rule (2.5), the two boundary intervals are removed, i.e.,
{[ Proceeding this way, we obtain after k subdivision steps
{[ 10]E0 We see that the union [.-Jsct3k B is indeed approaching the global attractor A = {0} for k --+ cx~. The speed of convergence obviously depends on the contraction rate of the global attractor. We will come back to this observation in Section 2.3. Convergence result. The abstract subdivision algorithm does converge to the relative global attractor A Q. In fact for the nested sequence of sets
Ok -- U
B,
k=0,1,2
(2.6)
BEBk
one can show the following result (see Dellnitz and Hohmann [8]). PROPOSITION 2.6. Let AQ be a global attractor relative to the compact set Q, and let 13o be a finite collection o f closed subsets with Qo - USEI3o B -- Q. Then lim h(AQ, Qk) = O, k-----~cx~
Set oriented numerical methods for dynamical systems
227
where h(B, C) denotes the usual Hausdorff distance between two compact subsets B, C c R n . 2.3. Convergence behavior and error estimate We begin by recalling the definition of a hyperbolic set (see, e.g., Shub [43]). DEFINITION 2.7. Let A be an invariant set for the diffeomorphism f . We say that A is
a
U hyperbolic set for f if there is a continuous invariant splitting TAR" = ESA G E A,
Df(EI~. ) _ Ef(x )s
and
D f ( uE, x) -- Euf(x),
for which there are constants c > 0 and )v E (0, 1), such that: (a) if v E E S~,then [[Df j (x)vi[ ~< C~j ]IV[[ for all j E N; (b) if v E Eta!, then [[Df-J (x)v[[ ~< c)U [[v[[ for all j E N. Since the estimates in Definition 2.7 are formulated in terms of the Jacobians, they are just valid infinitesimally for f . A consideration of the asymptotic behavior with respect to the diffeomorphism f itself leads to the definition of stable and unstable manifolds. DEFINITION 2.8. For x E R" and e > 0 we define the local stable (unstable) manifold by
W~~(x) - {y E N"" d ( f j (x), f J (y)) --+ 0 for j --+ ec and d ( f j (x), f J (y)) <~s for all j ~> 0},
W~' (x) -- {y E R"" d ( f j (x), f J (y)) --+ 0 for j --+ - e c and d ( f j (x), f J (y)) <<.e for all j ~< 0}. We now state part of the results which are known as the Stable Manifold Theorem for hyperbolic sets. A proof can be found in, e.g., [43]. THEOREM 2.9. Let A be a closed hyperbolic set for f . Then there is a positive e such
that for every point x E A, W s (x) and W u (x) are embedded disks of dimension equal to those of E~r and E n, respectively. The tangent space of W s (x) (W~ (x)) at x is E~r (EU). Moreover, W s (x) and W~' (x) satisfy the following properties: (i) There is a constant C > 0 such that d ( f J ( x ) , f J ( y ) ) < , c,kJd(x,y)
forallyEWS(x) andj)O,
d ( f - J ( x ) , f - J ( y ) ) <~c)vJd(x,y)
forall y E W~t(x) a n d j >~0,
where )~ is chosen according to Definition 2.7. (ii) The local stable and unstable manifolds are given by WS(x) - {y E N n" d ( f J ( x ) , f J ( y ) ) <~eforallj >~0}, W ~ ' ( x ) - {y E IR"" d ( f J ( x ) , f J ( y ) ) <. s forall j <~0}.
M. Dellnitz and O. Junge
228
We may use Theorem 2.9 to obtain a result on the convergence behavior of the subdivision algorithm in the case where the relative global attractor is part of an attracting compact hyperbolic set A. Define for a > 0
U~r(A)- {y E R n" i f y E W S ( x ) , x 6 A , thend(x, y) < a}. Obviously, A is a subset of U~(A). In the following result we assume for simplicity that 130 = {Q }. A proof can be found in Dellnitz and Hohmann [8]. PROPOSITION 2.1 0. Let AQ be a global attractor of the diffeomorphism f q relative to
the closed set Q, and suppose that A Q is an attracting compact hyperbolic set of f . Let p ~ 1 be a constant such that for each compact neighborhood Q of A Q we have h(AQ, Q) <~6 ~
Q_.C Up~(AQ).
(2.7)
Then the coverings Qk obtained by the subdivision algorithm for f q satisfy h(aQ, Qk) ~ diam(/3k) (1 + ot + Ot2 -+-...-q-otk),
(2.8)
where ot = Cio)~q/0min and C, )~ are the characteristic constants of the underlying hyperbolic set (see Theorem 2.9). REMARKS 2.1 1. (a) Geometrically it is evident that close to A Q both constants C and p are of order one. (b) Recently the estimate on h (A Q, Qk) in Proposition 2.10 has been used to develop an efficient global zero finding procedure (see Dellnitz et al. [ 13]). There the underlying idea is to view iteration schemes such as Newton's method as specific dynamical systems. COROLLARY 2.1 2. If the power q is chosen such that c~=
Cp• q
<1,
Omin
then we have for all k 1
h(AQ, Qk) <<,-1 --
diam(/3k).
2.4. Numerical examples EXAMPLE 2.13. We begin by considering a two dimensional dynamical system, the (scaled) H6non map
f (x) --
1 - ax 2 + X2/5) 5bxi
(2.9)
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229
Fig. 1. Successivelyfiner coverings of the global Hrnon attractor.
The computations are performed with b = 0.2 and a = 1.2. Starting with the square [ - 2 , 2] 2, we display in Figure 1 the coverings obtained by the algorithm after k = 6, 8, 10, 12 subdivision steps. For details concerning the implementation of the algorithm see Section 8. In Figure 2(a) we show the rectangles covering the relative global attractor after 18 subdivision steps. After this number of steps the diameter of the boxes is already 0.011. We remark that a direct simulation would not yield the same result. In Figure 2(b) we illustrate this fact by showing the attractor that appears if the transient behavior has been neglected. The reason for the difference lies in the fact that the subdivision algorithm covers all invariant sets in [ - 2 , 2] 2 - together with their unstable manifolds. In particular, the onedimensional unstable manifolds of the two fixed points (marked with circles in Figure 2(b)) are approximated- but those cannot be computed by direct simulation.
M. Dellnitz and O. Junge
230
Fig. 2. (a) Approximation of the relative global attractor for the H6non mapping after 18 subdivision steps; (b) attractor of the H6non mapping computed by direct simulation. The two fixed points are marked with o.
EXAMPLE 2.14. In this example we consider the following system of first order ordinary differential equations known as Chua's circuit,
(
2 -- o~ y - m o x -
1 3)
-~mlx
,
~--x-y+z, -
_r
In the computations we have chosen ot = 18, fl = 33, m0 = - 0 . 2 and ml = 0.01. We consider the diffeomorphism f given by the corresponding time-one-map, and approximate the relative global attractor inside Q = [ - 1 2 , 12] • [-2.5, 2.5] • [ - 2 0 , 20]. The results of the subdivision algorithm for k = 8, 11, 20 steps are displayed in Figure 3. In this figure we also show an approximation of the attractor obtained by direct simulation. With each of the set oriented computations we have covered the union of the global unstable manifolds of the three steady state solutions contained in Q. Again we refer to Section 8 for the details concerning the implementation of the subdivision algorithm.
2.5. The computation o f chain recurrent sets The subdivision algorithm can easily be modified in such a way that one can approximate the chain recurrent set within a given compact set Q c R n. Again we construct a sequence B0, B1 . . . . of finite collections of compact subsets of Q creating successively tighter coverings of the desired object. Set B0 = { Q J. For k = 1, 2 . . . . the collection Bg is obtained from B~-I in two steps: (i) Subdivision: Subdivide each set in the current collection Bk-1 into sets of smaller diameter;
Set oriented numerical methods for dynamical systems
231
Fig. 3. (a)-(c) Successively finer coverings of a relative global attractor for Chua's circuit; (d) approximation obtained by direct simulation.
(ii) Selection: Construct a directed graph whose vertices are the sets in the refined collection and by defining an edge from vertex B to vertex B', if
f (B) f-) B' =fi 0.
(2.10)
Compute the strongly connected components of this graph and discard all sets of the refined collection which are not contained in one of these components. REMARK 2.15. Recall that a subset W of the nodes of a directed graph is called a strongly connected component of the graph, if for all w, ~ 6 W there is a path from w to ~. The set of all strongly connected components of a given directed graph can be computed in linear time (Mehlhorn [36]). Intuitively it is plausible that the sequence of box coverings/3~ converges to the chain recurrent set of f . Indeed, under mild assumptions on the box coverings one can prove convergence, see Eidenschink [19], Osipenko [39].
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2.6. Numerical example We consider the following scenario and c o n c l u s i o n - the latter one is an application of the Wazewski T h e o r e m - which goes back to Conley [5]: Let ~ot denote a flow of an ordinary differential equation on 1~3 with the following properties: there is a cylinder of finite length such that outside the cylinder trajectories run vertically downward with respect to the cylinder. Assume further that there is some solution running through the cylinder which makes a knot as it goes from top to bottom. Then there must be a nontrivial invariant set inside the cylinder. Using the subdivision algorithm described above one can compute a covering of the chain recurrent set in the c y l i n d e r - see Dellnitz et al. [ 12] for details on how to explicitly construct the vector field with the desired properties. Figure 4, which has been produced together with Martin Rumpf and Robert Strzodka (both University of Bonn), shows the knotted trajectory and a coveting of the chain recurrent set in blue after 30 subdivision steps.
Fig. 4. Invariant set in a knotted flow. The covering of the chain recurrent set is shown in blue. The knotted trajectory that defines the flow is colored red.
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3. The computation of invariant manifolds We now present a set oriented method for the computation of invariant manifolds. Although the method can in principle be applied to manifolds of arbitrary hyperbolic invariant sets we will restrict, for simplicity, to the case where the underlying invariant set of (2.1) is a hyperbolic fixed point.
3.1. Description of the method The continuation starts at a hyperbolic fixed point p with the unstable manifold W u (p). We fix once and for all a (large) compact set Q c ]~n containing p, in which we want to approximate part of W u (p). To combine the subdivision process with a continuation method, we realize the subdivision using a family of partitions of Q. A partition 7") of Q consists of finitely many subsets of Q such that U BBE7~
O and
BNB ~-o
for all B, B t E 79 with B ~ B t.
Let 79e, g E N, be a nested sequence of successively finer partitions of Q, requiring that for all B E 79e there exist B1 . . . . . Bm E 79~+1 such that B = U i Bi and diam(Bi ) <~0 diam(B) for some 0 < 0 < 1. A set B E 79e is said to be of level ~. Let C E 79e be a neighborhood of the hyperbolic fixed point p such that the global attractor relative to C satisfies
Ac = WlUc(p) N C. Applying the subdivision algorithm with k subdivision steps to B0 = {C}, we obtain a covering Bk C 79e+k of the local unstable manifold WlUc(p) n C, that is,
Ac
-
WlUc(p) n C C U
B.
(3.1)
B EI3k
By Proposition 2.6, this covering converges to WlUc(p) N C for k --+ ec.
Continuation method. We are now in the position to describe a continuation algorithm for ..(k) ~.(k) the approximation of unstable manifolds. For a fixed k we define a sequence (50 , (51 . . . . (k) of subsets Cj c 7~e+k by (i) Initialization: C(ok) - Bk. (ii) Continuation: For j = 0, 1, 2 , . . . define
c(k) j+l -- {B E 7~+k" B n f ( B ' ) ~ O for some B' E Cj(k) }.
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234 Observe that the sets (k)
Cj--UB
BEC~k) form nested sequences in k, i.e.,
Cj(0) DCj(1) D ' "
for j - - 0 , 1 , 2
.....
3.2. Convergence behavior and error estimate
Convergence result.
Set W0 = WlUc(p) n C and define inductively for j = 0, 1, 2 . . . .
Wj+I = f (Wj) N Q. Then it is not too difficult to prove the following convergence result (see Dellnitz and Hohmann [7]).
(~)
PROPOSITION 3.1. The sets Cj
are coverings of Wj for all j, k = O, 1 . . . . . Moreover,
for fixed j, Cj(k) converges to Wj in Hausdorff distance if the number k of subdivision steps in the initialization goes to infinity. It can in general not be guaranteed that the continuation method leads to an approximation of the entire set W u (p) n Q. The reason is that the unstable manifold of the hyperbolic fixed point p may "leave" Q but may as well "wind back" into it. If this is the case then it can indeed happen that the continuation method, as described above, will not cover all of
W u (p) N Q. Error estimate.
Observe that the convergence result in Proposition 3.1 does not require the existence of a hyperbolic structure along the unstable manifold. However, if we additionally assume its existence then we can establish results on the convergence behavior of the continuation method in a completely analogous way as in Dellnitz and Hohmann [8]. To this end assume that p is an element of an attractive hyperbolic set A. Then the unstable manifold of p is contained in A. Choose
s
for some sufficiently small ~ > 0. Note that A = AQ. As in (2.7) let p >/1 be a constant such that for every compact neighborhood Q c Q of A a we have
h(AQ, Q) <<,3 ~
Q C Upa(AQ).
A proof of the following result can be found in Junge [26].
(3.2)
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PROPOSITION 3.2. Assume that in the initialization step of the continuation method we
have h(Wo, C(Ok)) <~r diamC'(k),,,0 for some constant ~ > O. If c- .j( k ) C Wos (Wj) for j --0, 1,2 . . . . . J, then
h(W ,Cj(k))<. diamC~k) max(~', 1
fi2
+...+
flj
(3.3)
for j -- 1, 2 . . . . . J. Here fl = C)~p and C and )~ are the characteristic constants of the hyperbolic set A (see Theorem 2.9). The estimate (3.3) points up the fact that for a given initial level k and )~ near 1 - corresponding to a weak contraction transversal to the unstable manifold- the approximation error may increase dramatically with an increasing number of continuations steps (increasing j).
3.3. Numerical examples EXAMPLE 3.3. As the first example we compute an approximation of a two-dimensional stable manifold of the origin in the Lorenz system
2 =a(y -x), ~=px-y-xz, ~ = - f l z + xy. In this computation we have chosen the "standard" set of parameter values, that is a = 10, p = 28 and fl = 8/3. With this choice a direct numerical simulation would lead to an approximation of the celebrated Lorenz attractor. (For illustrations as well as a discussion of topological properties of the Lorenz attractor the reader is referred to Guckenheimer and Holmes [21 ].) Since we want to compute the two-dimensional stable manifold of the origin, we proceed backwards in time and apply the continuation method to the diffeomorphism given by the time-(-T)-map. Starting in a neighborhood of (0, 0, 0) we approximate the stable manifold inside Q - [ - 2 5 , 25] 3. To demonstrate the continuation process, we begin with a rough approximation using the initial level e = 9 and k = 3 subdivision steps. In Figure 5 we display the coverings obtained by the algorithm after j = 0, 1, 3 and 5 continuation steps. We remark that in this case the stable eigenvalues are both real but the ratio of strong and weak contraction is relatively large. This is also reflected by the way the covering is growing (see Figure 5). A finer resolution (e = 21, k = 0, 10 steps, Q = [-120, 120] x [-120, 120] x [-160, 160]) is shown in Figure 6.
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Fig. 5. Continuation steps for the stable manifold of the origin in the Lorenz system for j = 0, 1, 3, 5.
Fig. 6. Covering of the two-dimensional stable manifold of the origin in the Lorenz system.
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EXAMPLE 3.4. As the second example let us consider a Hamiltonian system, the Circular Restricted Three Body Problem. Its equations of motion in a rotating frame are given by
~; -- l), Z -- //),
it = 2v + x + cl (x + # - 1) + C2(X -k- I-t), ~) = - 2 u + y + (el -k-r /b ~- (C1 -+- r
(3.4)
where #
1-#
m
Cl --
( ( x -q-/z -- 1) 2 -k- y2 _+_ Z2)3/2 '
c2 = -- ( ( x -q- ~ ) 2 + y2 + Z2)3/2
and tz = m 1/(ml + m2) is the normalized mass of one of the primary bodies. We use the value/~ -- 3.040423398444176 x 10 -6 for the sun/earth system here. We aim for the computation of the unstable manifold of a certain unstable periodic orbit. As it was pointed out in the error estimates in Section 3.2 a naive application of the continuation method would - due to the Hamiltonian nature of the system - not lead to satisfactory results in this case. We therefore apply a modified version of this method, see Junge [26]. Roughly speaking the idea is not to continue the current covering by considering one application of the map at each continuation step, but instead to perform only one continuation step while computing several iterates of the map. More formally, we replace the second step in the continuation method by: (ii) Continuation: For some J > 0 define
c(jk) = { B E 7)e+k 9 30 <~ j <~ J" B N f J (B') # 0 for some B' ~ C(ok) }. The convergence statement in Proposition 3.1 is adapted to this method in a straightforward manner. One can also show t h a t - as intended- the Hausdorff-distance between compact parts of the unstable manifold and the computed covering is of the order of the diameter of the partition, see Junge [26] for details. However, and this is the price one has to pay, one no longer considers short term trajectories here and therefore accumulates methodological and round-off errors when computing the iterates f J. A second advantage of the modified continuation method is that whenever the given dynamical system stems from a flow Ct one can get rid of the necessity to consider a time-T-map and instead replace the continuation step by (ii) Continuation: For some T > 0 define
C~k) = {B ~ Pe+k" 30 ~< t ~< T" B CqCt (B') 7~ 0 for some B' ~ C~k) }. This facilitates the usage of integrators with adaptive step-size control and finally made the computations for the Restricted Three Body Problem feasible. Figure 7 shows the result of the computation, where we set T = 7 and used an embedded Runge-Kutta scheme of order 8(7) (see Dormand and Prince[ 18]) with error tolerances set to 10 -9. See again Junge [26] for more details on this computation.
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Fig. 7. Covering of part of the global unstable manifold of an unstable periodic orbit in the Circular Restricted Three Body Problem. The blue body depicts the earth, the black trajectory is a sample orbit which leaves the periodic orbit in the direction of the earth.
4. The computation of SRB-measures An important statistical characterization of the behavior of a dynamical system is given by so-called SRB (Sinai-Ruelle-Bowen) measures. The important property of these invariant measures is, roughly speaking, that they lend weight to a region in phase space according to the probability by which "typical" trajectories visit this region. In this section we present a set oriented numerical method for the approximation of SRB-measures. The main idea of the approach is to define an operator (the Perron-Frobenius operator) on the space of probability measures whose fixed points are invariant measures, then to discretize this operator via a Galerkin method and finally to compute fixed points of the resulting matrix as an approximation to an invariant measure. Using spaces of piecewise constant functions on a partition of the underlying phase space this approach is commonly known as "Ulam's method", see Ulam [46]. There exist various statements about the convergence properties of Ulam's method, see, e.g., Li [35], G. Keller [28], Ding et al. [16],
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Ding and Zhou [ 17], Froyland [20]. In the following sections we are going to establish a convergence result for uniformly hyperbolic systems by combining a theorem of Kifer [30] on the stochastic stability of SRB-measures with results on the spectral approximation of compact operators.
4.1. Brief review on SRB-measures and small random perturbations Our aim is to obtain information about the statistical behavior of (deterministic) discrete dynamical systems of the form (2.1) where f :X --+ X is a diffeomorphism on a compact subset X C R n.
SRB-measures.
We denote by B the Borel a-algebra on X and by m the Lebesgue measure on B. Moreover, let .A4 be the space of probability measures on B. Recall that a measure # 6 A,4 is invariant if #(B) = # ( f - l ( B ) )
for all B E/3.
An invariant measure # is ergodic if # ( C ) 6 {0, 1}
for all invariant sets C 6 B.
Now we recall the notion of an SRB-measure. There exist several equivalent definitions in the situation where the underlying dynamical behavior is Axiom A, and we state one of them. DEFINITION 4.1. An ergodic measure # is an SRB-measure if there exists a subset U C X with m(U) > 0 and such that for each continuous function
lim - N--+c~ N
ap(f J (x)) j=0
f
~p d #
(4.1)
for all x 6 U. REMARKS 4.2. (a) Recall that (4.1) always holds for #-a.e. x 6 X by the Birkhoff Ergodic Theorem. The crucial difference for an SRB-measure is that the temporal average equals the spatial average for a set of initial points x 6 X which has positive Lebesguemeasure. This is the reason why this measure is also referred to as the natural or the physically relevant invariant measure. (b) The concept of SRB-measures in the context of Anosov systems has been introduced by Y.G. Sinai in the 1960s (e.g., Sinai [44]). Later the existence of SRB-measures has been shown for Axiom A systems by R. Bowen and D. Ruelle (see Ruelle [40], Bowen and Ruelle [3]). It should be mentioned as well that also Lasota and Yorke
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have proved the existence of these measures for a particular class of interval maps already in 1973, Lasota and Yorke [33]. M. Benedicks and L.-S. Young have shown that the H6non map has an SRB-measure for a "large" set of parameter values, Benedicks and Young [ 1]. More recently, Tucker proved the existence of an SRBmeasure for the Lorenz system, Tucker [45].
Stochastic transition functions. Although our aim is to consider deterministic systems it turns out to be more convenient to consider the stochastic context first. DEFINITION 4.3. A function p : X x 13 --+ [0, 1] is a stochastic transition function, if (i) p(x, .) is a probability measure for every x E X, (ii) p(., A) is Lebesgue-measurable for every A 6/3. Let ~y denote the Dirac measure supported on the point y 6 X. Then p(x, A) = ~ h ( x ) ( A ) is a stochastic transition function for every m-measurable function h. We will see below that the specific choice h -- f represents the deterministic situation in this more general set-up. We now define the notion of an invariant measure in the stochastic setting. DEFINITION 4.4. Let p be a stochastic transition function. If # E .M satisfies
~(A) - f p(x, A) dlz(x) for all A E B, then/z is an invariant measure of p. The following example illustrates the previous remark that we recover the deterministic situation in the case where p(x, .) = 8f(x). EXAMPLE 4.5. Suppose that p(x, .) = ~f(x) and let # be an invariant measure of p. Then we compute for A 6 B
#(A)--f p(x,A)d#(x)-f ,f(x)(A)dlz(x)f xA(f(x))d#(x) = # ( f - 1 (A)), where we denote by XA the characteristic function of A. Thus,/z is an invariant measure for the diffeomorphism f .
Small random perturbations. Now we assume that for every x 6 X the probability measure p(x, .) is absolutely continuous with respect to the Lebesgue measure m. Hence we may write p(x, .) as p(x, A) - / A k(x, y)dm(y)
for all A E B,
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with an appropriate transition density function k: X x X -+ R. Obviously,
k(x, .) ~ L1 (X, m)
and
k(x, y) ~ O.
In this case we also call the stochastic transition function p absolutely continuous. Note that
f k(x, y ) d m ( y ) - p(x, X) - 1 for all x 6 X. We now specify concretely the stochastic transition function p which is the theoretical tool for the derivation of a convergence result to SRB-measures. Recall that the purpose is to approximate the SRB-measure of a deterministic dynamical system represented by a diffeomorphism f . Hence the stochastic system that we consider should be a small perturbation of this original deterministic system. For e > 0 we set
ke(x, y) -- ~ X8 enm(B)
(y-x)
,
x, y 6 X.
(4.2)
Here B = B0(1) denotes the open ball in R n of radius one and XB is the characteristic function of B. Obviously ke(f(x), y) is a transition density function and we may define a stochastic transition function PE by
pe(x, A) -- f A ke(f(x), y)dm(y).
(4.3)
REMARK 4.6. Note that pe(x, .) -+ 8f(x) for e ~ 0 uniformly in x in a weak*-sense. Hence the Markov process defined by any initial probability measure/x and the transition function pe is a small random perturbation of the deterministic system f in the sense of Kifer [30].
4.2. Spectral approximation for the Perron-Frobenius operator The main purpose of this section is to describe an appropriate Galerkin method for the approximation of a specific type of transfer operator, namely the Perron-Frobenius operator. This operator is used for translating the problem of finding an invariant measure into a fixed point problem.
The Perron-Frobenius operator. DEFINITION 4.7. Let p be a stochastic transition function. Then the Perron-Frobenius
operator P : A A c -+ .Mc is defined by PI~(A) -- f p(x, A)dl~(x),
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where ,A//c is the space of bounded complex valued measures on B. If p is absolutely continuous with density function k then we may define the Perron-Frobenius operator P on L 1 by P g ( y ) --
f k(x y ) g ( x ) d m ( x )
for all g 6 L 1
REMARKS 4.8. (a) By definition a measure/z 6 .M is invariant if and only if it is a fixed point of P. In other words, invariant measures correspond to eigenmeasures of P for the eigenvalue one. Moreover, let )~ ~ C be an eigenvalue of P with corresponding eigenmeasure v, that is, P v -- )~v. Then in particular )~vCX) -- PvCX) --
f pcx x ) d r ( x ) -
v(x)
since p ( x , X) = 1 for all x 6 X. It follows that v ( X ) = 0 if )~ :~ 1. (b) Observe that in the deterministic situation where p ( x , .) = 8f(x) we obtain
f
p(x, A) dtz(x) - - / z ( f -1 (A))
(cf. Example 4.5). This is indeed the standard definition of the Perron-Frobenius operator in the deterministic setting (see, e.g., Lasota and Mackey [32]). Spectral information for the Perron-Frobenius operator cannot just be used for the approximation of SRB-measures but also for the identification of cyclic dynamical behavior, that is, there exist finitely many different compact subsets in state space which are cyclically permuted by the underlying dynamical system. In the stochastic setting this corresponds to the situation where there are disjoint compact subsets Xj C X , j -- 0 . . . . . r - 1, such that r-1
X--UX j , j=O and for which the stochastic transition function p satisfies p ( x , Xj+l modr) --
1
0
if x 6 X j, otherwise.
(4.4)
We now relate the cyclic dynamical behavior described by (4.4) to spectral properties of the corresponding Perron-Frobenius operator P. PROPOSITION 4.9. If the stochastic transition function p satisfies (4.4) then the following statements hold:
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(a) The r-th p o w e r p r o f the P e r r o n - F r o b e n i u s operator P has an eigenvalue one o f multiplicity at least r. Moreover, there are r corresponding invariant measures tXk ~ ,All, k = O, 1 . . . . . r - 1, with support on Xk, that is, s u p p ( # k ) C Xk. These measures can be chosen to satisfy #k = pklzo,
k = 0 , 1 . . . . . r - 1.
k k(b) The rth roots o f unity m r,
O, 1 , . . . , r -
1, where O r -
e 2rri/r, are eigenvalues
o f P.
A proof of this result can be found in Dellnitz and Junge [ 11]. The Galerkin method. We begin with the following observation which immediately follows from standard results on integral operators (see, e.g., Yosida [47, p. 277]).
LEMMA 4.10. S u p p o s e that the transition density f u n c t i o n k satisfies
fflk(x,
y)12dm(x)dm(y)
< ec.
(4.5)
Then the P e r r o n - F r o b e n i u s operator P : L 2 --+ L 2 is compact.
From now on we consider the case where P is given by a dynamical process with a transition density function k satisfying the condition (4.5). The aim is to use a Galerkin method for the approximation of such a Perron-Frobenius operator together with its spectrum. More precisely, let Vd, d >~ 1, be a sequence of d-dimensional subspaces of L 2 and let Qd :L2 __+ Vd be a projection such that Qd converges pointwise to the identity on L 2. If we define the approximating operators by Pd = Qd P then we have IIed -- P 112 ---->0
as d --+ co.
Since P is compact one can use standard results from operator theory in order to approximate the eigenvalues of P which are lying on the unit circle by a Galerkin method. For this we construct a Galerkin projection which preserves cyclic behavior in the approximation. Suppose that (4.4) holds and let {qg/}, j = 0, 1 . . . . . r - 1, i - 1, 2 . . . . , d j , be a basis of Vd with the following properties: (i) supp(9 J) C X j ( j -- O, 1 , . . . , r - 1, i -- 1, 2 . . . . . d j ) , dj (ii) ~
(4.6)
q)/(x) -- 1 for all x E X j, j - O, 1 . . . . . r - 1.
i=1
REMARK 4.1 1. In Section 8 we will show how to generate a basis satisfying (4.6) in practice. In that case, Vd will consist of functions which are locally constant.
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The Galerkin projection Qctg of g E L 2 is defined by (Qdg, 9 / ) = (g, ~o/) for all i, j,
where (.,-) is the usual inner product in L 2. The following result is a generalization of Lemma 8 in Ding et al. [ 16], where just the fixed point of P is considered. Its proof can be found in Dellnitz and Junge [ 11 ]. Recall that (-Or = e 2 z r i / r . PROPOSITION 4.12. Suppose that the Galerkin projection satisfies (4.6). Then the ap~ k - 0 , 1 , ... , r - 1 . proximating operators Pd -- Qd P also possess the eigenvalues o r, A combination of standard results on the approximation of spectra of compact operators (see, e.g., Osborn [38]) with Proposition 4.12 yields a convergence result for eigenvectors corresponding to eigenvalues of P of modulus one. COROLLARY 4.13. Suppose that P and its approximation Pa satisfy the hypotheses stated above. Then each simple eigenvalue e 2zcik/r of P on the unit circle is an eigenvalue of Pa and there are corresponding eigenvectors ga of Pd converging to an eigenfunction h of P. More precisely, there is a constant C > 0 such that for all d ~ 1 IIh - gd II2 ~ C IIed -- P II2.
4.3. Convergence result for SRB-measures Suppose that the diffeomorphism f possesses a hyperbolic attractor A with an SRBmeasure/ZSRB, and let PE be a small random perturbation of f . Then, under certain hypotheses on pe, it is shown in Kifer [30] that the invariant measures of pe converge in a weak*-sense to /ZSRB as e --+ 0. On the other hand one can approximate the relevant eigenmeasures of Pe by Corollary 4.13 and this leads to the desired result. THEOREM 4.14. Suppose that the diffeomorphism f has a hyperbolic attractor A, and that there exists an open set UA D A such that if x ~ f(UA) and y q~ UA.
ke(x, y) - 0
Then the transition function PE in (4.3) has a unique invariant measure zrE with support on A and the approximating measures
lZ~d(A) = fA ge converge in a weak*-sense to the SRB-measure IZSRB of f as e ~ 0 and d --+ oo, R.
lim lim /z5 --/ZSRB. e---~ 0 d--+ o o
(4.7)
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4.4. Numerical examples We present two examples for the set oriented numerical computation of invariant measures. Although the convergence result Theorem 4.14 is stated in the randomly perturbed context these numerical computations are performed using the unperturbed dynamical systems. EXAMPLE 4.15. Let us begin with a one-dimensional example, the Logistic Map f:[O, 1]-~ [0, 1],
f ( x ) =)~x(1 - x ) for ~. = 4. The unique absolutely continuous invariant measure/x of f has the density
h(x) =
rr ~/x (1 - x )
(see, e.g., Lasota and Mackey [32]). Discretizing the Perron-Frobenius operator on the space of simple functions on a uniform partition of [0, 1] we obtain an approximation of h in terms of a piecewise constant function. Figure 8 shows approximations to h using two different partitions with intervals of size 2 -4 and 2 -8 , respectively. EXAMPLE 4.16. As a more challenging task we consider the approximation of an invariant measure in the Lorenz system (see Section 3.3). We will tackle this by first computing a covering of the underlying invariant set. More concretely the continuation method is used to compute a tight covering of the unstable manifolds of the two nontrivial steady state solutions. Using the space of simple functions on the resulting collection of sets we then compute the stationary vector of the discretized Perron-Frobenius operator as an approximation to an invariant measure.
Fig. 8. Approximationsof the absolutely continuous invariant measure of the Logistic Map (black) using piecewise constant functions (gray).
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246
Fig. 9. Approximationof an invariant measurein the Lorenz system.The color depicts the density of the (discrete) invariant measure and ranges from blue (lowest density) over pink, green and red to yellow (highest density).
In Figure 9 we show the result of this computation for the time-0.2-map on subdivision level 30 (/3 = 1.2, Q = [ - 3 0 , 30] • [ - 3 0 , 30] • [ - 13, 67]). A color coding has been used to indicate the values of the invariant density on the covering.
5. The identification of cyclic behavior Suppose that the stochastic transition function of the randomly perturbed dynamical system satisfies the cycle condition (4.4). Then the purpose is to identify the components Xj.
5.1. Extraction of cyclic behavior k By Proposition 4.12 we know that the approximating operator P~ has the eigenvalues (.Or, k = 0, 1 . . . . . r - 1. The cyclic components can be approximated by certain linear combinations of the corresponding eigenvectors.
Two cyclic components.
In the simplest case, r = 2, there are two components X0 and X1 which are cyclically permuted by the underlying stochastic process. The idea is to find approximations of eigenmeasures #0 and # l - Pe/x0 of p2 with support on X0 and X1, respectively, see Proposition 4.9. By the same proposition we know that coo - 1 and o91 = - 1 are eigenvalues of Pe. Let v0 and Vl be corresponding (real) eigenmeasures. Then there are or0, otl 6 R such that vo = olo(#o + P~#o)
and
Vl = a l (#o - PE#O).
Set oriented numerical methods for dynamical systems R e s c a l i n g vo and Pl SO that vo(Xo) - 1)1 ( X o ) -
1
247
1 we can c o m p u t e / ~ o a n d / Z 1 by
1
/z0-z(v0+vl)
and
# l -- ~(v0 -- Vl). Z
Z
T h e s a m e p r o c e d u r e can be applied in order to find appropriate a p p r o x i m a t i o n s of the probability m e a s u r e s #0 and #1 for the G a l e r k i n a p p r o x i m a t i o n .
General case. For s = 0, 1 . . . . . r - 1 we d e n o t e b y / z e - P[lzo the invariant m e a s u r e of P[ with support on Xe (see P r o p o s i t i o n 4.9). LEMMA 5.1. For s ~ {0, 1 . . . . . r - 1} let r-1
(5.1) j=0 k be a specific choice f o r the eigenmeasures of Pe corresponding to the eigenvalues O)r, k - O , 1 . . . . . r - 1. Then r-1 1 Z s s O)r 1)k r k=0
#e+s modr.
B y this l e m m a we have to find eigenvectors v6 . . . . . vS_l of the discretized P e r r o n F r o b e n i u s o p e r a t o r w h i c h are a p p r o x i m a t i o n s of the e i g e n m e a s u r e s v ks in (5.1) for an s {0, 1 . . . . . r - 1 }. T h e n we can c o m p u t e
Us
mod r ~
1 r-1 Z
-r
s s OAr 1)k
k=O
for s = 0, 1 . . . . . r - 1, and the positive c o m p o n e n t s of u j provide the desired i n f o r m a t i o n about the support of # j o n X j ( j - - 0, 1 . . . . . r - 1). s In the case w h e r e the eigenvalues cork are simple the eigenvectors v os . . . . . Vr_ 1 are f o u n d as follows. S u p p o s e that we have a set of e i g e n m e a s u r e s Pk c o r r e s p o n d i n g to the eigenvalues cor,k k - - 0 , 1 , . . . , r - 1. Since the e i g e n v a l u e s are simple we k n o w that for each s 6 {0, 1 . . . . . r - 1 } there is a constant c~ E C such that Pk can be written as
H e n c e the task is to rescale Pk so that otks _ 1 for all k and this is d o n e by rescaling the p~'s by ( c o m p l e x ) factors so that for a particular s Pk(Xs)--i
for a l l k - 0 , 1
..... r-1.
With this choice it follows that Pk -- v~.
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248 5.2. Numerical examples
EXAMPLE 5.2. We reconsider Example 2.13 and set b = 0.2 and a = 1.2. Then the H6non map possesses a 2-cycle, and we can use the approximation procedure described above to identify the two components X0 and X1. In Figure 10 we show the approximations v0 and Vl of the two eigenmeasures of the Perron-Frobenius operator corresponding to the eigenvalues )~0 = 1 and ~,1 = - 1 . By L e m m a 5.1
1 UO = "~(VO-'[" Vl) 2
and
1 Ul = = ( v o - Vl) 2
are approximations of probability measures/x0 and ~1 which have support on X0 and X1 respectively. These are shown in Figure 11.
Fig. 10. Eigenvectors of the approximation of the Perron-Frobenius operator for the H6non map (a = 1.2, b =0.2).
Fig. 11. Approximations of probability measures with support on the two components of the 2-cycle (a = 1.2, b =0.2).
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249
Fig. 12. Approximation of the invariant measure for ot = - 1.7.
In the computation the box-covering was obtained by the continuation algorithm described in Section 3. The boxes were of size 1/21~ in each coordinate direction and the continuation was restricted to the square Q = [ - 2 , 2] 2 c IR2. This way we have produced a covering of the closure of the one-dimensional unstable manifold of the hyperbolic fixed point in the first quadrant by 2525 boxes. EXAMPLE 5.3. As the second example we slightly modify a mapping taken from Chossat and Golubitsky [4] and consider the dynamical system f :C --+ C, f ( z ) - - e -2zri/3
(
1)
([z 2 at- ot)z at- ~ ~,2 ,
for the parameter value oe = - 1 . 7 . For the computation of the box-covering we have used the subdivision algorithm as described in Section 2. Starting with the square Q = [ - 1 . 5 , 1.5] 2 we have subdivided Q seven times by bisection in each coordinate direction which leads to a box-covering by 3606 boxes. In Figure 12 we show the approximation of the invariant measure, that is, the eigenvector v0 corresponding to the eigenvalue )~0 -- 1 of the discretized Perron-Frobenius operator. In this case this operator additionally has .5, and . hence. we may . the eigenvalues 0 ) 6 . k - 1. use. L e m m a 5 1 to compute approximations v0 . . . . . v5 of the probability measures with support on the cyclic components X0 . . . . . X5 of a six cycle. These supports are shown in Figure 13.
6. The computation of almost invariant sets In the previous sections we have seen that we can approximate the physically relevant invariant measure or even cyclic behavior numerically by an appropriate Galerkin approximation. In practice - in particular in the area of molecular dynamics, Deuflhard et al. [ 14], Schtitte [42] - also the approximation of so-called almost invariant sets is of relevance.
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1 0.6 0.6
0.6
0.4
0.4
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0.2
0
0
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--0.2
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--0.8
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-!1
-11
P., y
-0.2
-0.8 -0.5
(a) uo
0
0.5
1
--1
(b) It1
(c) it2
1
1
08 ib
0.8 0.6
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~I
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.-0.8
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-1_
-0.5
0
0.5
(d) u3
1
-1
-1
-015
0
--1
0'.5
(e) u4
Fig. 13. Approximation of the cyclic components
r (f) u5
X 0 .....
X5
f o r ct - - - 1 . 7 .
Roughly speaking, these are sets in state space in which typical trajectories stay on average for quite a long time before leaving again. The concept of almost invariant sets can naturally be extended to the notion of almost cyclic behavior. For simplicity we will restrict to almost invariance but we will illustrate the existence of almost cyclic behavior by a numerical example. 6.1. Almost invariant sets The scenario we have in mind is the following: suppose that a dynamical system possesses two different invariant sets. Correspondingly the Perron-Frobenius operator has a double eigenvalue 1. Then these invariant sets merge while a system parameter is varied. Simultaneously one of the eigenvalues in one moves away from one. The aim is to relate the value of this eigenvalue to the magnitude of almost invariance which is still present in the dynamical system. EXAMPLE 6.1 (Dellnitz et al. [6]). Consider the following parameter dependent family of FourLegs maps Ts'[O, 1] ~ [0, 1]"
Set oriented numerical methods for dynamical systems
Tsx --
2x,
0~<x < 1/4,
s(x - 1/4),
1/4 <~ x < 1/2,
s(x - 3/4) + l,
1/2~<x<3/4,
2(x-
3/4 ~< x <~ 1.
1 ) + 1,
251
The graph of a typical T is shown in Figure 14. Obviously, for s = 2 the Perron-Frobenius operator for this map has a double eigenvalue one since both the intervals [0, 0.5] and [0.5, 1] are invariant sets. One can show that the Perron-Frobenius operator has isolated eigenvalues Xs ~ 1 for values of s which are arbitrarily close to 2. Moreover these eigenvalues approach 1 while s tends to 2; see Dellnitz et al. [6]. As in the case of SRB-measures we work in the following in the context of small random perturbations (see Section 4.1). DEFINITION 6.2. A subset A C X is 8-almostinvariantwith respectto p ~ .All if p ( A ) :/: 0 and
f a Pe(X, A) d p ( x ) - 3p(A). REMARK 6.3. (a) Using the definition of the stochastic transition function pe we compute for a subset ACX
p e ( x , A ) --
m(AABf(x)(e)) m(Bo(e))
1 3/4.
IIZ
I1~
0
i
1/2
3/4
Fig. 14. Graph of Ts for s = 4 - 1/8.
M. Dellnitz and O. Junge
252 Hence
_
1
[
m(A n Bf(x)(S))
p(A) JA (b) Recall that ps(x, .) ~
m(Bo(s)) r
dp(x).
for e ~ 0. Thus, we obtain in the deterministic limit
fAPO(
X, A) dp(x) = fA 8f(x)(A) dp(x) = p ( f - 1 (A) n A).
Therefore in this case ~ is the relative p-measure of the subset of points in A which are mapped into A. From now on we assume that ~. ~ 1 is an eigenvalue of Ps with corresponding real valued eigenmeasure v ~ .Mc, that is,
Psv = ~v. In this case v(X) = 0 (see Remark 4.8 (a)). In the following result, see Dellnitz and Junge [ 11 ], the value of the eigenvalue ~. is related to the number ~ in Definition 6.2. PROPOSITION 6.4. Suppose that v is scaled so that Ivl ~ M , and let A C X be a set with v(a) = 1. Then 8 + cr = ~ + 1,
(6.1)
if A is 6-alrnost invariant and X - A is a-almost invariant with respect to Iv [. REMARKS 6.5. (a) Observe that in the case where )~ is close to one we may assume that the probability measure Ivl is close to the invariant measure/z of the system. (b) In the numerical computations we work with the unperturbed equations rather than introducing noise artificially. Thus, it would be important to know whether the eigenvalues of P0 and Ps are close to each other for small s. First results concerning the stochastic stability of the spectrum of the Perron-Frobenius operator are obtained in Blank and Keller [2].
6.2. Numerical examples We illustrate the results by two numerical examples: first we identify numerically two almost invariant sets for Chua's circuit and then we present an almost invariant two cycle for the Hrnon map. EXAMPLE 6.6. Considering the time-0.1-map of Example 2.14 we cover - using the continuation method described in Section 3 - the unstable manifold of the origin by 10372
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253
Fig. 15. Illustration of the existence of two almost invariant sets in the Chua circuit: (a) Boxes corresponding to components of the approximating densities with value bigger than 10-4; (b) boxes corresponding to components of the approximating densities with value less than - 1 0 - 4 ; (c) superposition of the two almost invariant sets.
Fig. 16. Two almost invariant sets for Chua's circuit.
boxes. In addition to the eigenvalue one the discretized Perron-Frobenius operator does also possess the eigenvalue ~.1 - - 0 . 9 2 7 2 . We may conclude from this observation that there are two almost invariant sets. Indeed, a coarse numerical approximation of the corresponding regions in phase space leads to the result shown in Figure 15. The result of a more accurate computation is shown in Figure 16. A detailed numerical study of this particular example can be found in Dellnitz and Junge [9]. EXAMPLE 6.7. We reconsider the H6non map in Example 5.2 and set a = 1.272. For this parameter value the two cycle has disappeared, but in simulations the cycling behavior can still be observed for most iterates. Correspondingly we find that ~.1 = - 0 . 9 9 4 4 is an eigenvalue of the approximation of the Perron-Frobenius operator. Using the same notation as in Section 5 we show in Figure 17 the approximations of the eigenmeasures. In this case the box-covering consists of 3101 elements.
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Fig. 17. Eigenvectors v0, Vl of the approximation of the Perron-Frobenius operator and approximations u0, u i of probability measures which correspond to the two components of the almost 2-cycle for the H6non map (a -- 1.272, b -- 0.2).
7. Adaptive subdivision strategies The standard subdivision algorithm may approximate a part of the global attractor which is dynamically irrelevant in the sense that no invariant measure has support on this subset. The reason is that each box is subdivided in a step of the subdivision algorithm regardless of any information on the dynamical behavior. In particular, also those subsets of the relative global attractor corresponding to unstable or transient dynamical behavior are approximated by the standard procedure. On the other hand, if one is mainly interested in the approximation of the support of the (natural) invariant measure rather than in the precise geometric structure of the global attractor then this strategy may lead to unnecessary high storage and computation requirements. In the following we present a modified subdivision strategy (see Dellnitz and Junge [10]) which avoids this drawback: roughly speaking, - in the subdivision step we use the information on the actual approximation of the invariant measure to decide whether or not a box should be subdivided;
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- in the selection step we keep only those boxes which have a nonempty intersection with the support of the invariant measure. 7.1. Adaptive subdivision algorithm Let (Sk) be a sequence ofpositive real numbers such that 3k --+ 0 for k --+ cx~.The algorithm generates a sequence of pairs
(~0, U0), (B1, Ul), (B2, U2) . . . . . where the Bk's are finite collections of compact subsets of ]Rn and the discrete measures uk :Bk --+ [0, 1] can be interpreted as approximations to an SRB-measure #SRB:
uk(B) ~ /ZSRB(B)
for all B ~ Bk.
Given an initial pair (B0, u0), one inductively obtains (B~, Uk) from (Bk-1, Uk-1) for k = 1, 2 . . . . in two steps: (i) Subdivision: Define
{ B ~ ]~k-l" Uk-1 (B) >~Slr }.
]3k_ 1
Construct a new (sub-)collection/3~ such that B U B= U B~ i ~ B~~Sk+1
and
diam(B "+) ~< 0 diam(B~_l)
for some 0 < 0 < 1. (ii) Selection: Set B~ - - (Bk-1 \/3k_ + 1) U B fi'+ k . Using the space of simple functions on the collection Bk compute a fixed point fi'k of the discretized Perron-Frobenius operator. Set
Bk -- {B ~ Bk" ~ffk(B) > O}
and
Uk -- u'~lBk-
A result on the convergence of this method has been proven in the context of sufficiently regular stochastic transition functions, see Junge [27] for details. REMARKS 7.1. (a) In principle there is some freedom in choosing the sequence (3k) of positive numbers used in the subdivision step. Note that this sequence determines the number of boxes which will be subdivided and hence it has a significant influence on the storage requirement. In the computations we used 1
1
B~Bk
where Nk is the number of boxes in Bk.
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(b) One can think of more sophisticated ways of choosing the subcollection which is going to be refined in the subdivision step. In fact, one can show that one should aim for an estimate of the local error (between the true and the approximate invariant density) and subdivide only those boxes for which this estimate exceeds its average. See Guder and Kreuzer [23], Junge [26] for details on these alternative approaches.
7.2. Numerical examples In this section we illustrate the adaptive scheme by two numerical examples. First we consider the Logistic Map again. We will see that, as expected, the adaptive technique is particularly useful if the underlying invariant density has singularities. Additionally we consider the H6non map as a two-dimensional example and show the box refinement produced by the adaptive subdivision algorithm at a certain step. Again, we refer to Section 8 for details on the numerical realization. EXAMPLE 7.2. We have approximated the density h of the unique absolutely continuous invariant measure of the Logistic Map using: (a) piecewise constant functions on a uniform partition, and (b) the adaptive subdivision algorithm. Figure 18 shows the L 1-error between h and the approximate densities versus the cardinality of the partitions.
100
. . . . . . . .
i
. . . . . . . .
i
. . . . . . . .
i
. . . . . . . .
I
I n A
. . . . . . . .
I
. . . . . . . .
-
uniform partition Adaptive Subdivision algorithm
10 -1
10-2
..J
10-3
10 -4
10 -5 10 ~
. . . . . . . .
i 101
. . . . . . . .
I 10 2
,
,
,,
,,111
. . . . . . . .
10 3
i 10 4
,
i
......
i
. . . . . . .
10 5
number of boxes
Fig. 18. Ll-error between h and the approximate invariant densities versus the cardinality of the underlying partitions.
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257
Fig. 19. (a) A tiling of the square [-2, 2] 2 obtained by the adaptive subdivision algorithm; and (b) the subcollection/3 of boxes with discrete density bigger than 0.35 (see also (7.1)).
A more detailed analysis of this example using the adaptive algorithm can be found in Murray [37]. EXAMPLE 7.3. We apply the adaptive subdivision algorithm to the H6non map, see Example 2.13. In the computations we have chosen the parameters a = 1.2, b = 0.2, and considered the outer box/30 = {[-2, 2]2}. In Figure 19 we present a tiling of the square [ - 2 , 2] 2 obtained by the adaptive subdivision algorithm after several subdivision steps. The resulting box-collection B consists of the grey boxes shown in part (a) of this figure. We expect that due to the numerical approximation some boxes have positive discrete measure although they do not intersect the support of the real natural invariant measure. Having this in mind we negl~t those boxes with very small discrete measure and show in Figure 19(b) a subcollection 13 C 13 with the property that
Z
u(B) ~ 0.99
(7.1)
B~B
(see also Remark 7.1(b)). An approximation of a (natural) invariant measure obtained by the adaptive subdivision algorithm is shown in Figure 20. REMARK 7.4. For the choice of the parameter values we cannot explicitly write down a natural invariant measure. Hence it is impossible to compare the numerical results using analytical ones. Moreover, it is not even known for an arbitrary choice of parameter values whether or not the H6non map possesses an SRB-measure. However, as already mentioned before, M. Benedicks and L.-S. Young proved that the H6non map indeed has an SRB-measure for a "large" set of parameter values, see Benedicks and Young [ 1].
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258
Fig. 20. Illustration of a (natural) invariant measure for the H6non map. The picture shows the density of the discrete measure on/3, see (7.1).
8. Implementational details In this Section we are describing the details of the implementation of the set oriented algorithms. All of the algorithms described in this chapter have been implemented in the software package GAIO which can be obtained from the authors. 8.1. Realization of the collections and the subdivision step We realize the closed subsets constituting the collections using generalized rectangles ("boxes") of the form
B(c, r) = {y e ]1~n" lYi - cil ~ ri for i -- 1 . . . . . n }, where c, r ~ ~n, ri > 0 for i = 1 . . . . . n, are the center and the radius respectively. In the kth subdivision step we subdivide each rectangle B(c, r) of the current collection by bisection with respect to the j th coordinate, where j is varied cyclically, that is, j = ((k - 1) mod n) + 1. This division leads to two rectangles B _ ( c - , F ) and B+(c+,F), where r~ --
ri
for/r
j,
ri/2
for i -- J,
c/i --
I Ci
for/r
/ ci -4- ri/2
for i -- j.
j,
Starting with a single initial rectangle we perform the subdivision until a prescribed size of the diameter relative to the initial rectangle is reached. The collections constructed in this way can easily be stored in a binary tree. In Figure 21 we show the representation of three subdivision steps in three dimensions (n = 3) together with the corresponding sets Q~, k = 0, 1, 2, 3, see (2.6). Note that each collection and the
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259
Fig. 21. Storage scheme for the collections and the corresponding coverings Qk, k = 0, 1,2, 3.
corresponding covering Qk are completely determined by the tree structure and the initial rectangle.
8.2. Realization of the intersection test In the subdivision algorithms as well as in the continuation method we have to decide whether for a given collection Bk the image of a set B 6 Bk has a nonempty intersection with another set B ~ ~ 13k, i.e., whether (8.1)
f (B) N B' = f~.
In simple model problems such as our trivial Example 2.5 this decision can be made analytically. For more complex problems we have to use some kind of discretization. Motivated by similar approaches in the context of cell-mapping techniques (see Hsu [24]), we choose a finite set of test points in each set B E 13k and replace the condition (8.1) by f(x) ~ Bt
for all test points x 6 B.
(8.2)
Obviously, it may still occur that f ( B ) N B t is nonempty although (8.2) is valid. Distribution of test points. It remains to discuss how the test points are distributed inside each rectangle. To define the test points, observe that R(c, r) is the affine image of the standard cube [ - 1 , 1]n scaled by r and translated by c. Using this transformation it is sufficient to define the test points for the standard cube. Simple geometric considerations make it clear that one should obtain the best results for the test in (8.2) if most of the test points are lying on the boundary of the rectangle. An efficient choice for problems of dimension up to three turned out to be N test points on each edge distributed according to t(s
2s
1 N
1
fors
..... N
(8.3)
on [ - 1, 1]. As an additional test point we choose the center c = 0. Since an n-dimensional rectangle has n2 n-1 edges, we end up with p -- N n 2 n-1 + 1 test points per box.
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Rigorous choice of test points.
The numerical realization of the intersection test can be made rigorous in the sense that no boxes are lost due to the discretization. Indeed, to accomplish this it is sufficient to have estimates for the Lipschitz constants of the dynamical system f on Q. To be more precise let B be a collection of boxes B = B(c, r) = {x: Ix - cl ~< r} (where we write I x l - (Ixll . . . . . Ixnl) and x ~< y for x, y ~ N n, if Xi ~ Yi for i = 1 . . . . . n). We need to compute the set-wise image S'(B)-
{B' ~ B I f ( B ) n B' # 0}, A
for every B 6 B. Our goal here is to construct a set f ' ( B ) of boxes for which A
.T'(B) c .T'(B), so that we get a rigorous covering of f (B). To this end we will need to know local Lipschitz constants for f , that is, we require that for every box B in the current collection there is a nonnegative matrix L = L (B) 6 ]l~n x n such that
If(Y)- f (x)l <- Lly - xl
(8.4)
for x, y 6 B. If f is continuously differentiable then h = h (B) 6 ]t~n be a positive vector such that
Lij --max~B IOj 3~(~)1. Now let
Lh <.2r. Using the mesh widths h we now define a mesh
T'-- T'(B) -- {x: ( x i - c i ) EhiZ, i-- 1. . . . . n}. A
It is easy to see that for every y ~ B there is a mesh point x ~ T(B), such that lY - x l ~< h/2. On the other hand we are interested in a finite set of test points and indeed the only points x ~ T(B) we really need are those for which there is actually a y ~ B with lY - xl ~< h/2. So let
T(B) - ~(B) n {x I B n i n t B(x, h/2) ~ 9J} be the set of test points. Note that an additional constraint on h will be necessary in order to ensure that the test points are contained in B, which is necessary, since the local LipscAhitzestimate (8.4) on f is only valid for points in B. Finally we construct the collection .T'(B) by setting
.~(B) - {~ ~ t~ I B n 8 ( f
(x), r) -7(:0 for some x E T (B) }.
(8.5)
The idea of this construction is to look at the boxes B ( f ( x ) , r) r B corresponding to the images of the test points x and to collect in .TC'(B) all boxes which have nonempty
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261
f(B)
/ ..........
'....
/ /
" " i
1 W
Fig. 22. On the construction of ~-(B).
intersection with those boxes, see Figure 22. It is important to note that the construction of ;~(B) is a finite task: since the boxes B ( f ( x ) , r), x ~ T, have the same radius as the boxes in the collections 13, it suffices to consider the vertices of B ( f (x), r).
Adaptive choice o f test points. In order to reduce the numerical effort of the set oriented algorithms one has to reduce the number of test points per box as far as possible. We now show how to do that by considering local expansion rates of the map f . To this end we consider the singular value decomposition
D f ( x ) -- U ( x ) S ( x ) V T ( x ) of D f ( x ) , where U ( x ) = [ul(x) . . . . . u,,(x)] and V ( x ) = [vl(x) . . . . . v,,(x)] are real orthogonal (n x n)-matrices and S ( x ) ~ IR" • is a diagonal matrix having the singular values Crl (x) >~-.. ~> o',, (x) of D f ( x ) on the diagonal. The idea for an improved choice of the test points is to construct a mesh with respect to the basis of right singular vectors vl(c) . . . . . v,,(c) of D f ( c ) = U ( c ) S ( c ) V T ( c ) (where c denotes the center of the box under consideration) and to choose the mesh width hi, i = 1 . . . . . n, in relation to the singular value o-i (c). Let us suppose for the moment that for a box B the derivative D f ( x ) = D f ( c ) -- D f = U S V T is constant on a sufficiently large neighborhood A (B) of B. Then
f (x) - f (y) -- D f . (x - y) -- USV T (x - y), so that
If(x)where
f ( y ) ] ~< [ U [ S I V T ( x - y)[
IUI- (luijl). We choose mesh widths h 6 R", h > O, such that IUISh <~ 2r
(8.6)
and define the mesh
{x.
hiZ, i -
1..... ,/.
(8.7)
M. Dellnitz and O. Junge
262
A
Again it is easy to see that for every y 6 B there is a mesh point x E T (B), such that IVT(y -- x)l ~< h/2. We have to restrict ourselves to a finite set of test points again which can be written down as
(8.8)
T -- T(B) -- T'(B) N {x" B n int(VB(0, h/2) + x) # 0}. i
i
We construct 9t-(B) as in (8.5) and get that U ( B ) C .T'(B). Finally let us consider the general case where Df(x) is not constant on a box. Let
M(x) = ( m i j ( x ) ) i , j = 1..... n = n f ( x ) V ( c ) and set
M--(mij)i,j_l
n'
mij
.....
--
max x~A(B)
Imij(x)],
where A (B) is a sufficiently large neighborhood of B which we suppose to be convex in this case. We choose mesh widths h > 0 such that M h ~< 2r,
(8.9) h
and use the mesh as defined by (8.7) as well as the construction (8.5) for f ( B ) . It can easily be shown that the union o f b o x e s in f ( B ) covers f ( B ) , i.e., that f ' ( B ) C 9t-(B), see Junge [26] for details.
8.3. Implementation of the measure computation The feasibility of the computation of invariant measures even for higher dimensional systems relies on the fact that we first compute an outer covering 13 of the underlying invariant set by one of the set oriented methods presented in this chapter. As the ansatz spaces Vd for the discretization of the Perron-Frobenius operator we use the spaces of simple functions on the given collection/3. It is easy to see that the PerronFrobenius operator is then given by a stochastic matrix P = (pij) with entries m(f-l(Bi)nBj) Pij =
m(Bj)
'
Bi, B j E ]3.
For the computation of the Pij'S w e either use a Monte-Carlo approach (see Hunt [25]) or an exhaustion technique as described in Guder et al. [22]. The latter method is particularly useful when local Lipschitz constants are available for the underlying dynamical system. For the computation of certain eigenvectors of the resulting (sparse) matrix P an Arnoldi method is used (see Lehousq et al. [34]).
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Acknowledgments Figures 4, 6, 7, 9 and 16 have been produced using the software platform GRAPE, see Rumpf and Wierse [41 ].
References [1] M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measures for certain H~non maps, Invent. Math. 112 (1993), 541-576. [2] M. Blank and G. Keller, Random perturbations of chaotic dynamical systems. Stability of the spectrum, Nonlinearity 11 (5) (1998), 1351-1364. [3] R. Bowen and D. Ruelle, The ergodic theory, ofAxiom A flows, Invent. Math. 29 (1975), 181-202. [4] P. Chossat and M. Golubitsky, Symmetry-increasing bifurcation of chaotic attractors, Phys. D 32 (1988), 423-436. [5] C. Conley, Isolated Invariant Sets and the Morse Index, Amer. Math. Soc., Providence, RI (1978). [6] M. Dellnitz, G. Froyland and St. Sertl, On the isolated spectrum of the Perron-Frobenius operator, Nonlinearity 13 (4) (2000), 1171-1188. [7] M. Dellnitz and A. Hohmann, The computation of unstable manifolds using subdivision and continuation, Nonlinear Dynamical Systems and Chaos, H.W. Broer, S.A. van Gils, I. Hoveijn and F. Takens, eds, PNLDE, Vol. 19, Birkh~iuser, Basel (1996), 449-459. [8] M. Dellnitz and A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Numer. Math. 75 (1997), 293-317. [9] M. Dellnitz and O. Junge, Almost invariant sets in Chua's circuit, Internat. J. Bifur. Chaos 7 (11) (1997), 2475-2485. [10] M. Dellnitz and O. Junge, An adaptive subdivision technique for the approximation of attractors and invariant measures, Comput. Visual. Sci. 1 (1998), 63-68. [11] M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal. 36 (2) (1999), 491-515. [12] M. Dellnitz, O. Junge, M. Rumpf and R. Strzodka, The computation of an unstable invariant set inside a cylinder containing a knotted flow, Proceedings of Equadiff '99, Berlin (2000). [13] M. Dellnitz, O. Schtitze and St. Sertl, Finding zeros by multilevel subdivision techniques, IMA J. Numer. Anal. (2001) (to appear). [14] E Deuflhard, M. Dellnitz, O. Junge and Ch. Schtitte, Computation of essential molecular dynamics by subdivision techniques, Computational Molecular Dynamics: Challenges, Methods, Ideas, E Deuflhard, J. Hermans, B. Leimkuhler, A.E. Mark, S. Reich and R.D. Skeel, eds, Lecture Notes in Comput. Sci. Engrg., Vol. 4, Springer, Berlin (1998), 98-115. [15] P. Deuflhard, W. Huisinga, A. Fischer and Ch. Schtitte, Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains, Linear Algebra Appl. 315 (2000), 39-59. [ 16] J. Ding, Q. Du and T.Y. Li, High order approximation of the Frobenius-Perron operator, Appl. Math. Comput. 53 (1993), 151-171. [ 17] J. Ding and A. Zhou, Finite approximations of Frobenius-Perron operators. A solution of Ulam's conjecture to multi-dimensional transformations, Phys. D 1-2 (1996), 61-68. [18] J.R. Dormand and EJ. Prince, Higher order embedded Runge-Kutta formulae, J. Comput. Appl. Math. 7 (1981), 67-75. [19] M. Eidenschink, Exploring global dynamics: A numerical algorithm based on the Conley index theory, Ph.D. thesis, Georgia Institute of Technology (1995). [20] G. Froyland, Estimating physical invariant measures and space averages of dynamical systems indicators, Ph.D. thesis, University of Western Australia (1996). [21 ] J. Guckenheimer and Ph. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Berlin (1983). [22] R. Guder, M. Dellnitz and E. Kreuzer, An adaptive method for the approximation of the generalized cell mapping, Chaos, Solitons and Fractals 8 (4) (1997), 525-534.
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[23] R. Guder and E. Kreuzer, Control of an adaptive refinement technique of generalized cell mapping by system dynamics, J. Nonl. Dyn. 20 (1) (1999), 21-32. [24] H. Hsu, Global analysis by cell mapping, Internat. J. Bifur. Chaos 2 (1992), 727-771. [25] EY. Hunt, A Monte Carlo approach to the approximation of invariant measures, Random Comput. Dynamics 2(1) (1994), 111-133. [26] O. Junge, Mengenorientierte Methoden zur numerischen Analyse dynamischer Systeme, Ph.D. thesis, University of Paderborn (1999). [27] O. Junge, An adaptive subdivision technique for the approximation of attractors and invariant measures. Part II: Proof of convergence (2000) (submitted). [28] G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math. 94 (1982), 313-333. [29] H. Keller and G. Ochs, Numerical approximation of random attractors, Stochastic Dynamics, Springer, Berlin (1999), 93-115. [30] Yu. Kifer, General random perturbations of hyperbolic and expanding transformations, J. Anal. Math. 47 (1986), 111-150. [31 ] E. Kreuzer, Numerische Untersuchung nichtlinearer dynamischer Systeme, Springer, Berlin (1987). [32] A. Lasota and M.C. Mackey, Chaos, Fractals and Noise, Springer, Berlin (1994). [33] A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. [34] R.B. Lehoucq, D.C. Sorensen and C. Yang, ARPACK Users' Guide, SIAM, Philadelphia, PA (1998). [35] T.-Y. Li, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture, J. Approx. Theory 17 (1976), 177-186. [36] K. Mehlhorn, Data Structures and Algorithms, Springer, Berlin (1984). [37] R. Murray, Adaptive approximation ofinvariant measures, Preprint (1998). [38] J.E. Osborn, Spectral approximation for compact operators, Math. Comp. 29 (131 ) (1975), 712-725. [39] G. Osipenko, Construction of attractors and filtrations, Conley Index Theory, K. Mischaikow, M. Mrozek and E Zgliczynski, eds, Banach Center Publications, Vol. 47 (1999), 173-191. [40] D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math. 98 (1976), 619-654. [41] M. Rumpf and A. Wierse, GRAPE, eine objektorientierte Visualisierungs- und Numerikplattform, Informatik, Forschung und Entwicklung 7 (1992), 145-151. [42] Ch. Schfitte, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules, Habilitation thesis, Freie Universitfit Berlin (1999). [43] M. Shub, Global Stability of Dynamical Systems, Springer, Berlin (1987). [44] Y.G. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys 166 (1972), 21-69. [45] W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris S6r. I Math. 328 (12) (1999), 1197-1202. [46] S.M. Ulam, A Collection of Mathematical Problems, Interscience, New York (1960). [47] K. Yosida, Functional Analysis, Springer, Berlin (1980).
CHAPTER
6
Numerics and Exponential Smallness
Vassili Gelfreich* The Steklov Mathematical Institute at St. Petersburg, Russia Institut fiir Mathematik I, FU, Berlin, Germany E-mail:
[email protected]
Contents 1. A glance on "invisible" chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Chaotic dynamics of a symplectic integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Discretization as a rapid periodic forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Flow box theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Saddle points and separatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Splitting of complex separatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Exponential smallness of the splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Lower bounds for the splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Splitting of separatrices near resonant periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Fourier modes of analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: L e m m a on Cauchy integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C: Analytic solutions of finite-difference equations . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D: Analytic parameterization of separatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
267 271 274 278 285 287 289 291 295 298 303 303 304 307 309 311
*The author thanks the Alexander von Humboldt foundation. The work was partially supported by DFG priority program "Analysis, Modeling and Simulation of Multiscale Problems". H A N D B O O K OF D Y N A M I C A L SYSTEMS, VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 265
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1. A g l a n c e o n " i n v i s i b l e " c h a o s
In the recent decades the progress of computers and the development of new numerical tools have produced a great impact on the development of Mathematics (and not only, of course). The m o d e r n computers are quite fast, a n d - although it may seem to be paradoxical - it results that it often requires less time to compute something than to show that the results have something to do with the original problem. Any numerical method has two main sources of errors. The first one is related to unavoidable round-off errors. Their influence can be reduced by multiprecision arithmetics, which is quite fast now. One can repeat a computation with higher precision in order to estimate the importance of the round-off for the computer experiment. The second source of errors is due to the numerical method itself: an original problem has to be substituted by a problem suitable for computer treatment. We avoid a discussion of the round-off errors and try to give at least a partial answer to the following naive questions. Can a numerical method exhibit qualitative behavior different from the original system? If yes, how can we describe and measure this difference? The answer depends on the nature of the problem. We restrict our attention to the behavior of trajectories for an autonomous system of differential equations,
,;c=f(x),
xEDCR".
(1)
The vector field f is assumed to be analytic and bounded. We may say that the independent variable t is "time", and consider the system and its numerical solutions from the viewpoint of Dynamical Systems. The system of differential equations defines a (local l) flow ~ t which moves a point along its trajectory. Let x(t) be a solution of the differential equation. Then q~t (x0) = x(t), provided x (0) = x0. In order to compute a trajectory, it is necessary to choose an integration scheme, which allows to compute a trajectory for a discrete subset of time values, e.g., t = ks, k E N. The numerical trajectory is computed recursively: we define an initial value x0 and then compute the sequence
Xk+l = F~(xk),
k >10.
From the dynamical systems viewpoint the numerical trajectory is a trajectory of a dynamical system with discrete time. This dynamical system is defined by iterations of the map Fe. The numerical scheme is defined by an integrator. An integrator of order p and step-size s is a parametric map, Fe : D --+ R", such that Vx e D.
(2)
I We say that the flow is global, if it is defined for all t and all x 6 D. A flow is not always global - a trajectory can cross the boundary of D and leave the domain of the vector field. In some applications the domain is not bounded and the vector field grows at infinity. Then a trajectory can go to infinity in finite time. In particular, if D = Rn , this is the unique possible reason for a flow to be nonglobal. In such cases it is often possible to find an invariant compact subset, D' C D, then the restriction q~t iD' is a global flow.
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For example, Euler's method (first order, p = 1) seems to be the most classical one. It is not too difficult to construct an integrator of an arbitrary order p. The Taylor method is naive but very efficient: the system of differential equations (1) can be used to compute consequently an arbitrary large number of derivatives 2 of x(t) at t = 0; using the first p derivatives to compute the first p terms of the Taylor expansion of x(t) for t = e, one obtains an integrator of order p. There are many excellent books and papers, which describe constructions of integrators. The flow ~,t may have some special properties due to special properties of the vector fields itself. For instance if the vector field is Hamiltonian, the flow preserves areas. In this and similar cases it is reasonable to use an integrator which preserves the special properties of the flow. This gives a hope for a better qualitative matching between numerical and exact trajectories. All integrators belong to a remarkable class of close-to-identity maps: F~ (x) = x + O(e). This simple remark will play an important role later. In (2) the term O(e p+I) is the local error, which describes the error of one numerical step. One has to iterate the integrator to get an approximation for a trajectory. For a fixed k and small e
xk-- F (xo . Of course, the constant in the O-estimate generally increases with k. One has to make
k = e - I t steps to obtain x(t) for a prefixed value of t. Since the map Fe is t-close to identity, it can be checked that for a fixed positive time the distance between the exact and numerical trajectory is O(e p) provided the original trajectory and the numerical one do not leave the domain D. What happens for larger times? It is easy to compute. For example we consider the pendulum equation, m
y,
-- sin(x), which is Hamiltonian, i.e., it has the form
-
OH
~(x,
Oy OH -- - ~ ( x , Ox
y),
y),
2 This can be done either symbolically or by computing the numerical values of the higher order derivatives in a recurrent way based on Leibnitz rule (see, e.g., [5]).
269
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where H is the Hamiltonian,
y2 H (x, y) -- - f + cos x - 1. The coordinate x is usually considered as an angle variable. Then the phase space becomes a cylinder. The phase space of the pendulum consists of 3 parts: two zones of pendulum rotations (clockwise and counterclockwise), and a zone of pendulum oscillations. These zones are separated by two curves, which are called separatrices. Each of the separatrices is formed by a trajectory doubly asymptotic to the unstable upper equilibrium. So the dynamics of the pendulum is quite simple - all its trajectories (except 4 of them: 2 equilibria and 2 separatrices) are periodic. The flow of the pendulum is global. It has two remarkable properties: the map 45t is area-preserving; and the Hamiltonian H is an integral of motion, i.e., it is constant along trajectories. We study the pendulum numerically using the area-preserving Euler method, Fe" (x, y) ~ (xm, Yl),
where
xl - - x + eyl, Yt -- Y + e s i n x . The integrator coincides with the famous standard map, well known in the physical and mathematical literature. We showed 4 trajectories of the integrator on Figure 1 (do not forget that the phase space is the cylinder). It is clearly seen that the numerical scheme well reproduces the trajectories of the pendulum. But there is a small exception, namely, the separatrix of the
1
o
-1
-2
-3 --7i-
2
2
Fig. 1. Four trajectories of the standard map, s = 0.25
V. Gelfreich
270
pendulum. The pendulum separatrix is defined by H (x, y) = 0 and it consists of two invariant branches: one belongs to the upper half-plane, and the second branch belongs to the lower half-plane. On the other hand, one of the numerical trajectories of Figure 1 was started very close to the origin, and it draw both "branches" of the separatrix. On Figure 2 we enlarge a small neighborhood of the origin to see more details of these trajectory. The black set is filled by one trajectory only. It is often called a "chaotic layer". The form of this layer is apparently rather regular. But the picture is static, it does not reflects the dynamics. An observer can sit near a computer screen and look how this set is filled by the trajectory: this process seems to be rather chaotic. Some idea of a typical behavior of the trajectory of Figure 2 may be obtained from Figure 3. The trajectory irregularly switches between "oscillation" and "rotations". This is "chaotic" behavior. We give a more precise mathematical meaning to this in the next section. We see that in our case the chaos induced by the numerical scheme appears on extremely small space scales. It was called invisible [9]. One of the main goals of the present paper is to explain that such phenomena are exponentially small with respect to the step-size e and to provide an introduction in perturbation methods suitable for the study of exponentially small phenomena. The exponential smallness makes the chaos, induced by the numerical integrator, almost invisible. Nevertheless, it can lead to important qualitative differences between exact and numerical solutions.
1 e-05
5e-06
-5e-06
i - 1 e-05 -1 e-05
-5e-06
0
5e-06
1 e-05
Fig. 2. One trajectory of the standard map, e = 0.25, around one million iterates of a point (magnification of a part of Figure 1).
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271
Fig. 3. The x-component of a numerical trajectory (standard map, e = 0.25, initial conditions are from the chaotic zone of Figure 2).
2. Chaotic dynamics of a symplectic integrator The uniqueness of solutions and the continuous (smooth) dependence from initial conditions are the two basic properties of differential equations. That is, if one defines a position of a point at the initial moment of time, x(O) = x0, then the trajectory x(t) is uniquely defined for all t (provided the trajectory does not leave the domain of the vector field and does not escape to infinity in finite time). In other words, for any fixed time the flow ~ t is well defined and smooth. Of course, the integrator is also well defined and smooth (by its definition). Consequently, the dynamics of the differential equation and of its integrator are both completely deterministic. How can something "chaotic" coexist with the deterministic behavior? There is no way to determine the initial conditions exactly. As we will illustrate it in this section, an arbitrarily small error in initial conditions can lead to the complete loss of precision in comparatively short time. As a result an observer with limited precision of observation has no possibility to distinguish between a completely deterministic trajectory of a differential equation (or a numerical scheme) and a random process, similar to the flipping of a coin. Moreover, a numerical scheme can introduce chaotic dynamics into an originally regular system.
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Fig. 4. Smale horseshoe. The chaotic behavior of a dynamical system is described in the language of symbolic dynamics, introduced by the Russian mathematician V.M. Alekseev at the end of the 60s [ 1]. One of the simplest models, which exhibits chaotic motions, is the famous Smale horseshoe. It is also interesting, since it appears in many applications. Consider a smooth map F defined on a subset of the plane, which includes the unit square [0, 1] • [0, 1]. We will consider only those trajectories, which remain inside the square. So there is no need to describe the map outside the square. Let the restriction of the map F onto the square be described as a composition of two transformations (see Figure 4). The first one is just a linear transformation, described, for example, by the matrix
t0 ) 3
0
This map transform the square into a long horizontal rectangle. The second transformation bends the rectangle into the form of a horseshoe and puts it onto the original square. The second transformation is identical on the intersection of the rectangle with the square, the part "1". It rotates by re the part marked by "3" and translates it back onto the original square. Let us denote the original square by D, and the gray (closed) rectangles "1" and "3" by A and B, respectively, A, B C D. The initial conditions for trajectories which nether leave D belong to the intersection, K -- ~k~>0 Fk (D). EXERCISE. Show that the set K is a nonempty closed (forward-) invariant set. It is the interval [0, 1] times the Kantor set. 3 Let us consider the set of infinite sequences composed of two symbols A and B. Let ai E { A, B } denote the ith element of a sequence a. By the construction of the set K for any point, z0 6 K, there is a symbolic sequence a = {ai }/~--0 such that F i (zo) E ai. The inverse is also true. For any sequence a there is an initial condition, z0 = (x0, y0), such that F i (zo) E ai for all i. EXERCISE. Prove it. Hint: show that the s e t rqi-o F i (ai) is a nonempty closed set (like in the previous exercise). Any point of this intersection can play the role of z0. If we let k E Z in the definition of the set K and consider bi-infinite sequences a then the correspondence between the set K and the set of the sequences a is one-to-one, thanks to the hyperbolicity. 3 This is a comparatively elementary exercise, which is much easier to do yourself, than to understand arguments written by someone else.
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Let us suppose that a point zo corresponds to a symbolic sequence, a - - {ai }~=0' so that F i (ZO) E a i . Let zl = F ( z o ) , then F i (Zl) E a i + l . Consequently, the point zl corresponds to the shifted s e q u e n c e , {ai+l}i~=0 . One can say that FIK is semiconjugated to the shift on the set of symbolic sequences, or, in other words, we have constructed the symbolic dynamics. The set of the symbolic sequences can be supplied with a topology a n d - that is very i m p o r t a n t - with a probability measure. In Probability Theory the (Bernoulli) shift on the set of symbolic sequences of two symbols describes r a n d o m processes, like the flipping of a coin. We say that a map is chaotic if its restriction on a closed invariant set is semiconjugated with a Bernoulli shift defined on the space of infinite sequences. Let us suppose that the precision of an observer affords to decide if a point belongs to A or to B (except, perhaps, if it is too close to the boundary), but does not permit to measure positions with higher accuracy. If the initial conditions belong to the set K the observer can not distinguish a trajectory of a point zo 6 K and a randomly generated sequence. 4 The remarkable property of the Smale horseshoe is its stability to perturbations: the symbolic dynamics may be introduced even in the case of a notably d e f o r m e d horseshoe. We use the following example to illustrate, how a Smale horseshoe appears in a symplectic integrator. We consider the Hamiltonian flow, described by the Hamiltonian, 5
1
H -- x y - - ( ~ ( x + y)3
The dynamics of the corresponding flow is rather simple. There are 2 stationary points, one is at the origin and the second is at (1, 1). The separatrix of the saddle point (0, 0) is defined by the equation H (x, y) -- 0 and it has the form of a fish (see Figure 5). Inside the separatrix loop all trajectories are periodic, and outside all trajectories go to infinity both in positive and negative time. We use the following symplectic integrator: 6 F~, :(x, y) w+ (x l, Yl), where x l = ) ~ - l x + c ( x + )~y)2, yl -- )~y - c)~(x + ~,y)2,
and )~ = 1 + e, c = (1 - k - l ) / ( 1 + )~)2. The constant c defines the scaling. We have chosen it in such a way that the point (1, 1) is a fixed points of the integrator for all e. The origin is a saddle point of F~. The eigenvectors of the matrix D Fe (0) are horizontal and vertical. According to the H a d a m a r d - P e r r o n theorem the map Fe has one dimensional stable and unstable invariant manifolds, which are also called separatrices. The stable separatrix has a horizontal tangent at the origin, and the unstable manifold has a vertical tangent 4 On the other hand, in the Hamiltonian case KAM theorem implies that the most part of initial conditions
correspond to regular quasi-periodic behavior. The problem of existence of chaos on an invariant set of positive measure remains mainly open. 5 This is the "fish" Hamiltonian H = p2/2 + q2/2 - q3/3 after rotating by zr/4. We use it in this form to have more beautiful pictures. 6 The integrator F~ coincides with the area-preserving Hen6n map. The splitting of its separatrices for small e was described - as a special case- in [15].
V. Gelfreich
274
0.5
<
...............................................................
0
/i'i/,I
....
-0.5
-0.5
0
0.5
1
1.5
2
(a)
-1
-0.5
0
0.5
1
1.5
(b)
Fig. 5. (a) The separatrices of the differential equation; and (b) of the symplectic integrator.
(see Figure 5). We reconstructed these separatrices numerically for a comparatively large (e -- 1.95). It is seen that the separatrices of the integrator still look similar to the separatrices of the differential equation, but they have transversal intersections, which lead to oscillations of separatrices in a neighborhood of the saddle point. These intersections are called homoclinic points. The presence of transversal homoclinics leads to a horseshoe. We say that the separatrices of the integrator are split. We can choose (by an intelligent guess 7) a "rectangle" near the origin, and iterate it several times using a computer. On Figure 6 we see the rectangle itself (bold border) and its 8 consecutive images (iterations). It is clearly seen that the rectangle first shrinks in one direction and expands in the other, and on the 8th iterate it comes back being bent in the form of a horseshoe and intersects the original rectangle. This picture implies the chaotic dynamics. We used the large e to be able to draw a visible picture. The size of the horseshoe decreases rapidly with e, and the number of iterations required to return back to the original rectangle increases. A rigorous construction of the symbolic dynamics near split separatrices can be found in the Ya. Sinai book [25]. Now we switch to the methods which provide quantitative bounds for the discussed phenomena.
3. Discretization as a rapid periodic forcing A numerical method gives us an approximate trajectory- at least finite segments are close to the original trajectory. We can say that, except by round-off errors, the numerical trajectory is (exactly!) a trajectory of another system, which is close to the original one. Then The "intelligent guess" is mainly to chose the sides of the "rectangle.... parallel" to nearby segments of the stable and unstable separatrices.
7
275
Numerics and exponential smallness I
i
i
.,,~"'"-..-..... ,,"
9 t /
9 I 9 /
1.5
9
t /
,
, ,
t
i i i /
',
0.5
"iJi/ . . . . .
.
.
.
.
.
.
. . . . . . . . . . . . . . .
i!i ii -0.5 -0.5
0
0.5
1
1.5
2
Fig. 6. The 8th iterate of the map has the Smale horseshoe.
we can use perturbation methods to compare the behavior of the systems. Historically perturbation methods have been better developed for differential equation than for maps. So we show that an integrator may be considered as a small fast perturbation of the original system. It would be nice if the integrator was the e-time map for a system of differential equation of the same form as the original system (1). Unfortunately in general this is not possible. But it is possible to represent the integrator as a succession map for a nonautonomous (periodic in time) system of differential equations. Consider a system of nonautonomous differential equations,
where oF is periodic with respect to t" jg(x, t 4- T) -- f ( x , t) for all x and t from its domain. The succession map is the map which moves a point along its trajectory during the time equal to the period, this is x(0) w-> x(T), where x(t) is a solution of the differential equation. Sometimes the succession map is also called Poincar6 map of the nonautonomous system. Let f be analytic and bounded in a complex 8-neighborhood, D 4- i8. Let F~ be an analytic integrator of the differential equation (1) in D 4- i8.
THEOREM 1 (interpolating flow). For any 8' ~ (0, 8) there is eo > O, such that f o r all e E (0, so) and x E D + i8' the integrator Fe(x) coincides with a succession map o f a
V. Gelfreich
276
differential equation, Jc = f (x, t/e, e), defined by the nonautonomous vector field, f (x, t/e, e) = f (x) + e p g(x, t/e, e), where g (x, r, e) = g (x, r + 1, e) is 1-periodic with respect to the fast time r. The function g is smooth (C~ bounded by an e-independentconstant, and it is analytic in x on D + i6'. REMARK 1. If the differential equation is Hamiltonian and the integrator is symplectic the time-dependent vector field can be also made n o n a u t o n o m o u s Hamiltonian. The corresponding ideas go back to the works by J. Moser in the middle 50s, see, e.g., [3]. REMARK 2. The theorem remains valid if the smoothness in e is replaced by boundedness. That is enough if one is mainly interested in getting upper and lower bounds only. In fact, the smoothness in e has no important influence on the conclusions of all general theorems of this chapter. The function g may be also chosen analytic with respect to time. But the corresponding proof is much more complicated, and it does not add too much to our approach. PROOF OF THEOREM 1.8 First, we produce an explicit expression for trajectories in the extended phase space and then we prove that the corresponding velocities define a nonaut o n o m o u s vector field with the required properties. 9 It is convenient to consider the equation (1) in the extended phase space, (D + i6) x ( N / Z ) , formally adding the angle variable q):
Jc = f ( x ) , ( 9 = lie. If we choose an arbitrary point x0 from the domain of the local flow, then the corresponding trajectory of the extended flow connects the points (x0, 0) and (4~ e (x0), 1). We deform this curve to connect the points (x0, 0) and (Fe (x0), 1). We use the following trick. Let X0 : R ~ R be a C a m o n o t o n e function, such that
xo(r)-
1,
r ~< 1/4,
O,
r~>3/4,
and let X1 ('g) = 1 - x o ( r ) . 8 Here I rearranged the proof from [9]. Neishtadt [21] used a similar fact as something completely obvious. There are infinitely many ways of constructing the nonautonomous vector field. The simplest construction is to define a trajectory in the following way: it goes along a straight line from a point (x, 0) to the point (Fe(x), 1). This is not a completely satisfactory solution for our problem: (1) the nonautonomous vector field is not smooth (it can have jumps at the points with q9= 0 mod 1); (2) the perturbed trajectories are only e 2-close to the unperturbed ones. So we go by a more sophisticated way. 9
Numerics and exponential smallness
~
277
,I,~(x,) )
.F 1 d; 0
Fig. 7. Construction of the interpolating trajectory. EXERCISE. Write an explicit example of X0. Let us choose the point x l in such a way that the unperturbed trajectory, which starts at x l arrives to the point Fs (x0) after the time t = s, i.e., the point x l has to be chosen from the condition (bS(xl) = G(xo). We can simply let xl = ~-S(Fs(xo)). Since q~S(x) -Fs(x) + O ( e p + I ) , we have xl - x 0 = O ( s P + l ) . Now we can define the interpolating trajectory (see Figure 7),
2(t, xo, s) = Xo(t/s)~ t (xo) + XI (t/s) clgt(Xl),
t 9 [0, s].
(3)
In the extended phase space the equation (x qg) = (2(t x0 s) t_) defines a smooth curve, which connects the point (x0, 0) to the point (Fs (x0), 1). First, this curve follows the unperturbed trajectory of x0, then it smoothly switches to the unperturbed trajectory of x l. For an arbitrary point, (x, 99) E (D + i3') • (IR/Z), there is a unique x0, such that the trajectory with the initial condition at (x0, 0) passes through (x, qg). Indeed, the map x0 w-~ 2(t, x0, s), where t = sqg, is s-close to identity and analytic. Consequently, it is invertible and the inverse map is defined on D + i3' provided s0 is sufficiently small. The inverse map defines x0 as a function of x:
xo(x, s~o, s) - x + O(s). We define the vector field at the point (x, 99) as the velocity vector of the corresponding trajectory, i.e.,
f(~, ~, ~) -
0
~7
2(t,xo(x, s~o, s), s). t = e (fl
We can calculate the derivative explicitly:
f (x, qg, s) -- Xo(t/s) f (~t (xo)) + Xl (t/s) f (~t (x,)) + s -~ (x~(t/s)~' (xo) + xl (ts)~' (x,)),
t =s~o,
V. Gelfreich
278
where we used (O/at)c19 t (x) = f ( ~ t (x)). This defines a smooth vector field on the extended phase space, (D + i6') x (R/Z). For all points, except ~0 -- 0mod 1, the smoothness of f follows from the explicit formula. In a small neighborhood of ~p = 0 mod 1 the vector field f (x, qg, e) -- f (x) which is smooth. Let us estimate the time-dependent part of f . By the definition x0(r) + Xl (r) -- 1 for all r, then x~(r) + XI (r) -- 0. By the construction if a point (x, qg) belongs to an interpolating trajectory with the initial condition (x0, 0), then the point x belongs to a rectilinear segment, which connects the points q0e~~(x0) and q0e~~(x 1). The length of this segment is O(ep+l). Consequently,
cber
-- x + O(e p+')
and
cber
= x + O(eP+'),
since the distance from an internal point of the segment to its endpoints can not exceed the length of the segment. Substituting all this into the formula for f we obtain
f (x, r, E) -- f (x) + O(eP). Consequently, the equality,
g(x, r,e)- e-P(jg(x,r,E)- f(x)), defines a 1-periodic with respect to the fast time r function, which is smooth, bounded by a constant, and analytic in x on D + i6'. This finishes the proof of the theorem. E]
4. Averaging In the previous section we proved that the discretization is equivalent to a small rapid periodic perturbation of the original system, i.e., the integrator of the autonomous system :f -- f (x) coincides with the succession map for the nonautonomous system,
~c -- f (x) + eP g(x, t/E, e).
(4)
In such a situation, we say that the integrator is interpolated by the nonautonomous flow. The interpolating flow is not uniquely defined. In this section we use the averaging procedure to show that the integrator can be interpolated by an almost autonomous flow, the time-dependent term of which is exponentially small compared with the integrator stepsize. We achieve this by a close-to-identity coordinate change, l~
x -- y + ePu(y, t/e, e), 10 The construction of this section provides the coordinate change and the new vector field, which are generally not continuous with respect to ~. This is due to the construction itself. A trivial modification of the procedure leads to C c~ dependence. I leave this step as an exercise for an advanced reader.
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279
such that y satisfies the equation,
(5)
= f ( y , e) + ePg,(y, t/e, e) with f (y, e) = f (y) + O(e p)
and
~(y, t/e, e) = O(e -c/e)
for some positive constant c. The coordinate change is 1-periodic with respect to the fast time, r = t/e, and it is identical for r = 0 since we will choose u(y, O, e) = O. Hence Equations (4) and (5) have the same succession map. Thus Equation (5) provides the almost autonomous interpolation for the integrator. In (4) the vector field rapidly oscillates. The classical averaging theory provides a formula to decrease the time depending term using a sequence of close-to-identity timedependent coordinate changes. It was a nice observation by Neishtadt [21 ] that the classical procedure can be repeated O(e -1) times, which leads to an exponentially small time dependent term. 11 Now we formulate and prove the theorem, and then we discuss some of its consequences. THEOREM 2 (Neishtadt's averaging theorem). Assume that the functions f and g in (4) are analytic in x in a complex neighborhood D + i3. The function g is continuous as a function of all its variables and bounded in e. For all e ~ (0, eo) there is a coordinate change x = y + ePu(y, t/e, E),
u = o(1),
such that in the new variables Equation (4) takes the form -- f (y, ~) + ~P~(y, t/~, ~), where
=O(e f o r some positive constant c. The change of the variables and the new vector field are analytic f o r y ~ D + i~/2. PROOF. We. find the desired change of variables in the form of superposition of a large ( O ( e - I ) ) number of close-to-identity coordinate changes. Each of the changes is obtained using the classical averaging procedure. Let us describe the first step with more details. 12 I I It is remarkable that similar estimates can be proved by similar arguments for a vector-field quasiperiodic in time provided the frequencies satisfy Diophantine condition [24]. 12 Although some estimates may seam to be long, the proof itself is quite elementary.
280
V. Gelfreich
We look for the first coordinate change in the form, X -- ~C -Jr- 8 p+I llO(X, 75, 8),
where 75 =
t/s
(6)
is the fast time. The substitution into Equation (4) gives
x + s p+l Ow 9
Ow =
-~x 2 -+- e P 07:
f(.x+ sP+lw)+
sPg(Yc + sP+lw, 75).
From now on we do not write the explicit dependence on s. Note that, in front of the time derivative of w, one power of s disappeared due to the dependence on rapid time. The equation can be rewritten in the form, -~x
f (~ -+- e P + l w ) -k- e P g ( x -+- e P + l w ' 75) -- e P
Ow) Or
(7) '
where I denotes the unit n x n matrix. Now we choose the function w to kill the dependence on time in the "main order". Let us separate the "average" part of g: 1
(g(x, .))--
f0
g(x, r) d75,
and define w by the equation, Ow (s r) = g(~', r) - ( g ( s 075 '
.)),
which can be solved explicitly
w(s
r) --
fo l"(g(s
(8)
r) -(g(Yc, .)))dr.
The integral defines a periodic function of r, because the zero order Fourier mode of the function under the integral vanishes. In other words, subtracting the average we avoided secular terms. We have defined the coordinate change. Let us write explicit formulas for the pull back of the vector field and obtain the upper bounds. We exclude the time derivative of w from Equation (7) and obtain
x-
(l+sp+lOW - l ~) x
( f ( x + 6P+I 1/3)
+ s p g(Yc + s p+I w,
r) -
s pg(yc, r)
+
sP(g(s .))).
In order to finish the first averaging step we define the new principal autonomous part of the vector field adding the average of g:
Numerics and exponential smallness
281
Let eP~ denote the sum of all other terms from the right-hand-side of the equation:
-1 g' (x' T) -- ( I -f- 6p+l Ow
+
-
+ g(x + ~P+Iw, r) - g(x, "r) Ow - e -Ox ( f ( x ) + eP(g(x, .)))1. Then the function 2~ satisfies an equation of the form (4), where all letters should be equipped by a tilde. Now we describe the domain of the change and get bounds for the new vector field. 13 We have arrived to the place, where the analyticity of the functions plays. We use Cauchy-type estimates 14 for the derivatives. We reduce the domain by a small value ~, which will be chosen later. It will have the same value for all averaging steps. An attempt to "optimize" this choice does not lead to any essential improvement. The domain of the new vector field is D + i ~ with ~ = 6 - ~. It is easy to see from the definition (8) that ]lwllo+i~ ~ 211gllo§ Applying the Cauchy estimate we have 2 ~/-h-
oq~/3
IlgllD+i~. D+i6
The presence of x/-h-is due to the estimate of the matrix norm via the maximal element of the matrix, S?" ~" Let us assume that ~ satisfies
>14 x/ne :'+t ][gllo+is,
(9)
then
eP+lllWllD+ig ~ ~/(2x/-n)
1
003
and D+i~
This implies that the coordinate change (6) is well defined, and the following matrix function is bounded:
I + e p+l Ow ) - I Ox
~<2. D+i~
13 It is easy to see that ~ -- O(e) for real values of the arguments, consequently, the averaging moves the dependence on time to the next order of e. But for a fixed value of e "to be of the higher order" does not imply "to be less than", because of an unknown constant in the O-estimate. So we consider ~ in a complex domain, where its norm will be a half of the norm of g. This is not "much smaller" but at least it is really smaller! 14 Let a domain of an analytic function a contain a disk Iz - z01 ~< ~, then the derivative of a at the center of the disk is bounded by la'(zo)l ~< supl:-:.ol= ~ la(z)l/e~.
V. Gelfreich
282 The following estimates are obvious:
IIf(x + cp+-lw) - f (x)llo+ia ~ ~af O_q_i(~_.l_~/2)EPq'-lllll)[lO+i~, IIg(x +
Ep+111),-g)- - g(x, "C)IID+i~~
Ow
--ff--fx(f(x) + eP(g(x, .)))
Og ~X D+i(3+~/2) 8 p +
1 II w II o+ig,
Ow ~ D+i,~([[fllD+ig -+- 8Pllgllo+i~)'
D+i$
where we used that the arguments of f and g belong to D + i(6 + ~/2) since 8P+l [Iwllo+ig ~< ~/2. Using the Cauchy type estimates we obtain
[[f(x
+ E P + I w ) - f(x)llo+ia <. 4 ~
[[g(x + ~'+'
~) - g(x, -c)IID+i6
w,
Ep_+.111fllo+iallgllo+i~,
4~/-n p+l 2 ~< ~ 8 Ilgll D+i6,
Ow 2 C~-ff --~x ( f (x) + eP(g(x' "))) D,$ <~ IlgllD+i~(llfllD+i~ + ePllgllD+i~). We collect these estimates together to obtain the following bounds for the transformed vectorfield:
6,/~
II IIO-+-i~~
~
(l[ f [[O+i~ -+- 8 p IIg IIo-+-i6)[]g [ID-big,
II/11o+ia ~< Ilfllo+ia
+ ePllgllo+i~.
Now we are ready to repeat the averaging procedure using f and ~, instead of f and g. In this way we get a sequence f(J) and g(J), j = 1 . . . . . N, with f(0) and g(0) being from original Equation (4). How many times this procedure can be repeated? What are the bounds for the corresponding terms? The answer to the first question is simple: each step requires the reduction of the analyticity domain by ~. After N steps the total reduction is N~. In the theorem we asked the reduction to be no larger than 6/2, consequently, the maximum number of steps is N = Integer part of (6/(2~)). Note that the inequality (9) bounds the reduction ~ from below. To answer the second question we define two numerical sequences, Oj = IIf (j) II o~i~ and /Tj = [[g(J)II o(J~, J = 1 . . . . . N. We have
Oj+l ~
6~
~:
(Oj Jr-cPoj)Oj,
Oj+l <~Oj + ePoj,
j -- 1 . . . . . N.
(10)
Numerics and exponential smallness
283
It is a simple arithmetical fact that if
q - ~6 x / - f f e
(Oo4-2ePr/o)
< 1/2,
(11)
then the inequalities (10) imply the following majorant for r/j in the form of geometric progression: r/j ~ r/o q j,
Oj <~ Oo 4- 2 e P r/o ,
j -- 1 . . . . .
N .
(12)
The condition (11) gives the following restriction on ~"
(~3)
~> 12v/-ffe(00 4- 2ePr/0).
The last inequality implies (9). So all N averaging steps are well defined. Moreover, the inequality guarantees the decay of the time-dependent term of the vector field on each averaging step. We recall that 00 -- ]If I[, 7/o = ][g[[ describe the norms of the original vector field. Finally, we let y __ x(N),
/__ f(N),
~ __ g(N).
We choose ~ -- 12 v/-ffe(Oo 4- r/0). Then the maximal number of steps is N = N(e)-
Integer part of ( e - l g / ( 1 2 ~ / n ( O o
+ r/0))).
Then q -- 1/2, and we have from (12):
Ilgll D+i6/2 with c -- 3 / ( 1 2 ~
= r/N ~ r/oq N -- e-C/~llgllD+i~
(00 + 2eP r/0) log 2). This finishes the proof of the averaging theorem. U
REMARK 3. The following example shows that the time dependent term can not be eliminated by the averaging procedure completely. We consider a vector field on an annulus, 0 < r l < r < r2, written in the polar coordinates, i" -
g(qg, t / e , e),
where g(q0, r, e) - Z
gkl(e)eik~~
k,l
An averaging step preserves the form of the equation but changes the function g. Moreover, if e - ~- is a rational number, then the Fourier mode of the order (k, l) is not changed. Consequently, this Fourier mode can not be excluded by the averaging procedure. EXERCISE. Prove this. Hint: write the change (6) explicitly.
V. Gelfreich
284
The unremovable time-dependent term is exponentially small in e. This follows from the analyticity of the vector field in qg. Indeed, the domain of the function g contains a strip I Imqgl ~< p for a constant p > 0. The standard arguments (see Appendix A) show
sup [g(~p,e)[.
Jgkl(e)[ <~ e -Iklp
I Im~0l~
Then the unremovable "resonant" terms are exponentially small:
Igk;() I-
o(e-lklp/e) - O(e-plll/~) 9
This is essentially due to the fact that for a small e the number k is large. REMARK 4. It is interesting to compare the Neishtadt averaging theorem with the classical averaging results. In fact, the classical averaging theory uses exactly the same averaging procedure but the number of steps N is independent of e. We can follow the proof to see the following: there are positive constants Ao, A l and A2, such that N averaging steps are defined for e 6 (0, A o / N ) , after the Nth step of the averaging the time dependent component of the real vector field does not exceed
EPoN ~ CN 6p+N,
where
CN -- AIANN!.
Indeed, to estimate the first N steps we can choose the domain reduction, ~ = 6/(2N). The condition (13) provides a restriction on the maximal value of e: e < eo(N) - A o / N . We can assume that Ilfll > e P + l Ilgll (if necessary we can reduce A0, which reduces the maximal value of e). Then q < ~ - I l f [ ] . After N averaging steps the norm of the time dependent component of the vector field is bounded:
/TN ~ 00q N ~ ]lg II
16eN 6
II/11)u
The right-hand-side of the last estimate is approximately A I A N N! e N for some positive constants A1 and A2 due to the Stirling formula for the factorial. So the "classical" estimates for the time dependent term of the vector field have the form CNe N, where CN form a Gevrey-1 sequence. Such a situation is intimately related with exponential smallness. In order to obtain the exponentially small upper bound it is sufficient to realize that for a given e it is possible to make up to N ~ Ao/e averaging steps. Then it is easy to analyze which N leads to the smallest time dependent part. REMARK 5. If the nonautonomous equation (4) is (locally) Hamiltonian, then the change of coordinates can be made canonical. 15 15 To prove this it is necessary to look for the changes using generating functions on each averaging step. This is the unique essential difference of the proof for the Hamiltonian case.
Numerics and exponential smallness
285
Consequently, a symplectic integrator is exponentially close to the time-e map of an autonomous Hamiltonian vector field. The autonomous Hamiltonian flow preserves energy (the Hamiltonian). So the symplectic integrator preserves some "energy" with exponentially small error. In particular this implies that the original Hamiltonian H is preserved with the error of the order of e P during exponentially long time. So H is an adiabatic invariant. On the other hand, this explains, why the width of the stochastic zone for the symplectic integrator (see Section 1) of the pendulum is exponentially small. A rigorous proof requires an application of the KAM theory, to show the existence of invariant curves which bound the stochastic zone [21 ].
5. Flow box theorems The flow which corresponds to the differential equation, .;c = f (x),
x E D + ia c C",
has the following properties: q~~ series in t we obtain
= x and qf~
(14) = f ( x ) . Expanding ~ t (x) in Taylor
ci9e (x) -- x + e f (x) + O(e2).
By the definition the integrator is e P+l-close to this map and, consequently, Fc (x) = x + e f ( x ) + O(e2). We can rewrite it in the form Fe(x) = x + e f (x) + e 2 G ( x , ~),
(15)
where G is an analytic function of x, bounded by a constant independent of e. In particular, the integrator belongs to the remarkable class of close-to-identity maps. Let xo be a nonsingular point of the vector field, f ( x o ) :/: O. We compare the local dynamics of the flow and of the integrator in a neighborhood of this point. The famous flow box theorem states that the vector field can be straightened in a neighborhood of the nonsingular point. There is an open domain, U C D + i3, xo 6 U, such that after a coordinate change, x ~-+ y, the restriction of the vector field on this domain coincides with the constant vector field, which points in the direction of the first coordinate. 16 In the new coordinates the equations of motion take the form, =el,
e l = ( 1 , 0 . . . . . 0) E C " .
(16)
16 This coordinate system can be obtain as a pair "(time, initial conditions)": take an analytic section, which goes through the point x 0 transversally to the vector f ( x 0 ) ; the flow box coordinates of a point x E U equal to the minimal time, required for the point to reach the section, and the coordinates of the intersection of the trajectory with the hyperplane. This defines an analytic coordinate system due to analytical dependence on initial conditions and uniqueness of solutions.
V. Gelfreich
286
In other words, we can say that the vector field can be straightened in a neighborhood of its nonsingular points: all the (local) trajectories are rectilinear segments. In this sense the local dynamics is trivial. It is a remarkable fact that the integrator can be locally straightened too. There is a coordinate system, such that in the new coordinates the map Fe takes the form of a translation in the direction of the first coordinate axis: z ~ z +eel.
(17)
This fact has important consequences. For example, this implies that the integrator can be locally interpolated by an autonomous vector field. Let us precisely formulate the statement. Let x0 be a nonsingular point of the analytic vector field f , and let Fe be an analytic integrator of the form (15). Let e0 be sufficiently small. THEOREM 3 (discrete flow box). For any e ~ (0, eo) the integrator can be straightened in an e-independent neighborhood of xo by an analytic coordinate change, which is bounded from above by a constant independent of e. PROOF. The straightening procedure consists of two steps. First, we go to the flow box coordinates of the original flow. Then the original differential equation takes the form (16). Correspondingly, the integrator takes the form, Fe (Y) = Y + Eel -I- ~,2 ~ ( y , e). Next, we look for a close-to-identity coordinate change in the form y(z) = z + eu(z, e), which completes the straightening. This change is a solution of the conjugability equation
y(z + e e l ) - Fe(y(z)). This equation takes the form,
Z + eel + eu(z + eel, e) -- z + eu(z, e) + eel + e2G(z + eu(z, e), e). To clarify the structure of the last equation it is convenient to rewrite it:
u(z + eel, e) - u(z, e) = G(z + eu(z, e), e). E
(18)
This is a special type of finite-difference equations. It is a discretization of a first-order differential equation. This proposes the treatment for the equation. We rewrite it as an "integral" equation and apply a convergent iterative procedure in a suitably chosen ball of a Banach space. Consider this equation in the Banach space of all functions analytic in a small polydisk, Vr - - {Z E C n" Izkl < r, k = 1. . . . . n}, and continuous in its closure. This space is provided
Numerics and exponential smallness
287
with the supremum norm. It is shown in Appendix C that in this space the finite-difference operator has a bounded right inverse L. This means that any solution of the "integral" equation, . - c (?.(z +
--N(,,.
automatically satisfies Equation (18). There is a constant C such that [[G[[ v2, ~< C and [[D G [[v2, ~< C for all e 6 (0, e0) (if necessary we may reduce r and use the Cauchy type estimates to bound the derivatives). Consider two functions u and v from a ball, Ilvllv,., [lul[v, ~< R with radius R = r/eo. The choice of R is dictated by the following: if z E V~, then z + eu ~ V2~, and the argument of G does not leave the polydisk V2,-. Now we are ready to estimate the nonlinear operator ./V':
II
tlLtt
CliLtl
If C ]lL l[ < R, the ball is invariant with respect to the nonlinear operator A/'. If additionally eoC l[L [I < 1, the restriction of the operator on the ball is contractive, and the operator has a unique fixed point. This fixed point provides the desired solution for Equation (18). The required inequalities can both be satisfied by reducing e0 since we have chosen R = r/eo. []
6. Saddle points and separatrices In the previous section we established that the local dynamics away from singular points of the flow is trivial. It has qualitatively the same properties for the flow and for its integrator. The dynamics of the integrator is defined by iterations of the map Fe. Let us first recall some definitions related to iterations of maps. Let F be a diffeomorphism. A point x0 = F (x0) is called fixed point of the map F. Fixed points may be classified according to the spectrum of the matrix D F (x0). Hyperbolic fixed points are of special interest. A fixed point x0 is called hyperbolic, if the eigenvalues of the matrix D F(xo) do not belong to the unit circle. With the hyperbolic fixed point we can associate invariant objects, which play an important role in the dynamics. Let 3 > 0 be a small positive number. The set of points, which never leave a 3neighborhood of x0, is called a local stable manifold (or local stable separatrix),
Wi~oc(xo ) - {x " Vk ~ 0 I1Fk (x ) - xo II <~ 6 }. A local unstable manifold (or local unstable separatrix) is the local stable manifold of the inverse map F -1 ,
V. Gelfreich
288
If ~ is sufficiently small, a local manifold is the image of a unit disk under a smooth (analytic) immersion [22]. The dimension of the disk coincides with the number of eigenvalues of DF(xo) outside (for Wioc) or inside (for Wi~oc)the unit circle. In fact, iterations of a point on the local stable separatrix converge to the fixed point. This permits to define the global stable separatrix as the set of all points, whose iterations converge to the fixed point:
WS(xo)-- [x 9 lim F k ( x ) - - x o } . k --++ cx~ A (global) unstable manifold (or an unstable separatrix) is the stable manifold of the inverse map F - l , W" (xo) - {x"
lim Fk (x) -- xo ]. k--+- ~
The global unstable manifold may be obtained by iterations 17 of the local manifold: W u (xo) -- Uk>>,o Fk (WlUc)- Correspondingly, the global stable manifold may be obtained by the iterations of the local stable manifold by the inverse map. A local separatrix is the embedding of the unit disk. Consequently, it has no selfintersections. The map F is a diffeomorphism, and the corresponding global separatrix can not intersect itself. It is easy to see directly from the definition that if the map has two hyperbolic fixed points, their unstable manifolds do not intersect. On the other hand, a priori there is no obstacle for existence of intersections of stable and unstable manifolds. A point of intersection of two invariant manifolds associated with one fixed point is called homoclinic. As both manifolds contain the fixed point x0, this point is excluded when we speak about homoclinic points. A point is heteroclinic, if it belongs to the intersection of separatrices associated with two different fixed points. Now let x0 be a singular (stationary) point of the vector field, f (x0) -- 0. The singular point is called nondegenerate, if all eigenvalues of the matrix D f (xo) are different from zero. The stationary point is a fixed point for the corresponding flow. PROPOSITION 1. If xo is a nondegenerate singular point of the vector field f , then there is eo > O, such that for all e E (0, eO) the integrator has a unique fixed point, xE - Fe(xe), close to the singular point: x~ - xo + O(eP). This proposition is rather standard from classical perturbation theory. It can be proved using the implicit function theorem. The point x0 is a fixed point of the local flow, x0 = q~e (x0). The matrix Dq~ e (x0) is invertible due to the nondegeneracy of the fixed point, although its eigenvalues are e-close to zero. Then the equation, xe = F~(xe) = ~ (xe) + O(e p+l ) has a solution, xe = x0 + O(eP), due to the implicit function theorem. Singular points of the vector field can be classified according to the eigenvalues of the matrix D f (xo) too. One special class is of special interest for us. A singular point is called 17 If the map F is not globally defined, and, consequently, the dynamics is not global, one should be careful with this definition.
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289
hyperbolic, if the eigenvalues do not belong to the imaginary axis. If the vector field has a hyperbolic singular point, then it is a hyperbolic fixed point of the flow, and the integrator has a hyperbolic fixed point nearby. It is well known that the local separatrices are persistent with respect to small perturbations. In particular, the local separatrices of the integrator are e P-close to the separatrices of the differential equation (see, e.g., [7]). In the following we will mainly study integrators of differential equations on the plane (n = 2). We will call a point on the plane a saddle, if in the linear approximation it has one stable and one unstable directions.
7. Splitting of complex separatrices In the analytical case the geometry of the phase space can determine the separatrices splitting for all hyperbolic saddle points. Let a diffeomorphism have an analytic continuation up to an entire diffeomorphism F" C 2 --+ C 2. Let x l and x2 denote two hyperbolic saddle points, X l -- F (x l) and x2 -F (x2). We do not exclude the case x l = x2. According to the Hadamard-Perron theorem these points have one (complex) dimensional invariant manifolds. It is possible that the intersection W s (x l) A W u (x2) contains a point different from x l and x2. This heteroclinic (or homoclinic) point has to be one of the following types: (1) transversal heteroclinic (homoclinic) point; (2) point of finite order tangency; (3) point of infinite order tangency. We say that the separatrices are split if their intersection contains no points of infinite order tangency. If the separatrices split it is possible that they have no intersections at all, or there is a point of transversal intersection, or there is a point of finite-order tangency. In the last case the tangency may be of odd or even order. THEOREM 4 (Ushiki [27]). All separatrices o f all hyperbolic saddle points o f an entire diffeomorphism F" C 2 --+ C 2 are split. We formulate a nice geometrical corollary before proving this theorem. Let F" R 2 --+ ~2 be an analytic diffeomorphism. We say that the diffeomorphism F has a heteroclinic (homoclinic) connection, if there is a continuous invariant curve, whose ends are saddle points of F. We additionally require that all internal points of the curve are not fixed points for F. COROLLARY 5. If a real-analytic diffeomorphism F has an entire analytic continuation, F'" C 2 --+ C 2, then it has no homoclinic and no heteroclinic connections at all.
EXERCISE. Derive this corollary from the theorem. Hint: show that the heteroclinic (homoclinic) connection 1 is a subset of W S(xl) A W" (x2). The separatrices are one (complex) dimensional analytic curves, which intersect along one real dimension, so they also intersect along one complex dimension, i.e., each internal point of I is a point of infinite order tangency.
290
V. Gelfreich
PROOF OF THEOREM 4. The theorem is proved by contradiction. The stable (unstable) separatrix is the image of an entire map ~os(u)" C ~ C 2. We suppose that the intersection W s ( x 1) N W u (x2) contains a point of infinite order tangency. The supposition allows us to show that the union of separatrices W s ( x l) U W u (x2) is an analytic image of a Riemann sphere. Since the Riemann sphere is compact the union of separatrices is bounded. Then the entire map q9s is bounded, thus it is constant. Since its image is W s (x l) we obtain the contradiction. Let us implement this program. In Appendix D it is shown that the stable and unstable separatrices are images of the entire maps, q9s" C --+ C 2 and q9u" C ~ C 2. In particular, W s ( x l) - - q9s (C) and W u (x2) - q9u (C). These maps are one-to-one maps of C onto their images, and they are normalized in such a way that the images of the origin are the fixed points, Xl -- ~0s (0), x2 - q9u (0). Let U, V C C be the preimages of the set of all points of infinite order tangency of W s ( x l) and W u (x2) with respect to ~0s and q9~', respectively. The sets U and V are not empty due to the supposition. The analytic curves coincide in a neighborhood of a point of infinite order tangency, so the sets U and V are open. We can define an analytic diffeomorphism, h" U --+ V, by h - - ( ~ o U l v ) - J o (~oSlu). Now, we take two copies of the complex plane C and glue them by the diffeomorphism, h. The result is a one (complex) dimensional manifold, which we denote by M. This manifold is defined by two charts and the passage map, h. The functions q9s and ~p~ define in these charts a map, 7r" M --+ C 2, ~p(M) -- W s ( x l ) U W " (X2). This map is analytic (and continuous, of course). The first copy of the complex plane is embedded in M, but its image does not coincide with the whole manifold M since it does not contain the zero from the second copy of the complex plane: this point corresponds to x2, which is a saddle point and not a point of infinite order tangency. The manifold M is obviously connected. By the Koebe classification theorem its universal covering M is analytically isomorphic either to the unit disk, either to the complex plane, or to the Riemann sphe~. Since there is a (proper) subset of M, which is isomorphic to the whole complex plane, M is isomorphic to the Riemann surface. Consequently, M is compact. We can lift the map ~p onto the universal covering, M. The lifted map 7t defines an analytic map of M in C 2. The image is compact as it is an image of a compact set. On the other hand, the image coincides with the union, W s (x l) U W" (x2). Consequently, the union of separatrices is bounded. Each of the separatrices is the image of the entire map, q9T M C ~ C 2. It follows that each component of each of the maps is an entire bounded function, i.e., these maps are constant. This is the desired contradiction, because the separatrices consist of more than one point. D REMARK 6. The generalization of the theorem to the case when F is a diffeomorphism of C n and the invariant stable and unstable manifolds are one-dimensional is almost evident. This theorem has some interesting consequences for integrators. For example, let the original system of differential equations be a Hamiltonian system on the plane. Let this
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291
system have a saddle point and a homoclinic connection. The flow preserves the Hamiltonian function H. Let the integrator Fe have an entire analytic continuation (like in the example of Section 1). According to the theorem if the integrator has a homoclinic point, then this point is either transversal or finite order tangency. This is not compatible with existence of an analytic integral of motion (a nice elementary proof can be found in [6]). Consequently, an entire integrator of a plane Hamiltonian vector field has either nor integral of motion or nor homoclinic connections. For a symplectic integrator only the first alternative is possible, since the separatrices have to intersect due to the area-preservation property. EXAMPLE. This example shows that the conditions of complex analyticity is essential. There is an integrable integrator of the pendulum equation, Y - sin x, which has the form, Xk+l -- 2Xk + Xk-1 -- 82f(Xk, 82),
f (x, r
2 __ ~
arctan
e 2 sin x 2 - 62 COSx '
found by Suris [26]. The integral is known explicitly. The function 8
2
I (xk, xk-1, e 2) -- 1 - cos(xk - xk-1) + -~- (cos xk + cos xk-1) is constant along a numerical trajectory. The second component of a trajectory may be recovered by yk = (xk - X k - 1 ) / 6 . Then the integrator, Fe :(xk, Yk) ~ (xk+l, Yk+l), is symplectic (it preserves area). Unlike the integrator described in Section 1 it has a homoclinic connection, which converges to the pendulum separatrix when e --+ 0. Indeed, the integrator separatrix is not split, since it belongs to the set defined by i(x,
x -
= ,(0,
o,
-
It is easily seen from the explicit formula that
,,xx This example does not contradict to the theorem, because the analytical continuation of Fe is not entire. In fact, integrable integrators of integrable Hamiltonian systems are difficult to find, and they are known only as rare exceptions.
8. Exponential smallness of the splitting We consider an autonomous system of differential equations on the plane:
.;c=f(x),
xr
2,
(19)
V. Gelfreich
292
with the vector field analytic in a complex neighborhood, D + if 6 C 2. We assume that the system has a hyperbolic saddle, x0 E D, and there is a homoclinic trajectory o-(t) which forms a homoclinic connection to x0: l i m t ~ + ~ or(t) = x0. Since the vector field is analytic the homoclinic trajectory is also analytic and its domain contains a closed complex strip, /Tp-{t
~ C" IImtl ~
for some constant p > 0. An analytic integrator Fe has a saddle fixed point, xs = xo + O(sP). Its local separatrices are close to the separatrix of the differential equation. But it is quite possible that they form no homoclinic connection, i.e., they can split. In general the integrator Fe may have no homoclinic orbits at all. On the other hand, in many cases one can show that it has homoclinic trajectories due to preservation of the area or due to a symmetry. The whiskers of the integrator separatrices can be represented in parametric form (see Appendix D) by the solutions of the finite difference-equation, +
supplied with the boundary conditions lim cr~ (t) = x0
t---~ -~x~
and
lim cr+ (t) -- x0.
t---~ +cx~
These conditions do not uniquely define the solutions. It is possible to show that the parameterizations can be chosen close to the homoclinic trajectory of the differential equation in such a way that -
+
o-+ (t) - o- t) +
t e l 7 p , Ret < R , teFlp, Ret > -R,
for any fixed R > 0. The constants in the O-estimates depend on R, of course. How one can measure the splitting of separatrices? There are many alternative ways. For example, we can take a point cr (t) on the separatrix of the differential equation and erect a normal from it. Then the splitting distance d(t) is the distance between first intersections of the stable and unstable separatrices with the normal. Here "first" refers to an intersection closest to the fixed point inside the separatrix. Let us assume that for all s E (0, s0) the integrator Fs has a primary homoclinic point Xh, i.e., a point of the "first" intersection of the separatrix segments, { a - ( t ) : t < R} and {tr + (t): t > - R } . The primary homoclinic orbit is close to the separatrix of the differential equation: there is to 6 [ - R , R], such that Ff(xh) = tr(ks + to) + O(s p) for all integer k. THEOREM 6. Let R > O. Let cr(Flp) C D + if and let the integrator Fs be analytic in D + if. If Fs has a primary homoclinic orbit for all s E (0, so), then the splitting distance
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293
d(t) is exponentially small, d(t) - O ( e -2rrp/s) for all - R < t < R. The constant in the O-estimate depends on R and eo. In fact, the method, the basic ideas of which we explain later in this section, affords to estimate other characteristics of the splitting, such as areas of lobes formed by segments of split separatrices, or angles formed by the separatrices at a homoclinic point, or even describe how one separatrix oscillates near the second one. The most complete geometrical information may be obtained from the following theorem. THEOREM 7. Under the conditions of Theorem 6, there is an analytic system of coordinates (T, E), defined in an e-independentneighborhood of{or(t): - R < t < R}, such that in the new coordinates the following hold. (1) The integrator Fs takes the form of the translation, (T, E) w-~ (T + s, E); (2) The unstable separatrix is given by E = 0; (3) The stable separatrix is the graph of an s-periodic function 2zrT E = Oo(s) + O1 (s) sin ~ -l- O(e-4rrp/s),
where the coefficients, IO0(e)l, IOl (E)I = O(e-2Jrp/s), are exponentially small. (4) The coordinate change is bounded uniformly in s E (0, eo). REMARK 7. The constants Oo(S) and Ol (e) can vanish, and generally speaking the theorem provides only the upper bound for the splitting. For a class of examples it is possible to improve the method to estimate values of these constants and in this way to prove the transversality of the splitting. One special case is the standard map studied in the next section. REMARK 8. The theorem may be applied to an entire integrator of an entire vector field. In this case or(t) has complex singularities. 18 Let p0 be the distance from the real axis to the closest singularity of cr (t). Then the above theorems hold for any p < P0.
PROOF. 19 The idea of the proof is quite simple: in the (discrete) flow box coordinates any invariant object of the integrator is represented by an e-periodic pattern. The domain of the flow box coordinates is s-independent. This affords to describe the difference of two separatrices by an s-periodic function, which is analytic in an s-independent complex strip. The analysis of its Fourier series (see Appendix A) shows that the Fourier coefficients of all orders, except zero, are exponentially small. In fact, the function is almost sinusoidal. 18 If it was entire, we could use arguments of Section 7 to show nonexistence of homoclinic connection. 19 The proof is based on the ideas first formulated by Lazutkin [19]. A first general exponentially small upper bound for the splitting for close-to-identity maps was obtained by Fontich and Sire6 [8] in a closely related way.
294
V. Gelfreich
The larger is the domain of the flow box coordinates, the sharper are upper bounds. Because of that, we use special coordinates (T, E) instead of the flow-box coordinates of Section 5. The only essential difference is the larger domain, which includes an s-independent neighborhood of the set ~ (Hp) C C 2. The construction of these nonlocal flow box coordinates can be found in [10]. It is similar to the proof of Theorem 3. In the construction of this coordinate system, the complex time t defined on the separatrix of the differential equation, is used as a first order approximation of the first coordinate T. Because of this and the closeness of the integrator separatrix to o-, the domain of the coordinates (T, E) includes an s-independent neighborhood of the sets, {~r~(t)" IRet[ ~< R, IImtl ~< p}. The coordinate change can be normalized by the condition that on E -- 0 the coordinate T coincides with the parameter t on the unstable separatrix o--(t). Then the stable separatrix can be represented in the parametric form, (T, E) -- (t + ~ ( t ) , O(t)), where by definition (t) -- T (o-+ (t)) - t,
(20)
O)(t) - E(cr+ (t)).
These function are s-periodic. Indeed, O ( t + e) -- E ( c r + ( t + e)) -- E ( F s ( a + ( t ) ) )
-- E ( a + ( t ) )
-- O ( t ) ,
(t + 6) = T (~r+ ( t + e)) - t - s = T(Fs(a+(t)))
-t-
s -- T ( a + ( t ) )
-t--
~(t),
where we used that the integrator takes the form of the translation: E o Fs -- E and T o Fs -T + s. Moreover, these functions are analytic in the strip H p . Consequently, we can apply the lemma on Fourier coefficients (Lemma 1), which gives the estimate O ( t ) -- O0(s) + Ol (s) sin
2zc(t -- tl (S))
+ O(e-4Jrpls),
6
(t) -- q:0 (s) + O(e-2JrP/e), where the subscripts refer to the order of Fourier modes. These estimates can be differentiated with respect to s, the error being multiplied by s -1 . Moreover, [O)1 (s) I ~< const 9e -2Jrp/s. The lemma on Fourier coefficients provides an exponentially small upper bound for all Fourier modes except the zero order term. In general, this term can be of the order of O ( 6 P ) , then the separatrices do not have primary homoclinic intersections, which correspond to zeroes of the function 6).
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295
We assume the existence of the primary homoclinics, then the function O has zeroes. This implies that 169o1 does not exceed the sum of all other Fourier terms (by modulus), otherwise 69 would have no zeros. That is, it is exponentially small. Then we use the implicit function theorem to show that the stable separatrix is actually a graph, to see it we exclude t using the first of Equations (20). Finally, we translate the coordinate system in T-direction in order to kill the possible phase shift in the sinus function. D
9. Lower bounds for the splitting How sharp are the upper bounds of the previous section? They seems to be optimal, but proving this claim is an extremely difficult problem. There is a set of examples, most of which are polynomial, where lower bounds are available. The longest list can be found in the paper [15]. As an example we consider the standard map, already defined in Section 1. Let us describe the splitting of separatrices for the standard map with more details. The standard map has the form, Fe :(x, y) ~ (x l, Yl), where x l -- x + eyl, y m -- Y + e s i n x . It is a symplectic integrator of the pendulum equation. For small positive e > 0 the standard map has two fixed points, namely, (0, 0) and (0, Jr). The first one is hyperbolic and the other one is elliptic. Indeed, the matrix of the linear part at the origin is (1+~
1) 1 '
and its eigenvalues are X and ) - l , where
)~- 1 + e/2 + / ~
+ e2/4.
The stable W s and the unstable W" manifolds of this fixed point are analytic curves passing through (0, 0), the eigenvectors of the matrix being tangent vectors to these curves at (0, 0). The origin breaks each separatrix into two parts. We denote by W~ (WI') the upper part of the stable (unstable) separatrix. It is convenient to represent the unstable separatrix W I' in a parametric form using a solution (x, y ) = ( x - ( t ) , y - ( t ) ) of the finite-difference system,
x(t + e) - - x ( t ) -+-ey(t + e), y(t + e) -- y(t) + ~ sinx(t).
(21)
V. Gelfreich
296
~0
Fig. 8. Separatrix splitting for the standard map.
We impose the following boundary conditions on the function x - ( t ) " lim x - (t) - 0,
t-+-~
x - (0) - 7r.
(22)
The solution of Equation (21) is not defined uniquely by the boundary conditions (22). We study the solution, whose analytic continuation is entire and has a purely imaginary period 2zri. We assume that t = 0 corresponds to the first intersection of W~~ with the line x -- Jr (if the intersection of the stable and unstable separatrices is transversal, then there are infinitely many such intersections). Under these additional assumptions the solution of the problem (21) and (22) is unique. Originally, the solution of (21) is only defined in a complex half-plane Re t < - R and represents the local separatrix. Since the sine function is entire, iterations of Equation (21) allow to continue the solution up to an entire function. As in some other places we omit the explicit dependence of the functions x - , y - , x +, and y+ on e to shorten the notations. We define the parameterization of Wi~ by
(x+(t), y+(t)) -- (2zr - x - ( - t ) , y - ( - t ) + e s i n x - ( - t ) ) . Direct substitution shows that these functions satisfy the system (2 l) as well as the boundary conditions, lim x + (t) -- 0,
t---+-+-~
x + (0) -- Jr.
(23)
Since x - ( 0 ) = Jr we have x + (0) = Jr and y+ (0) = y - ( 0 ) that is t = 0 corresponds to a homoclinic point. Both parameterizations are close to the homoclinic solution of the pendulum equation, which is known explicitly: cr0(t) -- (4 arctane t, 2 / c o s h t ) , The complex time singularities of this separatrix, closest to the real axis, are at +i7r/2. They play an exceptional role in the estimates of the splitting.
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297
Since the standard map has an analytic continuation up to a diffeomorphism of C 2, the homoclinic point is either finite order tangency or transversal. On the other hand, the estimates of the previous sections show that the splitting is exponentially small, namely, it is O(e -2~rp/E) for any constant/9 6 (0, re/2). We will see that the constant in the exponent is arbitrarily close to the optimal value. The splitting angle is not a natural measure for the separatrices splitting. V.E Lazutkin proposed to study the homoclinic invariant defined by o9 _ det ( . f - ( 0 ) ~-(0)
.f+(0) ) ~ +(0) "
(24)
The homoclinic invariant is equal to the value of the symplectic form d x / ~ dy on a pair of vectors tangent to the separatrices at the homoclinic point. The coordinate-independent definition of the homoclinic invariant for a symplectic map on a symplectic two-dimensional manifold may be found in [ 14]. The homoclinic invariant has two remarkable properties: (i) it has the same value for all points of one homoclinic trajectory; (ii) it is invariant with respect to symplectic coordinate changes. THEOREM 8. The homoclinic invariant co o f the primary homoclinic point, z o - ( x - (0), y - (0)), equals
O9=-_
47r o9o _jr2/s 83
e
(1 + 0(o~
The coefficient, o)0 -- 1118.827706 .... was computed in [20] as a solution of an sindependent problem. COROLLARY 9. For all sufficiently small s > 0 the stable and unstable separatrices of the standard map intersect transversally at the homoclinic point zo (the first intersection o f the separatrices with the line x -- re), and the splitting angle is given by rr o9o _ rr2/ s
se
+
The lobe area is given by
2o9o 2/~ S--~e -~ (1+0(8)). ~8
The formula for the splitting angle was first obtained by Lazutkin [19]. The original proof was based on a detailed study of the analytic continuation of the function O(T) (see the previous section). Lazutkin's proof was not complete. A complete and self-contained proof of the formulas for the splitting of separatrices for the standard map can be found in [11].
298
V. Gelfreich
10. Splitting of separatrices near resonant periodic orbits Now we are going to deviate a bit from the mainstream of our chapter and discuss recent results on generic bifurcations of periodic orbits in a Hamiltonian system with two degrees of freedom. It is remarkable that almost all the methods and general results of the present chapter can be successfully applied to this problem [13,12]. We consider an analytic family of area-preserving maps Fe with an elliptic fixed point. We assume that for E = 0 the fixed point is resonant of an order n = 1,2 or 3. In each of these cases the fixed point can be unstable at the exact resonance, and close to the exact resonance there is a hyperbolic periodic orbit of period n. The resonant normal form is integrable and its separatrices form a small loop. Separatrices of the map F~ are close to the separatrices of the normal form but can intersect transversally. In particular Neishtadt averaging theorem and theorems from Section 8 imply an exponentially small upper bound for the splitting of separatrices. In this section we describe asymptotic formulae for the splitting of separatrices, which in particular imply lower bounds for the splitting in a generic situation. The splitting is exponentially small compared to E and can not be detected by Melnikov method. This problem is equivalent to studying a generic family of close-to-resonant elliptic periodic orbits in an analytic Hamiltonian system with two degrees of freedom. Indeed in a Hamiltonian system closed orbits are not isolated but form one-parametric families. As a rule, nearby members of the family belong to different energy levels. Here we consider an analytic Hamiltonian system with two degrees of freedom only. In such a system small oscillations about a family of closed orbits can be described by a time-periodic system with one degree of freedom which depends on a parameter. Let a periodic orbits be of elliptic type. It has a complex multiplicator # with I#1 = 1. We say that the periodic orbit is resonant if #" = 1 for some integer n. We take the smallest positive n to be the order of the resonance. The motion near the resonant periodic orbit can be described using a resonant normal form. The resonant normal form can be reduced to an autonomous system with one degree of freedom and its phase portrait can be drawn. If coefficients of the lowest-order terms in the normal form are in general position, then there are only finitely many different types of phase portraits [2]. The resonances of the order n = 1, 2, 3, and 4 have a special property: the linearly stable periodic orbit may become unstable at exact resonance. At resonance of the order n = 1 or 3 the elliptic periodic orbit is generically unstable. At resonance of the order n = 2 or 4 both stable and unstable cases are possible in general position depending on coefficients of the resonant normal form. Let us consider the problem with more details. It is more convenient to consider an equivalent problem formulated in terms of families of area-preserving maps, which are obtained by considering Poincar6 section near the closed orbits. Let Fe denote an one-parametric analytic family of area-preserving maps. Assume that there is a family of elliptic fixed points pe with a (complex) multiplicator #~, I#~1 = 1. Let /z 0n _ 1, we say that P0 is resonant. Here we study the resonances of the order n - 1,2, and 3. The case n = 4 has some special properties and it is considered. In a small neighborhood of its fixed point the nth iterate F n can be considered as a time-one map of an autonomous Hamiltonian system with one degree of freedom with an
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299
5>0
5=0
5<0
/
a) y2
X3
HI - Sx + -~- + A--~-
n-1
b) __ _ S X 2
n-2
H2
y2
X4
2 + -2--A
4
c) n _
3
H3 _
5_ x2_ +q-A~to ( x2
2
3 - 3 x y 2)
Fig. 9. Bifurcations of equilibria of the resonant normal forms (A > 0).
error, which is smaller than any power of s [2]. It is possible to simplify the form of the corresponding Hamiltonian by a canonical transformation. In this way the system is transformed to a resonant normal form. The lower order normal form Hamiltonian H,, depends on 2 parameters: 3 describes deviation from the exact resonance (and it is proportional to a power of s), and A depends on the map F0 only. It is not too difficult to provide an explicit formula for A and 3 as a function of several first coefficients of the Taylor series of F~ in two space variables and s. These formulae are not of big importance for our present purposes since we will formulate our results in terms of quantities, which allow coordinate independent definitions. The phase portraits and the main order terms of the normal forms are given on Figures 9 and 10. This is a complete list of non-degenerate cases for resonances of the order n = 1,2, and 3 [2].
V. Gelfreich
300
c~<0
c5=0
n-2
/~2 - ~-~- + -~- + A--~-
X2
y2
~5>0
X4
A>0
Fig. 10. Resonant normal form.
The phase portrait of the map Fe looks quite similar to its normal form, but still some explanations are necessary. In the cases n = 1 and 2 the elliptic fixed point pe exists only on one side of the resonance. Exactly at resonance the multiplicator of P0 is real ( + 1, respectively), and the fixed point is not elliptic but parabolic. In the case n = 1 shown on Figure 9 a the elliptic fixed point collides with a hyperbolic fixed point and disappear. This corresponds to a Hamiltonian saddle-center (BogdanovTackens) bifurcation. In the case n -- 2 we see a pitch-fork period doubling bifurcation. There are two possibilities: (a) (Figure 9(b)) the elliptic fixed point collides with a hyperbolic period-two trajectory, after that it becomes hyperbolic and the period-two trajectory disappears; (b) (Figure 10) the elliptic fixed point becomes hyperbolic and a period two elliptic trajectory arises. In the case n -- 3 (see Figure 9(c)) the elliptic fixed point exists on both sides of the resonance. At the resonance it collides with a period 3 hyperbolic periodic orbit. In each of these cases separatrices of the normal form make a small loop around the elliptic equilibrium. At any order the resonant normal form is integrable, and the corresponding stable and unstable separatrices coincide. This is not generically true for the map Fe - as it was already pointed in [2] - the separatrices of the map may intersect transversally. Detecting this is a very difficult analytical problem since in the analytical case the splitting of separatrices is exponentially small in e. Before providing the asymptotic formulae, which describe the splitting of the separatrices, we need in some preliminary definitions. There are many different ways in which one may quantitatively describe the separatrices splitting. We prefer Lazutkin homoclinic invariant, which is defined as a straightforward generalization of (24). Let the separatrices of the hyperbolic n-periodic orbit be parameterized by solutions of the following finite-difference equation ~ + ( t + h) - F n (~p+(t)),
~ + ( t ) ---->x~
as t --+ +oo,
where xe is one of the n points of the hyperbolic periodic trajectory. The " + " corresponds to the stable separatrix and the " - " to the unstable one. We let h -- log Xe and assume that
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301
the multiplicator of the hyperbolic periodic orbit )~e > 1. The image ~ + (R) is one whisker of the separatrix. Under an additional condition (27ri periodicity in t) these solutions are unique up to a substitution t ~ t + const. The derivative ~+(t) - d g r + / d t ( t ) defines a vector field tangent to the stable and unstable separatrices, respectively. These vector fields are Fn-invariant. It is easy to see that the vector fields are independent from the choice of the normalization constant in the definition of gr +. Let Xh denote a homoclinic point. Then Lazutkin homoclinic invariant, co'--e_
Ae+,
equals to the area of a parallelogram defined by the tangent vectors ~_ and ~+ at xh. Lazutkin homoclinic invariant has remarkable properties: (1) it takes the same value for all points of the homoclinic trajectory of Xh; (2) it is invariant with respect to canonical substitutions; (3) for a fixed Xh it is proportional to the sine of the splitting angle. We have chosen the homoclinic invariant in order to describe the splitting by one invariantly defined number, which is easy to compute with high precision. This permits a careful comparison of the theory with results of numerical experiments. Of course, inside the proofs there are expressions, which describe the behavior of the separatrices as curves, i.e., much detailed description of the separatrices is also possible by the same method. The main result of the present section is the following asymptotic formula for the Lazutkin homoclinic invariant. Let the map Fc afford one of the nondegenerate resonant normal forms shown on Figure 9. Assume e be sufficiently small, then Fe has exactly one n-periodic hyperbolic trajectory in a neighborhood of pe (only on one side of the resonance if n - 1 or 2). Separatrices of the hyperbolic periodic trajectory form a small loop around pe. There are at least two different primary homoclinic orbits. The homoclinic invariant of one primary homoclinic orbit is given asymptotically by
4re e _2rre/h (iOl co-- ~-5-
+ O(h)),
(25)
where h = log)~c, )~e is the multiplicator of the hyperbolic periodic orbit. It vanishes together with e, so the splitting is exponentially small compared to e. The preexponential factor 1691 is a constant, which depends on the map at the moment of the resonance only. If 69 does not vanish there are exactly two primary homoclinic orbits with different orientation of intersection of separatrices, and both of them are transversal. A similar statement is valid if the map Fc has the resonant normal form shown on Figure 10, but the asymptotic formula for the homoclinic invariant has to be replaced by: co-- 47r ~5- e _~r2/h
(IOI-+- O(h)),
(26)
where h - log )2, )~e < _ 1 is the multiplicator of the hyperbolic fixed point, which appears after the bifurcation. The only difference compared to the previous case is in the exponent. An important ingredient of the asymptotic formulae is the constant 69. The asymptotic formulae above imply transversality of the separatrices for small e only provided 69 does
V. Gelfreich
302
not vanish. This constant can be defined as a symplectic invariant associated with the corresponding map F0. The constant 69 can vanish. For example, if the map F0 is time one map of the corresponding normal form, or if F0 is integrable. Numerical experiments show that 69 may vanish in a nonintegrable case too. We conjecture, that for any map F0, any 69o E C and any natural N there is another map F0, which have the same Taylor expansion as F0 up to the order N and 69 (F0) -- 69o. In other words, the constant 69 is not defined by any final jet of the function F0. On the other hand there are efficient numerical method for its evaluation [20,16]. In some cases it is possible to prove analytically that 69 does not vanish [ 16], making in this way the proof of transversality of separatrices purely analytical even for concrete families and not only in generic sense. If the map FE has additional symmetries its separatrix splitting can be much smaller compared to the above-described general case. For example, consider n = 2 and let the map Fe be odd. Then the constant to(F0) = 0 due to the symmetry, and the splitting of separatrices is essentially smaller compare to the general non-odd case. Nevertheless an asymptotic formula for the splitting of separatrices can be derived by the same method for this case too. Instead of F~ we have to consider an auxiliary map G~ = - F ~ . Note that G~ = F~, so the separatrices of these two maps coincide. The corresponding statements and asymptotic asymptotic formula (26) remain valid for Ge if we use h = l o g ( - ) ~ ) . Note that the multiplicator of the hyperbolic fixed point of G~ equals to -~.e. Thus the new h is a half of the corresponding value for Fe, and the order of the splitting is much smaller (approximately square of the general case); moreover if t0(G0) # 0 then the map Fe has Exactly 4 different primary homoclinic orbits.
Definition of 69. We give a short description of a problem, which leads to the definition of the splitting constant 69. We consider the finite-difference equation
(27)
This system has a formal solution in class of formal series in powers of r - l . It is defined up to a translation r ~ r + r0 with r0 E •. The equation has two analytic solution (u+(r), v+(r)), which are asymptotic to the single formal solution as r --+ 4 - ~ , respectively. We consider their analytical continuation on complex r. They have the same asymptotic behavior as Im r --+ - c ~ . Moreover, their difference decreases exponentially as Im r --+ -cx~. The constant 69 is defined as the following limit
--
,im e2iTdet( "+")
Im r - + - e ~
~
(r).
(28)
V+ -- V -
It is not difficult to check that 1691 is a symplectic invariant of the map F0 - its definition is invariant with respect to area-preserving changes of coordinates.
Numerics and exponential smallness
303
11. Conclusion Many ideas described in this chapter comes back to the works by Poincar6 [23], who discovered the splitting of separatrices, including exponentially small splitting. 2~ H. Poincar6 also discovered the relation between dynamics of differential equations and maps. The perspectives are also related to his discoveries. There are a lot of nice open problems left: What happens in many dimensional systems? Can a numerical method induce Arnold diffusion in a system? How many trajectories (in measure sense) become chaotic? These and many other problems are waiting for a solution.
Appendix A: Fourier modes of analytic functions It is well known that the decay speed of Fourier modes is closely related to the smoothness of a functions. A domain of a periodic real-analytic function always contains a complex strip. The ratio of the strip width to the period determines the decay speed of Fourier modes. LEMMA 1 (on Fourier coefficients). If an h-periodic function 0 is analytic and bounded in the strip {t E C: I Imtl < b}, then its Fourier coefficients are bounded in the following way: [Okl ~ e -2zrblkl/h
sup Io~t~l f o r all k E Z. Ilmtl~
COROLLARY. If we additionally assume that b > h and that the mean value of O vanishes, then f o r all real t ~ I~:
Io
8e-2zrb/h
sup IO(t) [. [Imtl~b
In other words, the function 0 is exponentially small with respect to h on the real axis. PROOF. The proof is quite standard. The Fourier coefficients can be expressed in the form of the integrals, 1 foh e_2rckit/h f (t) dt. Ok -- -~
The function is analytic, consequently, the path may be moved up (if k > 0) or down (if k < 0), and we have
Ok
--
I f -+-ib+he -2rrkit/hf( t ) h- J q-ib
dt
--
l for7e-2Zrkir
20 Somedetails about the history of this discoverycan be found in the book [4].
f (~"_4_ib) ds
304
V. Gelfreich
The modulus of the integral is bounded by the product of the length of the path times the maximum of the modulus of the function under the integral:
I O k l - e -2rrblkl/h
If(t) l,
sup IImtl--b
which is the desired estimate for the Fourier coefficients.
D
To prove the corollary, we note that for t E IR oo
Io(,)l
oo
E
e-2zrb /h
k=-oc
k=l
2e-2rrb/ h
1
-
e -2rcb/h
sup
If(')l
sup IImtl~
If(t)l.
Ilmtl~
Similarly, O<3
10(t)l ~<
OO
2rrlkl h - I IOkl ~< 4 z r h - ' ~ k=-oc
k=l
Ikle -2rrbk/h
sup Ilmtl~b
If(t)]
4ge-2rrb/h
=
sup If(t)l h(1 - e-2rrb/h) 2 IImtl~
Taking into account that e -2rrb/h < e-2rr, due to b > h, we obtain the desired estimate for the function and its derivative.
Appendix B: Lemma on Cauchy integral The following 1emma provides a remarkable tool for studying analytical finite-difference equations. It plays a role similar to the role of the partition of unity in the smooth theory. This instrument was proposed by V.E Lazutkin at the very beginning of 90s. Let D C C be a domain with a piecewise smooth boundary OD. Denote by s the set of all complex-valued Lipschitz functions on 0 D. The norm of a function, X E/2, is defined by
IIx II - max Ix (x)[ + sup x#y
Iz(x)- x(y)l I x - yl
LEMMA 2 (Cauchy integral). Let D C C be a bounded convex domain with a piecewise smooth boundary, X E s and let g be a function analytic in D and continuous in the closure o f the domain. Then the integral h(x) - ~ 1
f
D
X(~)g(~) ~ -- x d~,
305
Numerics and exponential smallness
defines two analytic functions hint and hext defined inside D and outside D, respectively. Each of these functions has a continuous prolongation onto the closure of its domain. Moreover
Ihint,ext(X)[ ~ mDllxllsuplgl,
mD -- (1-+- IODI/(2Jr)),
D
where [OD[ denotes the length of the boundary. REMARK 9. If the support of g does not coincide with OD, then hint and hext define together a single analytic function on C \ supp X.
Proof It is evident that h (x) is analytic for x ~ supp X. Consequently, we only have to show that the restrictions of h on D and C \ D have continuous prolongations on the closures of the domains, and to check the upper bounds. For an arbitrary point, x0 E 0 D, we have
if
h(x) -- ~ i
-•
D
g(~e) d~ + X ( X ~
~- x
if
g(~)d~ "
D ~----~
The following equality is due to the famous Cauchy integral formula:
1 Z ~ -~g(~) - -d ~x- -
g(x) -
f~i
g(x),
x6D,
O,
xq~D.
Consequently, the second integral defines two analytic functions, which are obviously continuous in the closure of their domains. We have to check that a similar claim about the first integral is also true. Let us denote the first integral by
1 f ~pxo(X) -
~
L
o
X(~)-
X(Xo)
~ - x
g(~) d~,
xo6OD, xEC\OD.
This is an analytic function of x provided x r OD. Let ~'(x0) = ~0x0(x0). Then Lemma on Cauchy integral follows from the following 2 statements. (1) the function ~': OD --+ C is well defined, continuous and
[~'(x0)[ ~< [OD]/(2rc)[]X]]sup[gl forallx0 6 OD. D
(2) If x ~ x0 along a straight segment transversal to the boundary at x0, then limx ~ xo ~Oxo(x) = ~'(xo). Now we prove these two statements.
V.. Gelfreich
306
(1) We directly obtain the upper bound from the definitions of the corresponding norms:
I r
D
x (~) - x (x0) ~-x
laDI [g(~)[ IdOl ~ ~ Ilxll sup Igl. D
In order to check the continuity we take a small n u m b e r e > 0 and let is be an en e i g h b o r h o o d of x0 in 0 D. Let us estimate the difference
qgxo(XO)- ~oXl(Xl)
1 f~ g(~) (X(~)-X(Xo) X(~)-X(Xl))d ~ ~ - x0 ~ - Xl
-- ~ i
+ ~
if
g(~)(X(~)-X(Xo) X(~)-X(Xl))d~. D\ie
~--XO
--
~--Xl
Consider the first term. The difference in parenthesis does not exceed 2 IIx II. Consequently, the modulus of the first term is b o u n d e d from above by 2 -
sup Igl IIx lie. Y'f is
For a positive e I we can choose e in such a way that the first term is b o u n d e d by e ' / 2 . Let us fix this e < 1 and consider the second term. Let Ix0 - Xll ~< e / 2 , then x(~) - x(xo)
x ( ~ ) - x(x~)
-xo =
~ -xl
X(Xl) - X(Xo)
+
X(Xl) - X(~) xl - x o
- xo
xl - ~
~ - xo
is b o u n d e d by ( 2 / e ) l l x II Ix~ - x01. Consequently, the second term is b o u n d e d from above by 4 - IIx II Ix1 - xolJg. 8
If we choose x I sufficiently close to x0, the last expression is less than e ' / 2 . Consequently,
I~Oxo(xo) - ~Ox,(x 1)1 4 e', which finishes the proof of the continuity. (2) We have 1 ~xo ( X ) - - ~Oxo (XO) - - ~
D
g(~)(x(~)
- x(xo))
~ _ x
1 ) d~. -- xo
Since
1 --x
1 ~ --xo
=
x ~ X0
(~ - x ) ( ~
-x0)
and
x (~) - x (xo) m X0
~< IIx II,
Numerics and exponential smallness
307
the modulus of the integral is bounded from above by
II x II sup Igl
Ix-xol f IdOl 2zr J~D I~ - x l
If x goes to the boundary at the point x0 under a nonvanishing angle, the last integral grows logarithmically, and the product goes to zero due to the evident inequality l i m i t 0 t log t -- 0. Eli
Appendix C" Analytic solutions of finite-difference equations The equation, a(t+h)-a(t)
= g(t),
(29)
where g(t) is a given analytic function in D, provides the simplest example of a finitedifference equation. The function a(t) is a solution for the equation, if the equality holds for all t from the domain of the function g. If defined, the solution is unique up to addition of an h-periodic function. The analytical properties of the solution essentially depend on the geometry of the complex domain D. For example, if the domain D is open to the left, then a solution may be defined by the sum, DO
a(t)-Eg(t-kh)h, k=l
and if the domain is open to the right, DO,
a(t) -- - E
g(t + kh)h.
k=0
These sums define solutions for the finite-difference equation provided the sums are well defined and absolutely convergent. If in addition the sums converge uniformly in h E (0, h0), then the solutions converge to a primitive of g as h --+ 0. Unfortunately, in our case the geometry is different: the independent variable belongs to a square, D - - {t e C : IRetl < r , Ilmtl < r } . Let A ( D ) be the Banach space of all functions analytic in D and continuous in its closure. The space is equipped with the supremum norm. LEMMA 3. There is a bounded linear operator, L "A ( D ) --+ A ( D ) , such that a - L(g) is a solution of the finite-difference Equation (29).
V. Gelfreich
308
can represent the domain D as an intersection of two half-strips, D D - A D +, where
P R O O F . 2! W e
D - -- {t EC" R e t < r , Ilmtl < r } , D + -- {t ~ C" R e t > - r , I lmtl < r}. We can use the lemma on Cauchy integral to represent the function g as a sum g = g - + g+, such that the functions g+ are analytic and fast-decreasing in D +, respectively. We have already obtained the formulas, which solve the finite-difference equation with such right-hand-sides. Let us describe the procedure in more detail. Let us define an auxiliary smooth function X :R ---> R, such that x(r)
_10,
/
r~<-1/2,
1,
r ~> 1/2,
and let us define two smooth functions on the boundary 0 D by the following equations" X - ( t ) -- x ( R e ( t ) / r ) ,
x+(t)-
1 - x-(t).
Then we let h(t) -- g ( t ) c o s h ( t / r ) and
d~e, h - (t) -- ~ i1 f~ D X- ~(~)h(~) -- X
h +(t) -- ~
~ -- X 1 f D X+(~)h(~) d~.
Obviously, h - ( t ) + h+(t) = h(t) for all t E D. Moreover, according to the lemma on Cauchy integral the functions h i ( t ) are analytic and bounded in D +, respectively. Then the functions, g - ( t ) --
h-(t) cosh (t / r ) '
g+(t) --
h+(t) cosh(t / r ) '
define a representation, g(t) - g - ( t ) + g+ (t) for all t 6 D. We can separately solve the finite-difference equations, a - ( h + h) - a - ( t ) h
= g-(t),
a+(h + h) - a+(t) h
+ = g (t),
in the following way: oo
a-(t)-Zg-(t-kh)h, k--1
oo
a+(t)--Eg+(t
+kh)h.
k--0
Since the equation is linear, a(t) - a - ( t ) + a+(t) provides a solution for the original 2l The proof is based on the ideas of V.F. Lazutkin.
Numerics and exponential smallness
309
equation. This defines the linear operator L which solves the finite-difference equation in the square. E3 REMARK 10. Lemma 3 is valid in an arbitrary convex bounded domain D.
Appendix D: Analytic parameterization of separatrices In this section we describe a remarkable separatrix parameterization for a two-dimensional saddle. This parameterization is used in the proof of the theorem on the splitting of complex manifolds (Theorem 4). The parameterization is very useful for numerical approximation of separatrices. In a neighborhood of a hyperbolic fixed point the parameterization may be expanded in Taylor series. If the original map is entire the coefficients of the series decrease extremely fast (faster than consta k for any a > 0). This permits to use a relatively small number of terms for a high-precision approximation of a large piece of the separatrix. Without loss of generality we may assume that the saddle point is at the origin. Let F : (C 2, 0) ~ (C 2, 0) be a local analytic diffeomorphism, F (0) = 0. We denote the eigenvalues of the matrix D F (0) by ,kl and )~2. Let us assume that the origin is a hyperbolic saddle, i.e., 0 < 1~,21 < 1 < I'kl I.
LEMMA 4. There is a unique analytic parameterization, qg: (C, 0) --+ (C 2, 0), stable manifold, such that: r
~.
F((p(z)),
of the
un-
(30)
and qg(0) = 0, Ilqg'(o)II = 1. PROOF. We consider Equation (30) in the coordinate form. We can take the eigenvectors of the matrix D F (0) as a basis. In the new coordinates the linear part D F (0) is diagonal, so the map F has the form,
x2
)~2x2 + Gz(xl, x2)
where the Taylor expansions of the functions G1 and G2 do not contain constant and linear terms: Gk (0, 0) = 0, DGk (0, 0) = 0, k = 1, 2. Let us denote by (qgl, q92) the components of the vector function ~0 in this coordinate system. Then Equation (30) takes the form of the following system:
q)l(XlZ) -- )~lqgl(Z) + G1 (q)l(Z), q)2(z)), ~o:(Z,Z) -- ZZ~O2(Z) + G:(~o, (z), ~o:(z)).
310
V. Gelfreich
In the linear approximation the unstable separatrix coincides with the first coordinates axis. This proposes the following substitution: q)l (z) - z + ~Pl (z). This substitution leads to the system, ~1 ()~IZ) -- ~.1~1 (Z) + G1 (Z -+- ~1 (Z), q)2(Z)),
(31)
q)2()~lZ) -- ~.2992(Z) -+- G2(z -+- ~1 (z), q92(z)). We show that this system has a solution with grl (z) = O(z 2) and q92(z) = O(Z2). We rewrite the system in the form of an "integral" equation, and then we use the contraction map theorem in a suitable Banach space. We invert the linear part of the equation. The inverse is not unique, our choice leads to the equation (X3
k-1 k=l
(32)
oo
k=l where gk(z) - Gk(z + grl (z), 992(Z)). It can be verified by a direct substitution that any solution of this "integral" equation satisfies the original system (3 1). Let X denote the Banach space of analytic vector functions, g r ' D r --+ (C 2, 0), where Dr -- {z 6 C" Izl < r}, which are continuous in the closure Dr and have a finite norm, II~ II -
m a x sup k=l,2 zCDr
~k(z)
Z2
If gr = (lpl , (/91) E ~t~, the sums in the right-hand-side of (32) converge absolutely and uniformly defining an analytic function from X. It only remains to check that the nonlinear "integral" operator is contracting in a ball in Y provided the radius of the domain r is not too large. This is not too difficult. D COROLLARY. Let F" C 2 ~ C 2 be an analytic diffeomorphism, and let xo be its saddle point with multiplicators, 0 < 1~,21 < 1 < I)~l I. The stable and unstable separatrices of xo are images of entire maps, q)-, g)+'C --+ C 2. These maps satisfy the equations, ~o-(~.lz)-- F(q)-(z))
and
q)+()~2z)= F(q)-(z)),
and the initial condition, q)+(O)- xo. The parameterizations q)• are unique under the normalizing condition, IIDq9• (0)II -- 1. The proof of the corollary is quite elementary. Equation (30) extends the domain of qgfrom Dr to Dr' with r' -- I)~llr. Since I)~l] > 1 we can repeat the procedure to cover the whole complex plane.
Numerics and exponential smallness
311
In other words, the restriction of the map F on a separatrix is analytically conjugated with multiplication by the corresponding eigenvalue. This is quite simple dynamics. The complicated behavior of the separatrices is due to a complicated embedding of the separatrix into the phase space. If the map F = F~ appears in the process of discretization of a differential equation, : f (x), it is convenient to use other parameterizations for the separatrices, which allow a direct comparison with the separatrix of the differential equation. These parameterizations or- and cr + are solutions of the finite-difference equations,
Since the integrator has the form, Fc(x) = x + e f ( x ) + e2G(x, e), the last equation takes the form, cr+(t + e) - o'+(t)
= s(.+(,))+,G(-•
It is easy to check by a direct substitution that its solutions may be obtained by the following change of parameter: a-(t)-
~o- (etl~
O--q-(t)- ~-q- (et log&2/c).
This change of parameter moves the fixed point to -cxz in the case of the unstable separatrix and to +cx~ in the case of the stable one. The dynamics on the separatrix becomes a translation by e to the right. When e goes to zero, each of these parameterizations converges to a separatrix solution of the differential equation, 6 - f (o-). But this convergence is not uniform in the parameter, t. Moreover, for some values of t there is no convergence at all, because cr (t) can have singularities, even in the case when f and G are both entire. The approximation results are formulated in Section 8.
References [ 1] V.M. Alekseev, Quasirandom dynamical systems III. Quasirandom oscillations in one-dimensional oscillators, Math. USSR Sb. 78 (1) (1969), 3-50 (Russian). [2] V.I. Arnold, V.V. Kozlov and A.I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, 2nd edn., Springer, Berlin (1993). [3] G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the-identi~ symplectic mappings with application to symplectic integration algorithms, J. Statist. Phys. 74 (5-6) (1994), 1117-1143. [4] J. Barrow-Green, Poincard and the Three Body Problem, History of Mathematics, Vol. 11, Amer. Math. Soc., Providence, RI (1997). [5] H. Broer and C. Sim6, Hill's equation with quasi-periodic forcing: Resonance tongues, instabili~ pockets and global phenomena, Bul. Soc. Bras. Mat. 29 (1998), 253-293. [6] R. Cushman, Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors, Trans. Amer. Math. Soc. 238 (1) (1978) 45-55. [7] E. Fontich and C. Sim6, Invariant manifolds for near identi~ differentiable maps and splitting of separatrices, Ergodic Theory Dynamical Systems 10 (1990), 319-346.
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[8] E. Fontich and C. Sim6, The splitting of separatrices for analytic diffeomorphisms, Ergodic Theory Dynamical Systems 10 (1990), 295-318. [9] B. Fiedler and J. Scheurle, Discretization of homoclinic orbits, rapid forcing and "invisible" chaos, Mem. Amer. Math. Soc. 119 (570) (1996), 79 (Konrad-Zuse-Zentrum, preprint SC 91-5, (1991)). [10] V.G. Gelfreich, Conjugation to a shift and the splitting of invariant manifolds, Appl. Math. 24 (2) (1996), 127-140. [ 11] V.G. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map, Comm. Math. Phys. 201 (1) (1999), 155-216. [12] V.G. Gelfreich, Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps, Phys. D 136 (2000), 266-279. [13] V.G. Gelfreich, Splitting of separatrices near resonant periodic orbits, Math. Phys. Preprint Archive, No. 00-402, http://www.math.utexas.edu/mp_arc. [14] V.G. Gelfreich, V.E Lazutkin and N.V. Svanidze, A refined formula for the separatrix splitting for the standard map, Phys. D 71 (2) (1994), 82-101. [15] V.G. Gelfreich, V.E Lazutkin and M.B. Tabanov, Exponentially small splitting in Hamiltonian systems, Chaos 1 (2) (1991), 137-142. [16] V. Gelfreich and D. Sauzin, Borel summation and the splitting of separatrices for the Henon map, Notes Scientifiqes et techniques du Bureau des Longitudes, S067 (Mai 1999), 48 p. [ 17] E. Hairer and Ch. Lubich, The life-span of backward error analysis for numerical integrators, Numer. Math. 76 (4) (1997), 441-462. [18] B.M. Herbst, G.J. Le Roux and M.J. Ablowitz, Chaos in Numerics. Numerical Analysis, World Scientific, Singapore (1996). [19] V.F. Lazutkin, Splitting of separatrices for the Chirikov's standard map, VINITI no. 6372 (84) (1984) (Russian). [20] V.E Lazutkin, I.G. Schachmanski and M.B. Tabanov, Splitting of separatrices for standard and semistandard mappings, Phys. D 40 (1989), 235-348. [21] A.I. Neishtadt, The separation of motion in systems with rapidly rotating phase, PMM (USSR) 48 (2) (1984), 197-204 (Russian); English transl, in: J. Appl. Math. Mech. 48 (2), 133-139. [22] Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, MIT Press, Cambridge, MA (1971). [23] H. Poincarr, Les M~thodes Nouvelles de la M~chanique C~leste, Vols. 1-3, Gauthier-Villars, Paris (1892); Reprinted by: Dover, New York (1957). [24] C. Sim6, Averaging under fast quasiperiodic forcing, Hamiltonian Mechanics: Integrability and Chaotic Behavior, J. Seimenis, ed., NATO Adv. Sci. Inst. Ser. B Phys., Vol. 331, Plenum, New York (1994), 13-34 (Held in Toruri, Polland, 28 June-2 July 1993). [25] Ya.G. Sinai, Topics in Ergodic Theory, Princeton Univ. Press, Princeton, NJ (1994). [26] Yu.B. Suris, On integral maps ofstandard type, Funk. Anal. Prilozh. 23 (1) (1989), 84-85 (Russian). [27] Sh. Ushiki, Sur les liaisons-col des syst~mes dynamiques analytiques, C. R. Acad. Sc. Paris, Ser. A 291 (1980), 447-449.
CHAPTER
7
Shadowability of Chaotic Dynamical Systems Celso Grebogi 1, Leon Poon 2'3, Tim Sauer 4, James A. Yorke 5 and Ditza Auerbach 2 I Instituto de Fisica, Universidade de S~to Paulo, Caixa Postal 66318, 05315-970 Sao Paulo, SP, Brazil E-mail:
[email protected] 2 Institute for Plasma Research, University of Maryland, College Park, MD 20742, USA 3 Department of Physics, University of Ma~land, College Park, MD 20742, USA 4 Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA 5 Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
315
2. Basic concepts in shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
317
2.1. Continuous shadowability
........................................
317
2.2. Shadowing lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
318
2.3. Shadowing in nonhyperbolic systems
320
..................................
2.4. Glitches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Brittleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Homotopy continuation and brittleness
.................................
321 323 323
3.2. Test brittleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Numerical evaluations of brittleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
325 327
4.1. Brittleness of the forced damped pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Test brittleness of the double rotor system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Shadowing time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
327 329 332
6. Spatially unstable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
338
7. Implications for modeling
...........................................
342
8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
343
Acknowledgements
344
References
................................................
.....................................................
H A N D B O O K OF D Y N A M I C A L S Y S T E M S , VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 313
344
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Abstract In studying their systems, physical scientists write differential equations derived from fundamental laws. These equations are then used to understand, analyze, predict, and control the system's behavior, provided one is able to determine the solutions. As the role of nonlinearity grows in importance for the study of physics, solutions often cannot be obtained in closed form, and numerical solutions must be relied on. Computers are now an integral part of the physicist's modus operandi. A basic question always present when obtaining numerical solutions is to what extent they are valid. This question is especially meaningful when dealing with chaotic dynamics, since local sensitivity to small errors is the hallmark of a chaotic system. Floating-point calculations commonly used to approximate solutions of differential equations or compute discrete maps produce pseudo-trajectories, which differ from true trajectories by new, small errors at each computational step. Despite the sensitive dependence on initial conditions, the methods of shadowing have shown that for chaotic systems that are hyperbolic or nearly hyperbolic, locally sensitive trajectories are often globally insensitive, in that there exist true trajectories with adjusted initial conditions, called shadowing trajectories, very close to long computer-generated pseudo-trajectories. A dynamical system is hyperbolic if phase space can be spanned locally by a fixed number of independent stable and unstable directions which are consistent under the operation of the dynamics. In the absence of hyperbolic structure, much less is known about the validity of long computer simulations. Recently it was shown that trajectories of a chaotic system with a fluctuating number of positive finite-time Lyapunov exponents fail to have long shadowing trajectories. In other words, they are globally sensitive to small errors. Such hyperchaotic system has two positive Lyapunov exponents, although finite-time approximations of the smaller of the two fluctuate about zero, due to visits of the trajectory to regions of the attractor with a varying number of stable and unstable directions. The destruction of hyperbolicity caused by this phenomenon leads to global sensitivity- only relatively short pseudo-trajectories will be approximately matched by true system trajectories. Our discussion of the global sensitivity of trajectories for these non-hyperbolic systems is limited in this review to the comparison between physical models and computer simulations, but the same questions arise whenever comparing the time behavior of two systems evolving under similar, but slightly different dynamical rules. For example, a natural system and its theoretical model differ by modeling errors. In the presence of fluctuating Lyapunov exponents, global sensitivity may lead to trajectory mismatch, in particular when long times are considered. The result is that no trajectory of the theoretical model matches, even approximately, the true system outcome over long time spans.
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1. Introduction
Dynamical theories of physical systems are generally based on differential equation models. Typical systems of ordinary differential equations cannot be solved analytically, and hence one must generate solutions by numerical means. In any computer simulation, the numerical solution is fraught with truncation errors introduced by discretization and roundoff errors introduced by finite-precision calculation. A natural question arises as to whether the behavior of a numerical solution is similar to any "nearby" true solution of the system, that is, is the solution of the perturbed system uniformly close to the solution of the unperturbed system? More importantly, will the long-term statistics based on computer simulation yield quantities of physical interest? If not, then the model equations would be useless for any practical purposes. For example, a climate model that continues to repeat winter conditions all year long because of accumulated numerical errors will be useless for computing the mean annual temperature. This problem is even more pronounced in the case of chaotic systems. The trajectories of chaotic systems exhibit sensitive dependence on initial conditions: two trajectories with initial conditions that are extremely close diverge exponentially on the average from one another. As a result, a small truncation or roundoff error made at any step during the computation will tend to be greatly magnified by future evolution of the system. In view of this, it is natural to ask under what conditions the computed trajectory will be close to a true trajectory of the model. Previous studies have provided algorithms which can often be used to prove the existence of a true trajectory of the system that stays near or shadows the computer-generated trajectory. These works share the same goals as the shadowing theorems by Anosov [2] and Bowen [6], which show that shadowing is always possible in hyperbolic systems for arbitrarily long times. In nonhyperbolic dynamical systems, recent work [9,18,21,32-34] has concentrated on proving the existence of finite-length shadowing trajectories with the aid of computers. Essentially, the approach is to show that as long as the finite-length computer-generated trajectory remains sufficiently hyperbolic, then that piece of computergenerated trajectory is shadowable by a true trajectory. For example, a theorem by Sauer and Yorke [34] says that if certain quantities evaluated at points of the computer-generated trajectory are not too large, then there exists a true trajectory near the computer-generated one.
Consideration of simple examples of nonlinear maps [ 14,18,21 ] show that there are critical points of trajectories where roundoff errors or other noise can introduce new behavior. At such "glitches" the true trajectories diverge from the computed trajectory. Not only is the computed trajectory unshadowable, but the measure may be wrong and any long-term statistics based on them may show markedly different behavior from those of the model under study. In addition, the occurrence of a glitch is not an uncommon phenomenon, and for those simple systems, it can be attributed to folds caused by homoclinic tangencies and near tangencies of stable and unstable manifolds. Thus, the shadowability of chaotic systems is of paramount importance since it is a minimum requirement for the computation of long-term statistics. The other requirement is that the measure generated by a shadowing trajectory is the natural measure. The second requirement is necessary since there
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are cases where the true trajectory shadowing the numerical trajectory is not statistically "typical" [29]. In this review, we present some tools that will be useful in diagnosing whether certain dynamical systems are shadowable. It is our belief that in systems with high-dimensional chaos, one can only shadow the numerical trajectories for short period of times. We examine possible causes of failure in shadowability that is likely to occur in higher-dimensional chaotic systems, including spatially extended dynamical systems. In order to quantify the shadowability or lack thereof of a dynamical system, we need some concepts introduced by Dawson et al. [ 14]. They argued that only certain computergenerated pseudo-trajectories, called continuously-shadowable pseudo-trajectories, are meaningful in analyzing the shadowability of a system. In short, a continuouslyshadowable trajectory is a computer-generated trajectory that can be continuously deformed, or continuously perturbed, into a true trajectory, in such a way that the errors at each trajectory point are decreased monotonically to zero. Associated with the idea of continuous shadowability, we give a heuristic reasoning for the existence of a constant of proportionality between the error magnitude and the distance the pseudo-trajectory must move in phase space to be deformed into a true trajectory. This constant of proportionality is called the brittleness of the pseudo-trajectory and is easily seen to be a measure of the shadowability of a system. If the brittleness multiplied by the error magnitude of the pseudo trajectory is of the order of or larger than the extent of the attractor in phase space, then the system is unshadowable. In practice, the true brittleness cannot be computed precisely since it is a property of the true trajectory, and it is difficult to know the true trajectory exactly. Instead, we can use perturbations to the pseudo-trajectory and the Jacobian information generated from the simulation to compute a first-order approximation to the brittleness. We call this the test brittleness. Knowledge of the test brittleness is a useful diagnostic for continuous shadowability of the pseudo-trajectory, or lack thereof, as will be discussed later. In examining the brittleness of various pseudo-trajectories, we found that pseudotrajectories with high brittleness, and hence unshadowable, are usually accompanied by a Lyapunov exponent that "fluctuates about zero". By this we mean that given any long trajectory, computation of finite-time Lyapunov exponent using pieces of the trajectory would yield both positive and negative values for pieces of the trajectory of arbitrary length. Since the finite-time Lyapunov exponents quantify the expansion and contraction of phase space along the the trajectory over a finite stretch of time, this implies that the trajectory is visiting areas of phase space where the number of stable and unstable directions changes. The prototypical physical system is the double rotor system studied by Grebogi et al. [19]. In this system and for certain parameter values, a typical computer-generated trajectory undergoes expansion alternately in one or two perpendicular direction as it moves densely throughout the attractor. As a result, even rather short computer-generated trajectories of the double rotor are unshadowable. For systems with a Lyapunov exponent that fluctuates about zero, it would be advantageous to know how long one can expect to be able to shadow a pseudo-trajectory. Using ideas from Brownian motion theory with a reflecting barrier and applying it to the finite time Lyapunov exponent distribution, we were able to make order of magnitude estimates of the length of a pseudo-trajectory which can be shadowed [33]. The fundamental ques-
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tions of "how good" and "for how long" the solutions are valid is answered. The work in Ref. [33] answers these questions by establishing scaling laws for the shadowing distance and for the shadowing time in terms of physically meaningful quantities that are easily computable in practice. The scaling theory is verified against a physical model. In previous works, the failures of shadowability have been linked to nonhyperbolicity in the systems. However, in studying spatially extended systems, we discover a new effect, unique to these systems, which limits shadowability. In this review, we demonstrate how the presence of spatial instabilities can drastically curtail the size of systems that can be shadowed for long times. The outline of the review is as follows. We review some basic concepts in shadowing and discuss the ideas of continuous shadowability in Section 2. In the same section, we also outline the proof of the shadowing lemma and examine different kinds of glitches that result in unshadowability of pseudo-trajectories. The concepts of brittleness are explained in Section 3, and we also present an algorithm for computing the test brittleness. This algorithm is implemented numerically on the forced damped pendulum and the double rotor system, and we discuss the results in Section 4. In particular, we examine the role of the fluctuating Lyapunov exponent in the unshadowability of the system. The method of using finite time Lyapunov exponent distribution to compute the shadowing time and distance is explored in Section 5 along with some numerical results. In Section 6, we investigate the shadowability of spatially extended systems. In Section 7, we discuss the implications to modeling. We argue that, for certain chaotic systems, no model will produce trajectories that are close to the natural system. Finally, in Section 8, we present the main conclusions of the work.
2. Basic concepts in shadowing 2.1. Continuous shadowability Let f denote the map which represents one time step of the dynamics. For example, it may represent the time-T map induced by an ODE solver where the initial state is integrated forward in time by an amount T. Thus, if P0 is the present state of the dynamical system, then the correct state at time T later is given by f (p0). However, there will be a discrepancy between the solution produced by the ODE solver and the true state of the system after a time T. This discrepancy or one-step truncation error due to roundoffis usually bounded by some value g. If the ODE solver generates pi from P0, then IPl - f(P0)l < g and P0 and pi are two points of a g-pseudo-trajectory of the dynamical system f . Further numerical integration of the system would yield a g-pseudo-trajectory {p0 . . . . . PN } of length N + 1 where Ipi+l - f(Pi)i < g f o r / = 0 . . . . . N - 1. An important question that arises is whether there exists a true trajectory that "stays close" or shadows the pseudo-trajectory. By true trajectory we mean a trajectory that satisfies the map f , that is, f ( x i ) - x i + ! for i - 0 . . . . . N - 1. The true trajectory {xi}X0 is said to be an e-shadowing trajectory for the pseudo-trajectory {pi }L0 if there exists an e such that IXi -- Pi[ < e for i = 0 . . . . . N.
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If phase space is D-dimensional, we can define a long trajectory vector P = {P0 . . . . . PN} of length (N + 1)D and an error vector E = {Pl - f ( P o ) . . . . . PN -f ( P N - 1 ) } of length N D associated with P. A true trajectory vector X - {x0 . . . . . x u } would obviously yield a zero error vector by virtue of it satisfying the dynamics exactly. If we denote the original noise vector as E0, then we seek to deform this error vector continuously into zero. To accomplish that, we define an error function F such that F" P --+ E. If F -1 (E) goes from P to X continuously as E goes from E0 to 0, then the pseudo-trajectory {pi }U_0 is called continuously-shadowable. This might appear to be a more stringent condition for a shadowable pseudo-trajectory, but we believe that continuous deformability to a true trajectory is a minimum requirement for accepting a computer simulation as meaningful information. Continuous shadowability results from a perturbation of the true trajectory. If the pseudo-trajectory is not continuously-shadowable, probably there is no way to find the true trajectory. In fact, if a dynamical system satisfy the hypothesis of the original A n o s o v - B o w e n shadowing theorems [2,6,28] or the computer-assisted shadowing models [9,18,21,32-34], then the pseudo-trajectories are not only shadowable but continuously-shadowable. This is a sufficient condition for shadowing.
2.2. Shadowing lemma The shadowing lemma of Anosov and Bowen is a theoretical result for dynamical systems with hyperbolic structure. In Refs. [9,18,21,32-34] one gets shadowing as long as the pseudo-trajectory remains hyperbolic. An invariant set A is hyperbolic for f if for each x in A, the tangent space Tx splits into a direct sum of stable and unstable subspace Tx -- ESx 9 E u with the properties that: (i) the splitting varies continuously with x and is invariant under the action of D f , that s and D f (E u) -- E f(x) u , and is, V f (E s) = E f(x) (ii) there are constants C > 0, 0 < X < 1 such that: if v 6 E s, then [ D f n (x)v[ ~< CXn[v[ 9 if v ~ E Xu' then [Df-'~(x)v] <~ cXn[v[ The above conditions imply that vectors in a small neighborhood of E s (E x) will exponentially approach the forward (backward) iterated trajectory of x at a rate )~ which is uniform for all points in A. Furthermore, for each point x in A, we can define the stable and unstable manifolds of x. The stable manifold of x or W s (x) and the unstable manifold of x or W ~ (x) is defined by W S ( x ) _ {y ~ R,n. f n ( y ) __+ f n ( x ) as n --~ oo}, WU(x) -- {y E R m" f - n ( y )
__+ f - n ( x ) as n --+ co}.
Note that W s (x) is tangent to E~c and W u (x) is tangent to E~ at x. If the stable and unstable manifolds of a point x intersect, then the intersection is called a homoclinic intersection. If the stable and unstable manifolds are tangent at the intersection point, then it is called homoclinic tangency. One implication of hyperbolicity in a dynamical system is that homoclinic tangencies do not occur at any point x in the hyperbolic invariant set, that is,
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the angle between the stable and unstable manifolds, and hence the subspaces, is bounded away from zero. The shadowing lemma states that if a dynamical system is hyperbolic, then for each nonzero e, there exists an error magnitude 6 such that each 6-pseudo-trajectory can be e-shadowed. Furthermore, it was shown in Meyer and Hall [28] that each pseudo-trajectory from a hyperbolic system can be continuously shadowed within a distance of e. To see this, we present a brief outline of a proof of the shadowing lemma given in [28]. Since the shadowing lemma implies the existence of a shadowing trajectory of arbitrary length, we start with an extension of the long trajectory vector introduced above, namely, P ~ -{. . . . p - l , p0, p l . . . . } where the p,, 's are individual trajectory points. Let V be the space of all infinitely long vectors defined above, and the norm is given by IIPor II - sup,, Ip,, I where I" I denotes the Euclidean norm. Next, define a function G ' V --+ V such that ( ~ ( P x J ) ) , , - f ( ( P ~ ) , , - l ) - f ( P , , - l ) where f is a smooth map with hyperbolic structure and (.)i denotes the i th component of a vector in V. (Note that the i th component of P ~ , which is P,,, is also a vector, but it is finite-dimensional.) If X ~ is a fixed point of G, then ( ~ ( X o r x , , - f ( x , , - l ) . Thus, X ~ is a true trajectory. Por E V is a a-pseudo-trajectory if I]~(Por - Por < ~. Define the function f" such that .T'(Por -G ( P ~ ) - Poc and rename f ' ( P ~ ) as E ~ , then we can reformulate the shadowing lemma as: for every e > O, there exists a 6 > 0 such that if II.T'(P:v)[I- IIE~ -011 < ~, then there is a fixed point X ~ of ~ such that ]lPor - S o c l ] Ilbt--l(Eor -- f - l ( 0 ) l ] < e. In this suggestive notation, we can identify f" and E ~ as extensions of the error function F and the error vector E, respectively. In addition, this formulation also makes it plain that 9t-- 1 is continuous, and thus the ~-pseudo-trajectory Por is continuously shadowable. The key in showing that the pseudo-trajectory is continuously-shadowable is the use of the implicit function theorem in the form given in Ref. [22]. Before we can apply the theorem, certain conditions need to be satisfied, namely, we need to show that I I O ~ ( Y ~ ) l l c <~ g and II[D~'(Yor ~< K for all Y~r E Bo(P~o) (i.e., all Y~ with IIY~r - Pall < r/). Here, I1" IIc denotes a norm on the linear vector space s V), and it is defined by I I A I I c - maxllY~ll-1 IIA(Y~)II where A E s V). K and r/are constants that depend on the value of e. (Since D f [ D . T ' ] - l -- I, IID~-IIclI[D~] -111c/> IIIIIc -- 1, and thus IID~IIc and II[D~] -111c are bounded below by K - 1 . ) If we let p = rl/K 2 and o- -- 0 / K and if the above conditions are satisfied, then the implicit function theorem guarantees the existence of a domain s with B p ( P ~ ) C s C B,I(Por such that .T" is oneto-one on s Furthermore, B,~(.T'(P~)) -- B ~ ( E ~ ) C f(s (please refer to Figure 1). In particular, a proper choice of r/(which depends on e) will imply 0 6 B~ (Eor which means there is a unique true trajectory X ~ E s such that .T'(Xor - 0 . Moreover, the choice of 7/is such that II P ~ - x ~ II < rl ~< e. Thus, we have an e-shadowing trajectory. In particular, the implicit function theorem ensures the continuity of f - - I on B~ (Ecr This means that pseudo-trajectories (including P ~ ) in a small neighborhood of X ~ are continuously shadowed by the same true trajectory, namely X ~ . Recall that we cannot apply the theorem until we have shown that the derivative Dr" and its inverse [D.T']-l are bounded. This is where hyperbolicity comes in. Using conditions (i) and (ii) in the definition of hyperbolicity, one can show that the above values are bounded (see [28] for more details).
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F
F-1
F(~) Bo(P~o) Fig. 1. A schematic diagram depicting continuous shadowability.
2.3. Shadowing in nonhyperbolic systems While the shadowing lemma is a very important result, it is not widely applicable because most dynamical systems of interest are not hyperbolic. Furthermore, it is a prerequisite of the shadowing lemma that the pseudo-trajectory be in the hyperbolic invariant set A, but this is generally not the case when A has a fractal measure with respect to the Lebesgue measure. Motivated by these limitations, recent studies [9,18,21,32-34] have concentrated on computer-assisted proofs of the existence of finite-length shadowing trajectories for nonhyperbolic systems. One in particular, [34], was able to formulate a shadowing theorem which states that if certain quantities evaluated at points of the pseudo-trajectory are bounded, then there exists a true trajectory near the pseudo-trajectory. It should be noted that the shadowing lemma for hyperbolic maps follows as a corollary to the above generalized shadowing theorem. Furthermore, a key technique used in the proof is a method of reducing the noise in the pseudo-trajectory, thus generating a less noisy trajectory. This method, which was first introduced in [18,21], is called the refinement method (see Section 3.2 for more details). It was shown in [34] that repeated application of the refinement method will theoretically converge to a true trajectory near the original pseudo-trajectory. In showing that a finite-length pseudo-trajectory is shadowable, one is essentially showing that the pseudo-trajectory is sufficiently close to being hyperbolic, and thus continuously shadowable. To do this, we let Sn and Un denote complementary subspaces of the tangent space ~;~O N at each point Pn of a pseudo-trajectory {p,, }n=0, that is, Sn -q- Un - ]l~m for n = 0 . . . . . N. Assume that the Sn are consistent in the sense that the linearized dynamical system maps Sn to Sn+l within the truncation size 6, and the same for Un. A suitable choice of Sn and Un is to have them approximate the stable and unstable directions of Pn at each n, respectively. They do not need to have any relation to exact global stable and unstable manifolds, and in fact since the system may not be hyperbolic, these manifolds may not exist. The angle On
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between the subspaces [ 1] is defined implicitly by COS On " -
max 13T w , vESn, tUEUn
where v 3: denotes the transpose of v. The quantities which need to be measured to assure the existence of a nearby true trajectory depends on the angle 0,7 in the form of csc 0,,. Thus, if 0,7 is bounded away from zero, then the relevant quantities will have bounded magnitude, and there is a true trajectory near the pseudo-trajectory.
2.4. Glitches Shadowing would fail if the trajectory encounters regions of nonhyperbolicity. Failures of pseudo-trajectories to be continuously-shadowable are called "glitches" [ 14,18,21 ]. We say that a pseudo-trajectory has a glitch at point n if {Pi }tiz=0 can be continuously shadowed, but {pi }iz__+~ cannot. A schematic representation of one type of glitch is shown in Figure 2. Assume that q is a fixed point of the iteration. Assume that the pseudo-trajectory has no error until n, when error pushes p,7 across the stable manifold of q. True trajectories follow the unstable manifold of P,7-1, which separates exponentially from the trajectory located across the stable manifold of q, so that no continuous shadowing trajectory can exist. A second type of glitch can occur when the number of stable directions is not constant along the trajectory, making hyperbolic structures impossible. This can happen when the periodic points embedded in the chaotic attractor have varying numbers of stable eigendirections. If a trajectory spends arbitrary length of times near each of these periodic points,
stable manifold unstable manifold _
lr
oP~ _
V
Fig. 2. A near-tangency from a nonhyperbolic system. A small error in the computation of f (p,_ 1) can push p, across a stable manifold, resulting in a glitch.
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calculation of the finite-time Lyapunov exponents would yield varying number of positive Lyapunov exponents depending on which finite segment of the trajectory was used. This leads to at least one Lyapunov exponent that fluctuates about zero. The manner in which this fluctuation results in unshadowable pseudo-trajectories is nicely demonstrated in a theoretical model studied by Abraham and Smale [1] in 1970. In their example, there is an invariant set containing two fixed points. Let q l denote the fixed point with a single local expanding direction, and q2 the fixed point with a two-dimensional local expanding set. Typical trajectories wandering through the invariant set spend arbitrarily long times near each of the fixed points. The second largest Lyapunov exponent of such a trajectory is positive for trajectory segments near q2, and is negative for trajectory segments near q l, so this exponent fluctuates about zero. A ball of initial conditions beginning near q l will be contracted into a line segment (with small thickness) under evolution of the dynamics (refer to Figure 3). A computer-generated trajectory beginning in the ball, with truncation error 6,
Fig. 3. A diagram showing how a fluctuating Lyapunov exponent can lead to unshadowability of pseudotrajectories.
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will be displaced a distance of 8 from the line segment. When the region around the numerical trajectory develops a second expanding direction by visiting a neighborhood of q2, the numerical trajectory will be pushed away exponentially fast from the line segment of true trajectories, resulting in an unshadowable pseudo-trajectory.
3. Brittleness 3.1. Homotopy continuation and brittleness If a pseudo-trajectory is continuously-shadowable, we can introduce a homotopic parameter c~ such that F(P(a)) = E(c~). We now define or0 and o/f such that E(o~0) = E0 and E (otf) : 0 , and if we vary ot continuously from or0 to otf, then P (or) transforms continuously from P(ot0) = P0 to P(olf) : X where P0 is the original pseudo-trajectory and X is the true trajectory (see Figure 4 for a schematic illustration). Homotopic deformation of P0 continuously to X, and hence moving E0 to 0, requires the straight line approach which decreases the error magnitude in the direction of the error. For homotopy continuation, F satisfies the following differential equation [14]
d F (ot) d~
-- - F ( a ) .
(1)
We note that Equation (1) can be written as
d F ( P ( u ) ) _. DpF(P(ot)) dP(ot) = - F ( P ( o t ) ) do~ du '
(2)
where DpF(P(~)) is the Jacobian matrix of F evaluated at P(c~). This equation is a continuous version of Newton's method. If we approximate d P(~)/d~ ~ (P(~ + Aot) P(u))/Aot and if [DnF(P(~))] -1 exists, then Equation (2) can be rearranged to the form
(3)
P(ot + A o t ) - P(ot) - [D/,F(P(ot))]-' F(P(ot))Aot. In fact, this reduces to standard Newton's method if we set Aot -- 1.
Eo
F1
Po ("
E
X Fig. 4. A schematic diagram illustrating the dependence of the error vector F on the homotopic parameter or.
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Using the homotopy continuation method to move the error vector E to 0 implies that the direction of E (or) stays constant while its magnitude decreases as we homotope from or0 to oef. Consequently, we can write the error vector as E(ot) - ]E(oe)ls162 where d is an NDdimensional vector of norm 1~1 - 1. If we define a new variable ~ such that ~(c~) -- [E(ot)[, then we can rewrite Equation (2) as
dP = [ D p F ( P ) ] _ l ~ .
(4)
d~
For P sufficiently close to the true trajectory vector X, we can make the following approximation: AP
A~
[DpF(P)]-'[l ~ [DpF(X)]-'d.
(5)
Roughly speaking, we see that there exists a constant of proportionality (l[ Dp F (X) ] - 1s162 between the the error magnitude (A~ = IE01) and the distance ( A P ) the pseudo-trajectory P must move in phase space to be deformed into a true trajectory X. We call this constant of proportionality brittleness. If the function F is smooth, then one has that [D F]-1 = D F -1 . Thus, the brittleness is essentially a property of the true trajectory, and it measures the stretching of F -1 near the origin of the error vector E. Alternatively, brittleness can be seen as an indicator of how far the true trajectory will be perturbed if we vary the noise level. For small one-step errors, we can see from Equation (5) that the brittleness should be independent of the error magnitude. However, it is also clear that brittleness as defined in Equation (5) would depend on the error directions s162Technically, if we want the brittleness to be a true constant independent of error directions, we would define it as II[DpF(X)] -1 II where II 9 I[ denotes an operator norm. Since [ D F ] -1 is a linear operator defined on R No, that is, [DF] -l" ]t~ND ~ I~ ND, then II" II [30] is defined as the largest amplification of length that the operator [DF] -l is able to induce on a vector v in R N~ or
IIrDf]-' II----sup ~o
I[DF]-I(v)I
Ivl
However, applying the norm to [DF] -1 in Equation (5) involves taking the limit A~ --+ 0 and finding the maximum value of I[DpF(X)] -1 ill over all possible error directions s162so it is not really a workable definition. Instead, we start with a pseudo-trajectory, and when the pseudo-trajectory is continuously moved to a true trajectory by deforming its noise to zero, we call the maximum distance moved by any single trajectory point the shadowing distance. We then define brittleness 13 [14] of a pseudo-trajectory {pi }L0 as simply the ratio of shadowing distance over the magnitude of the one-step error or B =
shadowing distance one-step error
IPi
= max ~ i
-- Xi l
. t~
(6)
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A necessary condition for continuous shadowability is that the brittleness times the error magnitude (IE0l) of the pseudo-trajectory is smaller than the extent of the attractor in phase space. The brittleness of a pseudo-trajectory is thus a measure of its ability to be shadowed. If a pseudo-trajectory is created with noise level 10 -1~ for a chaotic attractor of unit size, and if the brittleness of the pseudo-trajectory is greater than 10 l~ then one cannot expect a true trajectory closely shadowing the pseudo-trajectory. For hyperbolic systems, pseudotrajectories of infinite length have finite (although possibly very large) brittleness [28]. For nonhyperbolic systems, one typically finds the brittleness to be an increasing function of the trajectory length. As the length of a trajectory of a nonhyperbolic chaotic process increases, the brittleness grows as the trajectory visits more nonhyperbolic regions of the dynamics. In regions near homoclinic tangencies, a small amount of noise may end up perturbing the trajectory points from the stable manifold onto the unstable manifold or vice versa. As a result, the perturbation (A P) of the true trajectory X as a response to the noise (A~) will be large. This, in turn, implies that that the magnitude of [DF] - l , and hence brittleness, will be large from Equation (5). The expected length between glitches is therefore related to the amount of hyperbolicity possessed by the system.
3.2. Test brittleness In practice, the true brittleness/3 cannot be computed precisely since it is a property of true trajectories, and our knowledge of the location of true trajectories is limited by numerical noise. However, we usually do not need a very precise value of brittleness since an order of magnitude estimate of brittleness will give us a very good indication of the shadowability of a system~ Thus, we can do a first order approximation of the brittleness, called test brittleness or/3, by performing one full Newton-step which converts the pseudo-trajectory {pi}N 0 into a less noisy trajectory {yi}N 0. The test brittleness is then defined [14] as /3 - max/[pi - Yi ]/6. Note that, unlike the true brittleness, test brittleness is a property of pseudo-trajectories, which is good since these are quantities generated in a computer simulation. The process of reducing the noise in a pseudo-trajectory using finite steps Aot in Equation (3), instead of reducing continuously, is called the refinement method [ 14,18,21 ]. In the refinement method, we use the Jacobians evaluated at pi to help us reduce the noise. Assuming the error at step i of the computation is 6i (that is, 6i = pi - f ( p i - l ) ) , N we need to perturb the pseudo-trajectory {pi}N 0 towards a less noisy trajectory {Yi }i=0 that satisfies the dynamics. If we called the perturbation (or correction) ci at each step i, t h e n Yi = Pi -+- ci. The requirement of a deterministic trajectory implies f (Yi ) = Yi+l or f (Pi + ci) = Pi+l + ci+l for i = 0 . . . . . N - 1. Using first order approximation, we have Pi+l + Ci+l = f (Pi -Jr-ci) ~-, f (Pi) -Jr-D f (pi)ci, a n d since f (Pi) = Pi+l -k- 6i+1, so
Ci+l ~ O f ( p i ) c i
+ 6i+1
for i = 0 . . . . . N - 1.
(7)
A straightforward iteration of the above equation is not feasible since any errors in ci will generally be amplified under iteration if f is chaotic. Instead, Equation (7) is split into two
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parts, one in the stable (contracting) subspace, and the other in the unstable (expanding) subspace. It will be a computationally stable iteration scheme if the stable component is iterated forwards and the unstable component is iterated backwards. In addition, we need to impose certain "boundary" conditions to guarantee uniqueness. In refinement, we are seeking an (N + 1)D-dimensional true trajectory X that satisfies a system of equations, F ( X ) -- 0, of N D dimension. In solving F ( X ) = 0, we arrived at Equation (7) which is N D-dimensional, but we do not have a unique solution until we specify D more conditions, which corresponds to an "initial" condition on the ci's. Since we split up the iteration scheme into two directions, we are imposing "initial" conditions on the ends of the finite-length trajectory or, more appropriately, "boundary" conditions. This can be done by choosing cu to be on the S-dimensional stable subspace and co to be on the U-dimensional unstable subspace where S + U -- D. Consequently, we are setting S components of co and U components of Cx to be zero, and since S + U = D, we have D more conditions on Equation (7). With this particular choice, we see that the true trajectory point x0, if it exists, would emanate from the unstable subspace of P0, and similarly, x u would emanate from the stable subspace of PN. This is a natural choice because, for uniqueness, we want the true trajectory and the pseudo-trajectory to agree outside the finite-length noisy trajectory. Thus, assuming noiseless dynamics before i - - 0 and after i = N , xi and pi would approach each other exponentially since the unstable (stable) subspace of po (PN) contracts for the inverse (forward) map. So if we write ci as the sum ci = si + ui of components in the stable and unstable directions, the above discussion implies we can set so = u U 0 and recursively solve -
-
-
Si +1 -- Sp ( O f (Pi)Si -k- Si)
(8a)
Ui -- l g p ( D f -1 (Pi+l)[Ui+l - 6i]),
(8b)
and
for Si and Ui, i = 0 . . . . . N - 1, where ,gp and lgp denote projections onto the stable and unstable subspaces, respectively. To project the vectors onto the stable and unstable subspaces, we need to determine the span of the stable and unstable subspaces. Specifically, we need to determine two sets of orthonormal vectors, {l)ij }j and {llOij }j, that span the stable and unstable subspaces at each step i, respectively. We begin by choosing an arbitrary set of U orthonormal vectors {wll . . . . . wlu} and an arbitrary set of S orthonormal vectors { v u l , . . . , VNS}. The basis for the stable and unstable subspaces at each step i can then be found by following the linearized map, that is, Df(pi-1)Wi-l,j Wij -- [ D f ( P i - l ) W i - l , j ]
for j -- 1 . . . . . U
(9a)
and Of-l(pi+l)Vi+l,j vij = ] O f _ l ( p i + l ) V i + l , j ]
for j = 1 . . . . . S.
(9b)
Shadowability of chaotic dynamical systems
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To get an orthonormal basis, we perform G r a m - S c h m i d t orthogonalization procedure on the set {~il . . . . , ~iu} to get {//-)i 1 . . . . . //)IU} (it might be necessary to use a more stable form of orthogonalization procedure [ 17] rather than G r a m - S c h m i d t procedure). Keep in mind that Equation (9b) is iterated backwards to find the set of v e c t o r s { v i j }j spanning the stable subspace since the stable subspace is expanding when it is iterated backwards. It should be noted that since one arbitrary set of orthonormal vectors was chosen at i -- 1 to span the unstable subspace and another set was chosen at i ---- N to span the stable subspace, we expect minimal refinement near the ends of the trajectory. The refinement method does a very good job of reducing the noise in the middle portion of the trajectory. The reason is that the computed span of the stable subspace approaches the true span of the stable space after only a few iterates due to the expanding nature of the unstable subspace under the map. The same is true for the stable subspace when it is iterated backwards. In any event, the problem of minimal refinement near the ends of the trajectory is not a major concern if one works with a sufficiently long pseudo-trajectory. After completing one full set of iterations using Equation (8) (one Newton-step), we can now compute the test brittleness, which is the ratio of the maximum magnitude of correction c i to the magnitude of one-step e r r o r ~i or .~
Icil
/3 -- max - - . i I~il
(10)
If extra precision is available (beyond that which {pi } was calculated), we can compute f ( p i - l ) in extra precision, and thus find ~i in extra precision (since f ( p i _ l) = Pi + ~i) and solve for the corrections ci. If not, we estimate the size of max/ I~il and choose ~i to have randomly varying directions. As we saw in Section 3.1, brittleness is dependent on the error directions, thus we need to amend the definition of test brittleness so that it takes on the maximum value over all possible error directions if extra precision is not available. Since this is not feasible in practice, one only needs to repeat the computations for sufficiently many different sets of error directions to get a good estimate of the test brittleness of the pseudo-trajectory.
4. Numerical evaluations of brittleness 4.1. Brittleness of the forced damped pendulum The first system we performed numerical experiments on is the forced damped pendulum ~i + v~ + sin y - F cos t. We discretize the system by taking the time-T map, where T -- 27r/n and n is the number of steps per cycle of forcing. In the computer simulations, we used the parameters v = 0.2 and F -- 2.4. It is known that a forced damped pendulum satisfying the above equation is chaotic [20].
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To numerically estimate the value of brittleness, we created a pseudo-trajectory of the forced damped pendulum with a constant one-step error of magnitude 3. The direction of the error at each point of the trajectory was assigned randomly. Using the same error directions, a set of six different pseudo-trajectories were produced with 6 = 10 -10, 10 -9 . . . . . 10 -5. The ODE solver took 30 steps per 2zr-cycle of the forcing, and we integrated the trajectory for eight cycles for a total of 240 steps. We then used extra precision (machine precision ~ 10-14) to deform each pseudo-trajectory into a nearby true trajectory up to the extra precision. The distance between the original pseudo-trajectory and the true trajectory, as a function of the 240 trajectory points (eight cycles of forcing), is graphed in Figure 5. The lowest curve, for instance, corresponds to error magnitude 6 -- 10 - l ~ From the figure, we see that the ratio of the maximum of the curve (e.g., 8.5 • 10 - l ~ to the error magnitude ~ (e.g., 10 - l ~ is essentially a constant with a value of 85. This constant is the brittleness of the pseudo-trajectory. The proportionality holds over a large range of noise levels, as long as the pseudotrajectory itself is not significantly changed. To make Figure 5, a relatively short trajectory (0 < t < 167r, covered by 240 discrete time steps of the ODE solver) was used, and the
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Shadowability of chaotic dynamical systems
329
added noise was small. Larger amounts of noise would change the pseudo-trajectory (because of the exponential divergence of chaotic trajectories), so that the comparison across noise levels would no longer be valid. This effect can be seen to a small degree in Figure 5 for large error values, beginning around step 200. From the discussion in Section 3.1, it was conjectured that for small one-step errors, the brittleness should be independent of the error magnitude. This independence is nicely demonstrated in Figure 5 where each curve has the same general shape, and one curve is essentially a scaled version of another. It was also found that the brittleness should depend on the error directions. To test this dependence, twenty different trials were performed with varying randomly-chosen one-step error directions. This resulted in brittleness estimates of between 15 and 135. Therefore, as discussed in Section 3.2, the brittleness should be defined to be the maximum of this magnification factor found over all possible error directions. In practice, we do not need a very accurate value of brittleness since an order of magnitude estimate of brittleness will give us a very good indication of the shadowability of a system. Along this line, we need to check that the test brittleness algorithm gives a reasonable estimate of the brittleness of a pseudo-trajectory. To do this, we generated a pseudo-trajectory in single precision and deformed it into a nearby true trajectory using double precision. The brittleness generated using this method is then compared to the value of test brittleness created using one full Newton step. We found that the numerical values of the brittleness and test brittleness are close to each other, differing typically by 10% when the trajectory is shadowable.
4.2. Test brittleness of the double rotor system We now examine a dynamical system for which even short pseudo-trajectories have extremely high test brittleness, and hence high brittleness. We believe that most nonlinear high-dimensional systems will be susceptible to this obstruction to the existence of nearby long true trajectories. The double rotor map [ 19] is a four-dimensional map which describes the time evolution of a mechanical system consisting of two connected massless rods lying on a horizontal plane (there is no gravity) as shown in Figure 6. The first rod, of length l l, rotates around a fixed pivot P1; the second rod, of length 212, pivots around P2, the opposite end of the first rod. The angles 01 (t), 02(t) specify the position at time t of the first and second rod, respectively. There is a point mass m l at the free end of the first rod (P2), and two equal point masses m2/2 at the ends of the second rod. Friction at the pivots is accounted for by vl, v2 which are the coefficients of friction at P1 and P2, respectively. A delta-function vertical impulse f (t), of magnitude p and always from the same direction, is applied to one end of the second rod (P3 in Figure 6) at times t -- T, 2T . . . . . The map describing the evolution of the system's four phase variables (the two angular positions and momenta) is given by: 0(,;+l) _ 01,,) + Mll 0<,,) A(") mod(27r) , 1 I + M12v2 2
-+- M21 I -Jr-M22 _ mod(27v), (,;+1) •(,l) fi(n) /~(,;+1) _ pll sin01 + L i l y I +L12v2 , ~l -Ii
(lla) (llb) (llc)
330
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2
,
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(n+l)
p12 sinA(n+l ) 0~n) . A(n) 12 v2 -+- L21 + ~22t;'2 ,
_
2
(lid)
where Ii denote the moments of inertia about the pivots. For simplicity, they are chosen to be equal, that is, II -- (m 1 -+-m2)l 2 -- 12 -- m212.
Mij and L i j a r e elements of constant 2 • 2 matrices L, M which depend on the lengths of the rods, the point masses, the period of the kicks, and the friction coefficients at the pivots. Explicitly, L and M are defined by 2
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In the numerical simulations, we chose T = ml = m 2 = Vl = 1;2 = 12 - - 1 and Ii = 1/~/~. Thus, the only parameter we shall vary is the forcing magnitude p. Interesting dynamical behavior is exhibited by this system for various values of the parameter p, and we see one aspect of this interesting behavior by applying the test brittleness algorithm to the double rotor system at two different parameter values. From Figure 7, we note the striking difference between the shadowable case (p = 9) and the unshadowable case (p = 8). In the vertical axis of Figure 7(a), we graph the "pointwise" test brittleness of
Shadowability of chaotic dynamical systems p I
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Fig. 7. First-order approximation of the shadowing distance per unit one-step error, or brittleness, as a function of trajectory point. The test brittleness is the vertical range of the graph. Results are shown for a 10000 point trajectory of the double rotor with parameters: (a) p = 9; (b) p = 8.
a length 10000 pseudo-trajectory for p -- 9, created with one-step error magnitude 10 -l~ Recall from Section 3.2 that the test brittleness for this pseudo-trajectory should be the maximum correction over all 10000 trajectory points after applying one full Newton step. For clarity, we have elected to plot the correction at each trajectory point, or "pointwise" test brittleness, to show that the corrections for p - - 9 are relatively small and roughly of the same magnitude. This result is in agreement with the behavior of a continuouslyshadowable pseudo-trajectory. The vertical extent of the graph is about 107, which is the test brittleness for this pseudo-trajectory. We should then expect the shadowing distance to be about 10 -3. Figure 7(b) shows the same information for a typical pseudo-trajectory of the double rotor with p -- 8. The test brittleness in this case is seen to be greater than 104~ and we note how the "pointwise" test brittleness fluctuates wildly anywhere from 10 ~ to 1043. This leads us to the prediction that at a minimum, 40 decimal digits of accuracy will be needed per iteration step in order to shadow a typical length 10000 trajectory of the double rotor with p = 8. As we increase the length of a pseudo-trajectory, we expect the brittleness to increase because the chances of encountering "glitches" and regions of nonhyperbolicity becomes higher. For an orbit of length 105, our estimate of the test brittleness is 10 ~~176 for p - 8. The explanation of the difference in shadowability for the cases p - 8 and p - 9 lies in the different "degrees of hyperbolicity" of the two systems. To quantify the difference, define the time-T Lyapunov exponents of a trajectory of a discrete dynamical system f by hi(f, T) = 1/Tlnri(DfT), where D f T denotes the Jacobian derivative of f r , and ri (Df T) denotes the i th largest principal component of D f T. These finite-time Lyapunov exponents quantify the expansion and contraction of phase space along the trajectory over a span of T iterates. For some dynamical systems, the number of time-T Lyapunov ex-
332
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et al.
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Fig. 8. Estimates of the four time-T Lyapunov exponents of the double rotor, for T = 100, 200,300. A dozen simulations were done for each T: (a) In the p = 9 case, there is consistently only one positive finite-time Lyapunov exponent; (b) for p -- 8, trajectory segments alternate between one- and two-dimensional expanding subspaces.
ponents that are positive can change along the trajectory or pseudo-trajectory. When this happens, we find large values of brittleness, and we expect shadowing to fail. A numerical study of the behavior of finite-time Lyapunov exponents for the two parameter values of the double rotor, p -- 8 and p = 9, are shown in Figure 8. For p -- 8, the finite-time Lyapunov exponents show fluctuation between one and two positive exponents. We could say the second exponent is "near zero". For p -- 9, there are consistently two positive exponents. One could describe the former case as "more nonhyperbolic" than the latter case. A close examination of the dynamics of the double rotor reveals an explanation for the fluctuating number of positive finite-time Lyapunov exponents in the p -- 8 case. There are many periodic orbits embedded in the attractor whose local behavior varies in a qualitative way. Some of the periodic points have one expanding direction and three contracting directions, while others have two expanding and two contracting [ 19]. As a trajectory (or pseudo-trajectory) moves densely through the attractor, its number of positive finite-time Lyapunov exponents varies as it moves among the varying type of periodic orbits. The fact that a chaotic attractor can contain fixed points of varying number of expanding directions was noted by Abraham and Smale [1]. Although the double rotor with p - 8 is the first physical example for which this behavior has been demonstrated, we believe it can be commonly found, especially in higher dimensional systems.
5. Shadowing time Although the dynamical reasons and conditions for the loss of validity of chaotic solutions have been identified [2,6,14,18,21], the central and most practical question of all for phys-
Shadowability of chaotic dynamical systems
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ical scientists remained to be answered: If the numerical solutions are valid, "how good" are they and "for how long" are they valid? In this work, we answer these questions by establishing fundamental scaling laws in terms of physically meaningful quantities that are easily obtained in practice when doing computer simulations of physical systems. We answer the "how good" question by obtaining a quantitative rule governing the shadowing distance, the pointwise distance from the shadowing trajectory to the pseudo-trajectory; and we answer the "for how long" question by obtaining a quantitative rule governing the shadowing time, the length of true shadowing trajectories. We find that the expected shadowing distance and time have power law dependencies on the size of the one-step error made in the computer simulation. The exponents of the power laws depend on the mean and variance of the Lyapunov exponent nearest to zero, quantities that are easily computable in practice. The greater the finite-time fluctuation about zero, the smaller the power law exponent, resulting in large shadowing distances and valid trajectories of limited length. We begin with a statistical description of the pointwise shadowing distance. We would like to restate that, for our purposes, a pseudo-trajectory is a discrete list of numbers generated according to a computer-implemented evolution rule, such as a Runge-Kutta approximation to the solution of a differential equation. Typically, at each of a number of discrete steps, there is a small discrepancy between the rule and the governing equation, due to the truncation error of the rule or the rounding properties of the computer. In the case of a chaotic trajectory, sensitive dependence usually causes the pseudo-trajectory to lose all correlation with this particular true trajectory. Shadowing theory shows that, under certain conditions, there is another true trajectory, with a slightly adjusted initial condition, that follows very closely to (shadows) the pseudo-trajectory. The pointwise shadowing distances are the stepwise distances between it and the true shadowing trajectory. The maximum such distance may be considered the "global error" of the original calculation. Typical distributions of pointwise shadowing distances are shown in Figure 9. The two plots show histograms of shadowing distances for simulations of two different dynamical systems ((a) p -- 8.2, (b) p = 8.7). The exponential shape of the histogram of log distances suggests that the distances themselves obey a power law fit. These physical systems are taken from the family of kicked double rotors, which are hyperchaotic systems with two positive Lyapunov exponents for certain parameter settings (see [ 19] for more details). The leftmost histogram in Figure 9(a) corresponds to a pseudo-trajectory created by integrating the double rotor map with kick strength p = 8.2, and artificially adding errors of size ~ -10 -16 at each step (at each kick). The true trajectory that shadows was computed in higher precision using the refinement technique, as discussed in Section 3.2 [ 18], and log distances between each point of the two form the exponential distribution shown. In Figure 9(b), the kick strength p has been increased to 8.7, and the exponent of the exponential distribution is much larger. In Figure 9, the distribution of shadowing distances over several orders of magnitude occurs because the trajectory experiences nonhyperbolicity due to the varying number of stable and unstable dimensions. Since this nonhyperbolic behavior is reflected by a Lyapunov exponent fluctuating about zero, we use a diffusion approximation to explain the quantitative aspects of the distributions in Figure 9 in terms of the finite-time Lyapunov exponents of the system. Our answer [33] to the "how good" question is that the shadowing distances y follow a power law distribution cy -2m/~2, where m and ~r are the mean
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0 le-20 le-16
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Fig. 9. (a) Distribution of pointwise shadowing distance for a trajectory of the kicked double rotor with p -- 8.2. The three histograms, from left to right, correspond to one-step errors of 10-16, 10-14, and 10-12, respectively. (b) Same as (a), but for p = 8.7.
and standard deviation o f the time-1 Lyapunov exponent closest to zero. 1 We hypothesize
the exponential distribution of log shadowing distances of Figure 9 to follow from a biased random walk with drift toward a reflecting barrier. When the pseudo-trajectory lies in hyperbolic regions of the attractor, shadowing theory guarantees the existence of a nearby true trajectory. The true trajectory is found by adjusting the points in a consistent manner along the stable and unstable directions. When a nonhyperbolic region is entered, this consistency of adjustments is interrupted by a normally expanding direction becoming momentarily contracting (or vice versa), causing an excursion away from the reflecting barrier at log 6, as shown in Figure 9. The one-step error 6 serves as a reflecting barrier since new errors are created on each step, so that the correct trajectory can never be expected to lie closer than 6 to the pseudo-trajectory. The time-t Lyapunov exponents of an m-dimensional system trajectory, as discussed in Section 4.2, are the m averages )~i of the logarithm of local expansion rates along the trajectory of length t, so that an infinitesimal sphere of radius dr at the beginning of the trajectory would evolve to an ellipsoid with axes ~'it dr after t time units. Distributions of the four time-100 Lyapunov exponents for the kicked double rotor, gathered over a long trajectory, are graphed in Figure 10. Our diffusion model uses the finite-time Lyapunov exponent closest to zero as the per-step innovation. For the kicked double rotor, we consider only the second largest Lyapunov exponent, since this one reflects the varying number of unstable dimensions along the trajectory. To obtain the exponential distribution of log shadowing distances shown in Figure 9 in terms of the finite-time Lyapunov exponent closest to zero, we consider the transition i The mean m is computed as the mean of the second-largest time-100 Lyapunov exponent, which is an approximation to the (infinite-time) Lyapunov exponent. The standard deviation o- was computed as ten times the standard deviation of the distribution of the second-largest time-lO0 Lyapunov exponent.
Shadowability of chaotic dynamical systems
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~ ,,,~
9 d~
.0
~ ,-~
c13 ~ v,,~
.
-4 -2 0 2 time- 100 Lyapunov exponents
-4 -2 0 2 time- 100 Lyapunov exponents
(a)
(b)
Fig. 10. Distribution of time-100 Lyapunov exponents for the double rotor: (a) p = 8.2; (b) p = 8.7.
probability P for a continuous diffusion process to be given by Kolmogorov's equation 0 P/Ot - (o-2/2)02 P / O x 2 + mO P/Ox, where the innovations have mean - m and variance o-2 [16]. The time-invariant equilibrium distribution is found by setting OP/Ot --O" together with the assumptions 0 -- P(co) = dP/dx(cx~) due to the drift - m < 0, it follows that the equilibrium is given by an exponential distribution [33] 2m e_2mx/a2
(12)
P (x ) -- -ff~
This is consistent with the empirical distributions of Figure 9, where the distribution of log distances x = log l0 (Y) have a roughly exponential shape. A quantitative test of the fitness of the diffusion approximation for shadowing distances is shown in Figure 11. The incredibly close agreement between the exponent measured from the shadowing distance distributions and 2m/o- e calculated from finite-time Lyapunov exponents supports the diffusion model explanation of shadowing distance. The fact that shadowing distances obey a power law distribution with exponent -2m/o-e as a function of one-step error magnitude allows us to infer shadowing time. Shadowing trajectories exist as long as the shadowing distance is small, compared to the size of the attractor. Breakdowns in shadowing (called "glitches" in [14,18,21])occur when the reverse happens, namely an excursion far from zero under the diffusion approximation. Therefore times between glitches are analogous to first passage times of the shadowing distance to approach the order of the attractor length in phase space. The first passage time can be computed from the parameters of the diffusion process. A standard Laplace transform calculation yields the expected time [33] o-2 --
-
1)
ln3 m
(13)
C. Grebogi et al.
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Parameter rho Fig. 11. Comparison of three exponents. CROSSES represent the exponent from the power law fit of pointwise shadowing distances from Figure 9. DIAMONDS represent 2m/a 2, calculated from finite-time Lyapunov exponents shown in Figure 10. BOXES represent the exponent from the power law fit of shadowing times, which are the slopes of the line segments in Figure 12. There is no fitting constant in this plot.
for the shadowing distance to reach 1. Our answer to the "for how long" question is that for small 6, the expected shadowing time r is governed by the power law ("g) ~
6 -2m/Cr2.
(14)
To compare actual shadowing times of the kicked double rotor with the power law (14), we have made lengthy calculations summarized in Figure 12. Shadowing trajectories were calculated for between 500 and 10 000 pseudo-trajectories 2, whose mean shadowing time is plotted as a function of one-step error magnitude. For each fixed kick-strength parameter p, log shadowing time shows straight-line behavior as a function of log 6, supporting the power-law conjecture (14). The slopes of the least squares fits from shadowing times in Figure 12 are plotted as small boxes in Figure 11 for each of the parameters p. According to our heuristic argument above, the slopes should be the power law exponents of (14), and in particular should match -2m/cr 2 measured from the finite-time Lyapunov exponent distributions. Indeed it does for the middle of the parameter range. For p = 8.7 the fit is not as good; a possible explanation is that using the Lyapunov statistics from only one Lyapunov exponent loses validity when the mean m moves away from zero, as it does for larger p. For 2 The number of pseudo-trajectories used to create a data point in Figure 12 depended on the shadowing times. A data point in the upper-right comer of Figure 12 represents an average of around 500 runs; data points at the bottom of the figure represent an average of about 10000.
Shadowability of chaotic dynamical systems 1e+07
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digits of accuracy Fig. 12. Squares represent mean number of steps for which a pseudo-trajectory of the kicked double rotor with specified one-step accuracy can be shadowed by true trajectory. Straight line on this log-log plot supports a power-law model for shadowing time. The lines drawn are least squares fits, whose slopes are plotted as squares in Figure 11. The top line corresponds to p = 8.7. Other lines connect data points corresponding to smaller p with decrement of 0.1; lowest line corresponds to p = 8.1.
8.1 ~< p ~< 8.3 the exponent from - 2 m / a 2 is an overestimate because the terms neglected in moving from (13) to (14) have more effect when m approaches 0. Our derivations for shadowing distance and shadowing time are first-order approximations that depend on the existence of one finite-time Lyapunov exponent that is significantly closer to zero than the others. The fundamental conclusion to be drawn from Figure 12 is that to obtain a long trajectory which is even approximately correct is for some systems virtually impossible. Dynamical systems like the kicked double rotor that have a finite-time Lyapunov exponent lying close to zero, relative to the variance of its distribution, possess obstructions to the existence (not to mention explicit computation) of true shadowing trajectories close to long pseudo-trajectories. Figure 12 shows that the limit for double-precision (10 -15) shadowable pseudo-trajectories is a few thousand; nor does the situation improve very much for higher precision. The slope of the lowest line, corresponding to p = 8.1, is almost flat (slope ~ 0.006), which is to be expected from the power law (14) when m/or 2 ~ 0. When the scaling exponent m / ~ 2 is close to zero, and increasing the one-step accuracy of the computation results in virtually no improvement in the lengths of shadowable trajectories, it is far from obvious how a long computer simulation should be interpreted.
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The fact that for some systems, long computer-generated pseudo-trajectories are not matched by true trajectories was first pointed out in [14]. In the present article we have shown explicitly how this phenomenon is caused by a Lyapunov exponent fluctuating about zero, and described quantitatively how shadowing breaks down depending on the proximity of the exponent to zero. Although we have demonstrated fluctuating Lyapunov exponents only for a mechanical system (kicked double rotor), we expect it, and the accompanying global sensitivity of trajectories, to be a common feature of higher-dimensional chaotic dynamical systems.
6. Spatially unstable systems In the previous sections we discussed how the shadowing lemma can fail due to the nonhyperbolicity of the system. Additional difficulties may arise in shadowing numerical trajectories of spatially extended dynamical systems. In particular, we focus on systems with a mean flow for which small disturbances are amplified and spread in extent as they develop into large nonlinear waves downstream. This type of behavior was studied by Reynolds in the transition to turbulence in a fluid flowing down a pipe [31 ]; small patches of turbulent fluid (coined slugs) appear intermittently in the laminar phase and grow in size and volatility as they are convected down the pipe. In open flow systems, noise is often of paramount importance in the development of such nonlinear waves, through the presence of a convective instability. There, a single local disturbance is amplified as it is convected through the spatial extent until eventually it flows out of the system; no permanent structure is produced. However, a persistent noise can have a devastating effect on the eventual dynamics through the appearance of sustained complex wave patterns downstream [ 15]. These noisesustained structures have been characterized by effectively one-dimensional spatial models in a variety of experiments, such as viscous films flowing down an incline [27], TaylorCouette flow with an imposed axial flow [5] and channel flow in the presence of a periodic array of stationary obstacles [35]. Convective instabilities can also arise in discrete spatial arrays of nonlinear oscillators. A prototypical system in which this type of instability gives rise to intermittent spatiotemporal dynamics can be constructed by forming a one-dimensional array of coupled elements, each of which is described by an identical, low dimensional chaotic model [24]. The elements interact with their nearest neighbors asymmetrically, modeling diffusive interactions in the presence of a preferred direction of propagation, or a mean flow. For convenience, consider discrete time systems, which are typically modeled as coupled map lattices [36]. In a linear array of one-dimensional maps, the scalar state uj(n -k- 1) (site j , time n + 1) is determined according to the evolution equations
uj(n -k- 1 ) - - ? ' o f ( u j ( n ) ) + ?'l f ( u j - l (n)) + ?'2f (uj+l (n))
(15)
for j = 2 . . . . . N - 1, where V0 = 1 - Y'l - ?'2, and the nearest neighbor coupling constants are non-negative and satisfy the condition ?'I + ?'2 < 1. The nonlinear mapping is taken to be the quadratic map f ( u ) = 1 - au 2 with the parameter 0 < a ~< 2 chosen within the chaotic regime where the Lyapunov number )~ satisfies I)~l > 1. The shadowability of
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trajectories of the quadratic map was analyzed in Ref. [21]. They found that numerical trajectories with a noise level of 8 can be e-shadowed within e = v/~ for time stretches of the order of 6-1/2 iterations. Below we will show how coupling these maps on a spatial array leads to more severe limitations on the shadowability of noisy orbits. In fact the numerical trajectories will be shadowed only within 0(61/x) where N is the number of elements and 3 is the noise level. The type of boundary conditions for the array (periodic or open) strongly influence the eventual motion that will be observed [3,15]. In order to mimic the effect of an open flow, the boundary conditions are: +
uN(n +
1)
=
-
=
(1 -
•
y,)f(uN(n))+ y,f(uN-, (n)).
(16)
Spatially uniform states make up a class of solutions of the coupled system which satisfies the open boundary conditions. For such states the elements are synchronized and evolve coherently with u j (n) ~- u (n) for all j , where u (n) is a solution of the uncoupled dynamics of an individual element. Are there any non-uniform solutions? For simplicity, consider first the case of unidirectional coupling (V2 = 0). The evolution of the nearest neighbor deviations A j ( n ) - - l U j + l (n) - uj(n)l satisfy the inequalities A l ( n 4- 1) ~< 2a(1 -- y1)Al(n),
Aj(n 4- 1) ~< 2a(1 - yl)Aj(n) 4- 2ayiAj_l(n)
(17)
for j = 2 . . . . . N - 1 where the bound on the derivative If'(u)l ~ 2a was used. The goal is to show that eventually the deviations A j (n) must decay to zero at large n. Assuming the downstream coupling is sufficiently strong so that p = 2a(1 - gl) < 1,
(18)
the first of the above inequalities implies Al (n) ~< p" A1 (0).
(19)
Using this exponential bound on the decay of A l (n), the maximal deviation of the next two sites can be obtained recursively as A2(n) ~< pAe(n - 1) + 2aylp ''-I AI(0) ~< p2A2(n - 2) + 4 a y l p " - ' A l ( O )
<~ A2 (0) p" + 2a Y1np"- l A 1(0).
(20)
By applying a recursion of the above form to the inequalities in Equation (17), bounds for the deviations between neighboring sites further downstream can be obtained. The third
C. Grebogi et al.
340 deviation is bounded by
A3(n ) ~< pnA3(O ) + 2ay1A2(O)(n -- 1)p n-1 + ~1 (2ayl) 2A 1(O)n(n - 1) p n - 2 9
(21)
Continuing this recursive procedure it is clear that for large n, the leading order term in the bound of A m ( n ) is of O ( n m-1 p n - m + l ) . Thus, the deviations between the values at adjacent sites decay to zero asymptotically. Indeed the uniform states are the only types of solution for an array with unidirectional coupling, provided that condition (18) holds. It turns out that for typical initial conditions, it can be shown that this condition can be relaxed somewhat to the condition of Equation (23), below. First, one can rederive the inequalities of Equations (19)-(21) asymptotically for large n, by replacing the maximum derivative 2a by the average expansion rate 2, in Equation (18). Introduction of weak backward coupling Y2 makes the derivation more complicated since backward propagating fronts are present in the dynamics. These fronts have been shown to decay upstream [4] and numerical simulations suggest that synchronized states are the only true solution of the chain, provided Equation (23) holds. In order to evaluate the difficulty in shadowing numerical trajectories, we show how the true synchronized solution is destroyed in the presence of noise. First, the stability of the true solution is obtained by solving Equation (15) to linear order in the deviations •Uj (n) -- u j ( n ) - u(n) about the synchronized chaotic state u(n) [3]. The linearized equations are
6ul(n + l) = f'(x(n))[(1 -- y2)3ul(n) + y26u2(n)], (~UN(, -~ 1)
-
-
f'(x(n))[(1
-- ~I)~HN(?/) -~- ~l(~ttN-l(n)],
6uj(n + l) -- f'(x(n))[yo6uj(n) + yl6uj-l(n) + V26uj+l(n)]
(22)
for j = 2 . . . . . N - 1. One unstable eigenvalue of this system is the Lyapunov number )~ of the single site dynamics f (u), corresponding to the sensitivity of the chaotic state to uniform perturbations. By uniform perturbations we mean that each site experiences disturbances of the same magnitude. The other N - 1 eigenvalues are given by )~q ~- ~.[Y0 + 2 ~ ] - ~ c o s 69q] for q -- 1 . . . . . N - 1, where (~)q -- 7rq/N [3,4]. Thus, the uniform state is linearly stable towards non-uniform perturbations provided 12.[[V0 + 2 ~ ~ - j ]
< 1.
(23)
The stability of the uniform state towards non-uniform perturbations implies that a local disturbance applied to the dynamics at a particular instant will eventually decay. This tells us nothing about the size of the transients. In the case where the uniform state is convectively unstable, yet linearly stable, a single local disturbance may grow exponentially in a moving reference frame. Eventually, the disturbance flows out of the system through the downstream boundary, restoring the uniform chaotic state.
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341
When is the synchronized state convectively unstable? Suppose the synchronized state is disturbed at site i by an amount 6ui (0). The deviation of a site located n13 downstream from the synchronized state n times steps later evolves according to 6Ui+vn(n) ~ 6ui(O)e hA(v)
(24)
for large n. The exponent A (v), or comoving Lyapunov exponent, quantifies the growth of disturbances in a reference frame moving downstream with constant speed v. If A (v) > 0 for a range of speeds 13, then the synchronized state is convectively unstable. For unidirectionally coupled arrays (Y2 = 0) the comoving Lyapunov exponent is given explicitly as (see Ref. [23]) lmv
13
A (v) = log I&l - (1 - v) log ~
1 -
- v log - - . ?'l
(25)
?'J
It turns out that for v -- ~'1, the comoving Lyapunov exponent is exactly the Lyapunov exponent of the uncoupled single element dynamics. In fact, there is a range of speeds around Fl for which A (v) > 0. We conclude that a unidirectional array of synchronized chaotic elements is convectively unstable yet linearly stable provided condition (23) holds. In general, even if 72 > 0, the spatially uniform state exhibits a convective instability [3,4] provided that condition (23) holds. Weak noise in the dynamics can destroy the coherent state in a convectively unstable array. Disturbances at the boundary will grow to have significant amplitude for a convectively unstable array of sufficient size. In the presence of noise, only the elements near the "upstream" edge will be essentially synchronized, evolving chaotically, while the remaining elements will display complex incoherent time evolutions. In Figure 13, 16 consecutive time steps of the eventual dynamics of an array of 100 sites are overlayed, starting from random initial conditions. A random variable, uniformly distributed on the interval [ - 1 0 -l~ 10 -1~ was added at each time step and at each site to the true dynamics of Equation (15). The coherence length, or the number of elements that are essentially synchronized near the upstream edge can be seen from Equation (24) to grow with reduced noise as log a, where a is the maximum magnitude of the added noise. Therefore, in order to e-shadow a numerical trajectory of an N element array, the noise level cr decreases exponentially with the array size. In other words, for a given noise level the presence of spatial instabilities drastically limits the size of systems whose numerical trajectories have anything to do with the true ones. In other words, a numerical trajectory of an array at the parameters of Figure 13 can be shadowed for a long time to within 10 -7, only if we limit the size of the system to consist of at most 30 elements (assuming that the only noise present is that due to roundoff of O(10-15). Computationally, one way of obtaining more realistic trajectories for large arrays is through applying a sparse array of local feedback controllers, separated by distances that are smaller than the coherence length [4].
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342
1.0
0.5
"~
0~
-0.5
m
.0
0
I
I
I
I
20
40
60
80
100
Fig. 13. 16 consecutive time steps in the eventual d y n a m i c s of an array with Yl = 0.75, T2 = 0.05, a* -- 1.6, at noise level cr -- 1 0 - 1 0 , are overlayed.
7. Implications for modeling Thus far, we have concentrated on the link between the model and its numerical solutions, but how do these shadowing results impact on the link between the model and nature? In this section, we argue that obstructions to shadowing due to fluctuating Lyapunov exponents impose a severe limitation on modeling. In particular, this means that such a system cannot be modeled faithfully even if one is able to observe and record all data generated from the system, and solutions to any model of the system, even when an exact solution is obtainable, will not reflect the solutions of nature. We begin by examining what one considers a good deterministic model of nature. A necessary requirement for such a model is robustness under small perturbations. Any model has parameter values that are not known with exact precision. One can easily generate two different versions of the model using slightly different parameter values. For chaotic systems, it is well known that the outcome of the system is sensitively dependent on the initial conditions, that is, a slight difference in the initial conditions can result in vastly different outcomes. In view of this, we consider a model robust if the set of all possible outcomes of the two versions of the model are very similar. To illustrate what we mean, consider the simple case where two very closely related models are used to emulate a physical system in nature. Denote these models as model A and model B, and if the differences between the two are small, we can regard one as a slightly different version of the other. Some possible differences between models A and B could be: (1) a small change in one of the parameter values, (2) a slightly different external influence on each, or (3) a different noise level in the models.
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343
For "well-behaved" physical systems, we expect the set of all possible outcomes from model A to agree closely with the set of all possible outcomes from model B. Specifically, we are interested in outcomes that evolve in time or trajectories. Then, robustness implies that for every trajectory of A, there exists at least one trajectory of B that stays uniformly close to, or shadows, the particular trajectory of A and vice versa. Difficulties appear when trajectories from one model fail to be shadowable by trajectories from the other. The problem becomes critical when no trajectory of A follows closely any trajectory of B (or vice versa) for all but short periods of time. Since trajectories from closely related models do not agree, then either model is useless in representing the physical system. If this problem persists for various configurations of A and B (that is, for a wide parameter range of both A and B), then this poses a major obstacle to modeling the particular physical system. In our study of the double rotor map, we have exactly this problem of the model not being robust in a certain parameter range, namely around p = 8. It was found that the shadowing time for trajectories from slightly different models is very short. Thus, this kind of behavior makes it difficult to rely on any model trajectories for acceptable periods of time, and deterministic modeling has encountered a major stumbling block. The specific cause of this particular problem stems from the fluctuating Lyapunov exponent. Since the double rotor map is a four-dimensional dissipative map and for the parameter ranges in which the system is chaotic, the window in which there is a fluctuating Lyapunov exponent is thus not large, and we only see the modeling problem around p = 8. However, we believe there is no such constraints in much higher-dimensional systems, and thus we conjecture that this modeling difficulty is more common in higher dimensional chaotic systems [25,26].
8. Conclusions
We have examined the various levels of difficulties that can obstruct the shadowability of model trajectories. Along the way, we have defined certain quantities like brittleness and shadowing time that should be of use in diagnosing whether certain dynamical systems are shadowable. These ideas and concepts were applied to the double rotor map which leads to the recent discovery that fluctuating Lyapunov exponent poses a major obstacle to deterministic modeling. Furthermore, we believe this difficulty to be more common in higher dimensional chaotic systems as the possibility of having one or more finite-time Lyapunov exponents fluctuate about zero increases with the dimensions of the system. This raises the interesting question: How common is this problem? It is unclear at this point how many physical systems suffer from this problem, but we believe one should at least check whether any of the Lyapunov exponents fluctuate about zero when modeling is undertaken for any chaotic system. Given the vast array of models ranging from climate modeling to turbulent fluid flow which are both high-dimensional and chaotic, it is imperative that the limits to deterministic modeling be examined carefully if scientists are to obtain faithful portrayals of nature. We should mention that, in addition to the work in Ref. [9], Chow and collaborators also published in [7,8,10-13].
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Acknowledgements This research was supported by grants from FAPESP and CNPq, both Brazilian agencies, and from ONR and NSE
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [ 16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
R. Abraham and S. Smale, Proc. Sympos. Pure Math., Vol. 14, Amer. Math. Soc., Providence, RI (1970), 5. D.V. Anosov, Proc. Steklov Inst. Math. 90 (1967), 1. I. Aranson, D. Golomb and H. Sompolinsky, Phys. Rev. Lett. 68 (1992), 3495. D. Auerbach, Phys. Rev. Lett. 72 (1994), 1184. K.L. Babcock, G. Ahlers and D.S. Cannell, Phys. Rev. Lett. 67 (1991), 3388; A. Tsameret and V. Steinberg, Phys. Rev. Lett. 67 (1991), 3392. R. Bowen, J. Differential Equations 18 (1975), 333. S.N. Chow, X.B. Lin and K.J. Palmer, Differential Equations: Proc. EQUADIFF Conference, C.M. Daftermos, G. Ladas and G. Papanicolaou, eds, Marcel Dekker, New York (1989), p. 127. S.N. Chow and K.J. Palmer, Dynamics Differential Equations 3 (1991), 361. S.N. Chow and K. Palmer, J. Dynamics Differential Equations 3 (1991), 361; J. Complexity 8 (1992), 398. S.N. Chow and K.J. Palmer, J. Complexity 8 (1992), 398. S.N. Chow and E.S. Van Vleck, Random Comput. Dynamics 1 (1992), 197. S.N. Chow and E.S. Van Vleck, SIAM J. Sci. Comput. 15 (1994), 959. S.N. Chow and E.S. Van Vleck, Contemp. Math. 172 (1994), 97. S. Dawson, C. Grebogi, T. Sauer and J. Yorke, Phys. Rev. Lett., 73 (1994), 1927. R.J. Deissler, J. Statist. Phys. 40 (1985), 371; R.J. Deissler, Phys. D 25 (1987), 233. W. Feller, An Introduction to Probability Theory and its Applications, Wiley, New York (1957). G. Golub and C. Van Loan, Matrix Computations, Johns Hopkins Univ. Press, Baltimore (1989); Y.C. Lai, C. Grebogi and J.A. Yorke, Nonlinearity 6 (1993), 779. C. Grebogi, S. Hammel, J. Yorke and T. Sauer, Phys. Rev. Lett. 65 (1990), 1527. C. Grebogi, E. Kostelich, E. Ott and J.A. Yorke, Phys. D 25 (1987), 347; R. Romeiras, C. Grebogi, E. Ott and W.P. Dayawansa, Phys. D 58 (1992), 165. E.G. Gwinn and R.M. Westervelt, Phys. Rev. Lett. 54 (1985), 1613; K. Hockett and P.J. Holmes, Ergodic Theory Dynamical Systems 6 (1986), 205. S. Hammel, J.A. Yorke and C. Grebogi, J. Complexity 3 (1987), 136; Bull. Amer. Math. Soc. 19 (1988), 465. P. Hartman, Ordinary Differential Equations, Wiley, New York (1964). K. Kaneko, Phys. D 23 (1986), 437; R.J. Deissler and K. Kaneko, Phys. Lett. A 119 (1987), 397; K. Kaneko, Phys. Lett. 111 (1985), 321. Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, Berlin (1984). Y.C. Lai and C. Grebogi, Phys. Rev. Lett. (1999). Y.C. Lai, C. Grebogi and J. Kurths, Phys. Rev. E (April 1) (1999). J. Liu and J.P. Gollub, Phys. Rev. Lett. 70 (1993), 2289. K. Meyer and G.R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer, New York (1992). J.I. Palmore and J.L. McCauley, Phys. Lett. A 122 (1987), 399; J.L. McCauley and J.I. Palmore, Phys. Lett. A. 115 (1986), 433. L. Perko, Differential Equations and Dynamical Systems, Springer, New York (1991). O. Reynolds, Philos. Trans. Roy. Soc. 44 (1883), 51. J.M. Sanz-Serna and S. Larsson, Appl. Num. Math. 13 (1993), 181. T. Sauer, C. Grebogi and J.A. Yorke, Phys. Rev. Lett. 79 (1997), 59. T. Sauer and J. Yorke, Nonlinearity 4 (1991), 961. M.E Schatz, R.P. Tagg, H.L. Swinney, P.E Fischer and A.T. Patera, Phys. Rev. Lett. 66 (1991), 1579. I. Waller and R. Kapral, Phys. Rev. A 30 (1984), 2047; K. Kaneko, Prog. Theor. Phys. 72 (1984), 480.
CHAPTER
8
Numerical Analysis of Dynamical Systems* John Guckenheimer Mathematics Department, Malott Hall, Cornell Universi~, Ithaca, NY 14853, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Classical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Limit sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Error estimation and verified computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Computation of invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Invariant toil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Stable and unstable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Chaotic invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Statistical analysis of time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Bifurcation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Continuation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Numerical methods for computing bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
347 348 348 352 355 358 358 365 366 369 372 374 374 379 381 385
*This work was partially supported by grants from the Department of Energy, Air Force Office of Scientific Research and the National Science Foundation. H A N D B O O K OF D Y N A M I C A L S Y S T E M S , VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 345
Numerical analysis of dynamical systems
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1. Introduction
This paper presents a brief overview of algorithms that aid in the analysis of dynamical systems and their bifurcations. The viewpoint is geometric and the goal is to describe algorithms that reliably compute objects of dynamical significance. Reliability has three facets: (1) the probability that the algorithm returns an answer for different choices of starting data, (2) whether the computed object is qualitatively correct, and (3) the accuracy with which the objects are computed. Numerical analysis has traditionally concentrated on the third of these topics, but the first two are perhaps more important in numerical studies that seek to delineate the structure of dynamical systems. This survey concentrates on exposition of fundamental mathematical principles and their application to the numerical analysis of examples. There is a strong interplay between dynamical systems theory and computational analysis of dynamical systems. The theory provides a framework for interpreting numerical observations and foundations for algorithms. Apparent discrepancies between computational output and theoretical expectations point to areas where phenomena have been overlooked in the theory, areas where algorithms produce misleading results, and areas where the relationship between theory and computation is more subtle than anticipated. Several examples of simple systems are used in this article to illustrate seeming differences between computation and theory. Geometric perspectives have been introduced relatively recently to the numerical analysis of ordinary differential equations. The tension between geometric and more traditional analysis of numerical integration algorithms can be caricatured as the interchange between two limits. The object of study are systems of ordinary differential equations and their flows. Numerical solution of initial value problems for system of ordinary differential equations discretize the equations in time and produce sequences of points that approximate solutions over time intervals. Dynamical systems theory concentrates on questions about long time behavior of the solution trajectories, often investigating intricate geometry in structures formed by the trajectories. The two limits of (1) discretizing the equations with finer and finer resolution in time and (2) letting time tend to infinity do not commute. Classical theories of numerical analysis give little information about the limit behavior of numerical trajectories with increasing time. Extending these theories to do so is feasible only by making the analysis specific to classes of systems with restricted geometric properties. The blend of geometry and numerical analysis that is taking place in current research has begun to produce a subject with lots of detail and richness. Interesting examples from diverse applications infuse the subject and establish mathematical connections between other disciplines. Thus, the development of better algorithms and software can have far reaching consequences. This paper takes a pragmatic view of this research. The focus here is on understanding the mathematical properties observed in numerical computation and on assessing the capability of theory, algorithms and software to elucidate the structure of dynamical models in mathematics, science and engineering. Issues that have been investigated from this perspective are presented and a few pointers are provided to the rapidly growing literature.
348
J. Guckenheimer
2. Numerical integration 2.1. Classical theory Systems of ordinary differential equations Jc -- f (x),
(1)
f : IRn --+ R n,
define vector fields. Vector fields on manifolds are also defined by systems of the form (1) in local coordinates [148]. The existence and uniqueness theorem for ODEs [86] states that a Lipschitz continuous vector field (1) has a unique flow q0 : IRn x IR --+ I[~n defined in a neighborhood of ]Rn x 0 with the properties that r 0) - x and 4~(x, t) = f ( ~ ( x , t)). The time t m a p ~t :IR n ~ IR'z is defined by ~bt(x) -- cI)(x,t). The curves cI)(x,t) defined by fixing x and letting t vary are trajectories, denoted by xt. There are seldom explicit formulas for 4} in terms of f . Iterative n u m e r i c a l integration algorithms are used to compute trajectories with discrete time approximations that march along the trajectories. Numerical integration is a mature subject, but still very active - especially with regard to algorithms designed for special classes of equations. The subject is supported by extensive theory and abundant software. Several excellent texts and references are Henrici [91 ], Gear [66], Hairer et al. [86], Hairer and Wanner [87], and Ascher and Petzold [ 10]. Part of the intricacy of the subject lies in the fact that no single integration algorithm is suitable for all problems. Different algorithms reflect trade-offs in ease of use, accuracy and complexity. The basic concepts of numerical integration are explained here only briefly. Explicit Runge-Kutta algorithms are described, followed by a short survey of refinements, alternate approaches and terminology for numerical integration of ODEs. Explicit Runge Kutta methods construct mappings ~Ph from the vector field (1) that depend upon a parameter h, called the time step of the method. Several partial steps are taken from an initial point, and the values are combined to produce the map gth. The primary goal is to produce a family of mappings 7th depending on h, whose Taylor series expansion in the time step h agrees with that of the flow map q~h to a specified degree. Each function evaluation is called a stage of the method. The method is said to have order d if the Taylor series agree to degree d 4- 1. Each stage is performed at a point that depends upon the preceding stage. The scheme for an s-stage method has the following form: kl -
f (x),
k2 = f (x + haz l k l ) ,
ks -- f (x 4- h(aslkl 4- as2k2 4-...-Jr- as,s-lks-l)), r
-- x + h ( b l k i + b2k2 + . . .
+ bsks).
(2)
Formulation of higher order Runge-Kutta methods for system (1) is based upon repeated differentiation of this equation. With each differentiation, substitution of f (x) for ,f on the right hand side yields expressions for the derivatives of trajectories in terms of derivatives
Numerical analysis of dynamical systems
349
of f . For example, the second and third derivatives of x ( t ) are given by Y- Dvf/c- Dxff, X (iii) ~- Dxx f f 2 + (Dx f ) 2 f .
Taylor expansion of ~h in the system of equations (2) gives expressions in the derivatives of f and the coefficients aij and b j of the method. Equating the degree d -+- 1 expansions of ~Ph and x ( h ) obtained from repeated differentiation of system (1) produces a system of polynomial equations for aij and b j . The number of equations obtained in this manner grows faster than d. For d ~< 4, there are order d methods with d stages [86]. Order d methods with d > 4 require more than d stages. As d increases, the complexity of solving the equations for order d methods grows rapidly. The fourth order method k l = h f (xo), k2 = h f (xo + kt /2), k3 = h f (xo + k2/2), k4 = h f (xo + k3 ), 1 xl - x0 + -7(kl -+- 2k2 + 2k3 -+- k4) b
is a "standard" choice [86]. The simplest Runge-Kutta method is the forward Euler method defined by Eh (x) = x + h f (x). This is a single stage explicit Runge-Kutta method, but it receives limited use for two reasons. First, it only has order one. For example, if f (x) = x, then Eh (x) = (1 -+- h)x and ~Ph(x) = exp(h)x = (1 + h + h 2 / 2 ) x + O(h3). If I --+ oo and h --+ 0 so that lh = t, then the iterates E~ (x) converge to ~Pt(x). This expresses the convergence of the method. In the example f ( x ) -- x with x(0) - 1 and t - 1, Eti/l(1) -- (1 + 1/l) l while ~Pl (1) -- e. Taking logarithms, we estimate the difference log(e) - log(Etl /t(1)) -- 1 -- I log( 1 + 1/ l) -- l / 2 + o(l). The order gives the degree of the lowest order term in the difference between the Taylor series expansion of ~Pt(x) and its computed value from 1 steps of steplength h with lh = t. Computing the value of the vector field to single precision (seven decimal digits) of accuracy for moderate times with a first order method can be expected to take millions of time steps. This can be reduced to a few tens of time steps with a fourth order method. Even with fast computers, the performance of the Euler method is awful. The second limitation of the Euler method is its instability for stiff systems. This is exemplified by the example ~ = -)~x with Eh(x) = x - h)~x and ~ t ( x ) = exp(-)~t). All of the solution trajectories tend to zero as t ~ cx~. However, if h)~ > 2, then the trajectories of the numerical method are unbounded, with oscillating signs. This phenomenon persists in multidimensional linear systems with negative eigenvalues of large magnitude. Many examples, in particular those obtained by discretization of partial differential equations, have rapidly decaying modes whose eigenvalues place stringent limits on the time steps with which the Euler method gives trajectories that even qualitatively resemble the trajectories of a vector field.
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Explicit, fixed time-step Runge-Kutta methods are only one group of widely used methods. We give here a list of criteria that are used to distinguish and classify numerical integration algorithms. Explicit vs. implicit. In explicit methods, the next time step is computed by direct evaluation of function(s) of previously computed data. In implicit methods, the next step is computed by solving a system of equations, often using Newton's method. The use of implicit methods is motivated by the difficulties of solving stiff systems. Compare the explicit Euler method with the implicit Euler method, defined by x,7+l = x,, + h f (Xn+l). This differs from the explicit Euler method in that the function evaluation takes place at the still unknown next point along the approximate trajectory. The equation for x,,+l is implicit since x,,+l appears on the right hand side of the equation. For the linear example ,f - -)~x with large )~ > 0, the equation is readily solved, giving the formula x,,+l = x,,/(1 § h)~). When )~ > 0, ]xnl ~ 0 monotonically for any initial condition and any positive step size h. The limitations on step length that were necessary to achieve stability for the explicit Euler method have disappeared, at least for this system. All explicit Runge-Kutta methods applied to a linear equation yield polynomials that are unbounded as the step length increases. Therefore, they all become unstable when applied to the equation ,f = -)~x with large enough )~ > 0. When the desired time span for an integration is long enough compared to the step length required for stability by explicit integrators, the differential equation is called stiff. Development of stiff integrators was a particularly active research area in the 1970s. The guiding criterion that was applied to this work was Dahlquist's concept of Astability [35], namely that the integrator should remain stable for all positive step sizes when applied to a linear system with negative eigenvalues. Explicit Runge-Kutta methods are not A-stable, a fact that provided strong motivation for improvement of implicit methods. One-step vs multi-step. One-step methods only use information from the last computed step while multi-step methods use information from several previously computed steps in determining the next step of the integrator. Multi-step methods have the advantage over one-step methods that higher order accuracy can be achieved with a single function evaluation at each time step. On the other hand, theoretical interpretation of one step methods is easier since they can be regarded as giving approximations to the flow map 4~. A kstep multi-step method can be viewed as a discrete mapping on a product of the phase space with itself k times, but it is difficult to single out the class of mappings on this larger space than correspond to multi-step methods. A k-step method also needs a way to compute the first k steps, for example by using a Runge-Kutta algorithm. Implicit multi-step methods called backwards differentiation formulae are used widely as integrators for stiff systems [ 10,87]. Fixed step vs. variable step. The most common type of adaptation in numerical integrators is the use of prediction-correction to adjust step size. Variable time step algorithms incorporate criteria for assessing the accuracy of each computed time step. With RungeKutta methods, accuracy is commonly assessed by formulating methods of different orders that share intermediate time steps. By comparing the solutions with the principal terms in the asymptotic expansions of the truncation error for each method, an estimate of the error
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for the time step can be made. If the estimate is larger than a predetermined error criterion, the time step is reduced and the step is repeated. Typically, there are also criteria used to determine when time steps can be increased while maintaining the desired error criteria. For the most part, heuristic arguments and tests with sample problems form the basis of adaptive strategies that are used to vary time steps. As with multi-step methods, variable step methods are hard to interpret as discrete approximations to a flow. The use of variable time step methods is an area in which practice is far ahead of theory. There are few theorems describing the qualitative properties of adaptive time step algorithms viewed as dynamical systems. Extrapolation is a technique that can be used to improve the order of accuracy of either explicit or implicit integrators. Many integrators have asymptotic expansions in step size h for the errors made in computing 4~t as h tends to zero. When numerical computations of qSr are performed with different step sizes, the sequence of computed values can be fit to the beginning of the asymptotic expansion for the errors. These can then be extrapolated to the limit h = 0, giving a higher order estimate for 4~t. The extrapolation process is independent of the integrator that is used, so that high orders are achievable using simple low-order integrators. Most implementations of extrapolation methods are based upon integrators for which only even terms appear in the asymptotic expansion of the error. The methods can readily vary their order adaptively by selecting the number of intermediate time-steps and the segment of the asymptotic expansion that is fit. These properties give these methods more flexibility in automatically adapting to ongoing computations than high order RungeKutta methods. There is renewed interest in Taylor series methods for numerical integration at this time, some of it motivated by work on verification discussed below in Section 2.3. Series solutions of trajectories are easy to construct theoretically: substitution of a power series expansion x(t) = ~ ai ti into the system (1) yields a recursive system of equations for ai. Implementation requires that the series expansion of f ( x ( t ) ) is computed, and this is not straightforward. On the one hand, finite difference approximations of the derivatives of f are no more accurate than the Runge-Kutta methods. On the other hand, symbolic differentiation of f produces long complicated formulas that are expensive to evaluate. But there is a third way. Automatic differentiation [73] is a technique for computing derivatives with only round-off errors that makes the computation of highly accurate Taylor series approximations to solutions practical. The series can be computed readily to sufficiently high order that the radius of convergence and truncation errors of the Taylor polynomials can be estimated, enabling choice of time steps based entirely upon information at the initial point of the step. Moreover, the methods produce dense output in that the Taylor polynomials give the value of the trajectory at all intermediate times to uniform order. In Runge Kutta methods, there is no procedure to directly evaluate the trajectory at fractional time steps while maintaining the order of accuracy of the methods. A posteriori tests of the Taylor polynomials can form the basis for adaptive reduction of step size. For example, error criteria can be formulated in terms of the difference between the vector field evaluated along the Taylor polynomial approximation to a trajectory and the tangent vectors to these approximate solutions [32]. While Taylor series methods long ago were demonstrated to work extremely well on a broad range of examples [32], they have not yet been widely adopted. The advent of improved programming languages and environments may change this situation.
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Complementary to the classification of numerical integrators as explicit/implicit, onestep/multi-step and adaptive/fixed-step are questions about whether numerical integrators preserve mathematical structure found in special classes of problems. The most intense effort has been devoted to the development of symplectic integrators [ 134]. A Hamiltonian vector field is one that has the form of Hamiltonian's equations in classical mechanics: D - O H / O q and ~) - - O H / O p with Hamiltonian function H" IR" x R 'Z --+ R. The flow of a Hamiltonian vector field is symplectic, meaning that it preserves the two form ~ d p i / x dqi and energy preserving, meaning that H is a constant of the motion. A symplectic integrator is one for which each time step is given by a symplectic map. The differences between symplectic integrators and other methods become most evident when performing very long time integrations. A common feature of non-symplectic integrators is that the value of H changes slowly along trajectories, but eventually drifts far from its original value [151]. Symplectic integrators do not usually preserve energy either, but the fluctuations in H from its original value remain small. On a deeper level, KAM theory implies that quasiperiodic motions are frequently observed in symplectic flows [ 134]. Symplectic integrators define maps that satisfy assumptions of the KAM theory while nonsymplectic integrators generally do not. The construction of symplectic integrators is still sufficiently new that it is early to tell how prevalent they will become as the methods of choice for investigation of conservative systems.
2.2. Limit sets Classical theories of numerical integration give information about how well different methods approximate trajectories for fixed times as step sizes tend to zero. Dynamical systems theory asks questions about asymptotic, i.e. infinite time, behavior. Only recently has there been emphasis upon understanding whether numerical methods produce good approximations to trajectories over arbitrarily long periods of time [ 143]. We investigate the question as to when the limits of step size tending to zero and time tending to infinity can be interchanged in numerical computations, but there are additional questions that give a different perspective on long time integration. Two phenomena shape our discussion on the limitations of long time integration. The first phenomenon is based upon a slow drift of numerically computed trajectories from those of the underlying vector field. Consider the explicit Euler method applied to the harmonic oscillator k
~
~y~
m
X.
The flow trajectories are circles, but the non-zero trajectories of the numerical method 2 .2 satisfy x,,+l + Yn+l - (1 + hZ)(xn2 nt- y2) n and are all unbounded. The second phenomenon is closely related to structural stability of hyperbolic invariant sets. Hyperbolic invariant sets have the property that trajectories do not remain close. There is a bound 6 > 0 so that no two distinct trajectories in the invariant set remain within distance 6 of each other. Given this fact, it is unreasonable to expect that a numerical computation will remain close to the trajectory of its initial condition for all time. Nonetheless, there is a sense in which
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numerical trajectories can give excellent approximations to trajectories within the invariant set. The concepts of pseudoorbits and shadowing described in this section help explain this apparent paradox. Further discussion of infinite time behavior of flows and approximating numerical methods will be facilitated by the following definitions and concepts from dynamical systems theory: 9 I n v a r i a n t set.
A is an invariant set if 4~t(A) = A for all t. A is forward invariant if 4~t(A) C A for all t > 0. A is backward invariant if ~bt(A) C A for all t < 0. 9 co-limit set o f a trajectory.
y is in the co-limit set of the trajectory x ( t ) if there is a sequence ti -+ e~ so that x ( t i ) y. 9 u - l i m i t set o f a trajectory.
y is in the or-limit set of the trajectory x ( t ) if there is a sequence ti --+ -cx3 so that x ( t i ) --+ y. 9 W a n d e r i n g point.
x is a wandering point if there is a neighborhood U of x and a T > 0 so that t > T implies x (t) r U. 9 N o n - w a n d e r i n g set.
The non-wandering set is the complement of the set of wandering points. 9 (Uniformly) hyperbolicstructure.
A hyperbolic structure of a compact invariant set A is an invariant splitting of tangent spaces T A R " - - E s G E u | E c so that E c is the one dimensional space tangent to the vector field and for t large Ddct expands vectors in E u at an exponential rate while contracting vectors in E s at an exponential rate. The wandering set of a flow is open and the nonwandering set is closed. One of the goals of dynamical systems theory is to decompose the nonwandering set into disjoint closed subsets, called basic sets, which have dense orbits. When this can be done, the entire phase space can be partitioned into the stable sets of the basic sets. The stable set of a basic set is the set of points whose co-limit is in the basic set. Similarly, the unstable set of a basic set is the set of points with or-limit set in the basic set. The geometric characterization of structural stable dynamical systems advanced by the seminal work of Smale [ 140] gives a large class of systems for which these decompositions have a particularly nice form. On a compact manifold M, structurally stable systems have a finite number of basic sets A i , each of which possess a uniformly hyperbolic structure. Chaotic dynamical systems display sensitive dependence to initial conditions: nearby trajectories diverge from one another, typically at exponential rates. In the presence of sensitive dependence to initial conditions, it is hardly reasonable to expect that a numerical method will accurately track the trajectory of its initial condition for long times, since trajectories of nearby initial conditions do not remain close to the chosen one. Any error made in a single step of a numerical integration will be amplified by the inherent divergence of trajectories. This fact underlies fundamental limitations in the accuracy of numerical integration over long times. In hyperbolic invariant sets, it is inevitable that errors in the numerical solution of the initial value problem grow exponentially. This is even true for
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iteration of diffeomorphisms where there is no truncation error of numerical integration, only round-off error in the evaluation of the diffeomorphism. The effects of sensitivity to initial conditions prompt new perspectives on the initial value problem. Over short times, we expect numerical integration to be accurate. What positive results can be established about long time integration? In the case of hyperbolic invariant sets, there is a satisfying theory whose ultimate conclusion is that numerical trajectories approximate actual trajectories of a different initial condition. The concepts of shadowing and pseudoorbit [20] have been used to explore these issues. For a discrete dynamical system defined by the mapping F : ~ n ~ ]Kn, a 6-pseudoorbit is a sequence of points xi with the property that Ixi+l - F(xi)l < 3. On each iterate, there is an error of at most 6 in the location of the next point relative to the location of the mapping applied to the current point of the pseudoorbit. If there is a point y whose trajectory has the property that IFi (y) - x i l < e, then we say that the trajectory of y e-shadows the pseudoorbit. The extension of the shadowing concept and this theorem to a flow q~ requires allowance for time "drift" along trajectories. If (xk, hk) 6 ~n x R is a sequence of points and time increments, it is a 3-pseudoorbit if q~(xk, tk) - xk+l < 3 for each k. The pseudoorbit eshadows the orbit of y if there are times tk with [xk - q~(y, tk)l < e and Ihk - (tk+l -- tk)] < e [30]. For a numerical iteration or integration, we can view the algorithm as producing a pseudoorbit. The one step accuracy of the method determines a ~ for which the numerical trajectory is a pseudoorbit. We can ask for which systems, which one step methods grh, which initial conditions x and which e there is a point y so that 9 (y, nh) e-shadows the numerical trajectory ~p~'(x). The qualitative characteristics of the invariant sets of a flow 9 are a big factor in determining whether they satisfy shadowing properties. Hyperbolic basic sets do. Here is the statement of a result for discrete time systems. THEOREM 1 [20]. Let A be a hyperbolic invariant set o f a C 1 diffeomorphism F. If e > O, there is a ~ > 0 such that every 3-pseudoorbit f o r a trajectory in A e-shadows a trajectory of FinA. Numerical trajectories that start near a hyperbolic attractor A will stay near A and they will be shadowed by trajectories within the attractor. Thus, the shadowing property of hyperbolic sets enables us to recover long time approximation properties of numerical trajectories when they are computed with sufficient accuracy for fixed, moderate times. This theorem is very satisfying mathematically, but we note with caution that there are few examples of hyperbolic attractors that arise from physical examples. The chaotic attractors that have been observed in applications seem not to be hyperbolic and structurally stable. The discrete Henon mapping is the example that has been studied most intensively [89]. The details of its dynamical properties are much more subtle and complex than those of hyperbolic attractors [20], but they appear to be typical of chaotic attractors with a single unstable direction. Additional complexity is present in partially hyperbolic attractors in which the dimension of the unstable manifolds of points vary within attractors [2,127]. Hammel et al. [88] have investigated the shadowing properties of one dimensional mappings and the Henon mapping. They demonstrate that very long sections of trajectories have the shadowing property, but that one cannot expect it to hold for infinite time.
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Coombes et al. [30] implemented methods for shadowing trajectories of vector fields. Convergence of a numerical integration algorithm proves that it provides shadowing of trajectories for fixed, finite times. Nonetheless, the theory of these algorithms says little about the precision of a pseudoorbit required to provide an e-shadow of a trajectory. The ability to shadow trajectories for long periods of time is closely related to their Liapunov exponents (discussed in Section 2.3) and exponential dichotomies. If there are no Liapunov exponents close to zero, then an infinitesimal neighborhood of a trajectory x(t) can be decomposed into unstable directions that diverge from x(t) and stable directions that converge towards x(t). Deviations from x(t) in the stable directions (eventually) become smaller in the forwards direction while deviations from x (t) in the unstable directions become smaller in the backwards direction. Heuristically, to find an orbit shadowed by a pseudo-orbit, take the trajectory that matches the projection of the pseudoorbit onto the stable directions at its beginning and onto the unstable directions at its end. Several authors, including [28-31,88], have formalized this conceptual framework to give explicit estimates for the shadowing constants of a vector field. Consider a set of points (xk, tk), k = 0 . . . . . N, along a trajectory of the vector field f with flow 45. These points satisfy the equations
9 (xk, tk) --xk+l = 0
(3)
for = 0 . . . . . N - 1. We can view the left hand side of these equations as a map F:IR (n+l)(N+l) -+ R N. A one-step numerical method for integrating the vector field gives an approximation to this map. The analysis of Coombes et al. [30] applies Newton's method and its extensions to analyze how much the solutions of the system (3) change with perturbations of q}. To deal with the flow direction itself, these authors constrain the points xk to lie on a fixed set of cross-sections to the vector field. The important quantities in determining the shadowing data are the magnitude of a right inverse to the map F and the C 2 norm of the vector field in a neighborhood of the trajectory. The right inverse of F is not determined uniquely: there are essentially n degrees of freedom that specify a trajectory. To obtain a right inverse whose norm is relatively small, the trajectory is decomposed into its expanding and contracting directions. The contracting coordinates are chosen at the beginning of the trajectory and the expanding coordinates are chosen at its end. Using their methods, Coombes et al. [30] demonstrate shadowing of very long trajectories in the Lorenz system [ 113].
2.3. Error estimation and verified computation Numerical integration algorithms are fundamental tools for the investigation of dynamical systems, but the results they produce are seldom subjected to verification or rigorous error estimation. Indeed, the exponential divergence of trajectories in systems with sensitive dependence to initial conditions sets limits on the time for which one can expect a numerical trajectory to remain close to the actual trajectory with the same initial condition. Naive attempts at estimating the errors of numerical integration tend to introduce artificial instability coming from varied sources such as the rectilinear geometry inherent in interval arithmetic [33]. This wrapping phenomenon amplifies the expected exponential growth of
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errors, typically producing pessimistic results. This state of affairs creates a tension between simulation and the mathematical theory of dynamical systems. On the one hand, numerical integration seems necessary in the investigation of systems that do not have analytically explicit solutions. On the other hand, the difficulty in estimating the errors in these integrations makes it hard to use simulations in rigorous analysis. There has been recent progress in attacking this issue, and the number of successful examples in which numerical computation gives rigorous results about dynamical systems is growing steadily. The variational equations ~(t) - Dfx(t)~(t) of a trajectory x(t) for system (1) give an infinitesimal picture of how nearby trajectories differ. The exponential growth rates of solutions to this time varying linear system of differential equations are the Liapunov exponents of the trajectories. Their existence is discussed in the next section. The computation of Liapunov exponents must contend with two phenomena. The first phenomenon is that the rates of expansion and contraction along a trajectory may vary dramatically. This is particularly evident in canard solutions to systems of differential equations with multiple time scales [47,48,57], also discussed in the next section. The second phenomenon is that the directions of expansion and contraction may twist along a trajectory. Changes in twist are intimately involved in the bifurcations that take place in chaotic invariant sets. Both of these phenomena are common, so general algorithms for computing Liapunov exponents should take them into account. Suppose that (xk, tk) is a sequence of points along a trajectory and that Dk is the Jacobian of the flow map from (xk, tk) to (xk+l, tk+l). Denoting by al . . . . . an the singular values of Ju = D N . . . DzDo, the Liapunov exponents are defined as the limits of log(a/) / tN as N --~ oe. Thus computation of the Liapunov exponents requires computation of the singular values of Ju. Performing this computation by first computing the product and then computing the singular values is ill-conditioned in general. If the largest Liapunov exponent is separated from the remaining ones, the Ju tend to rank one matrices and small perturbations of the Di produce large relative changes in the magnitudes of its smaller singular values. To accurately compute the smaller singular values, two basic strategies have been proposed [136,152]. The first is to form exterior powers of the Jacobians that represent its action on subspaces. The dominant singular values of exterior power i will be the product of the largest i singular values, so the Liapunov exponents can be recovered from the ratios of the largest singular values of the exterior powers. The second strategy is perform a matrix decomposition of Ju by working with its factors iteratively. For example, the QR factorization of Ju can be computed by first computing the QR factorization Do = QoRo. Then one computes the QR factorization of Dl Q0 = Ql RI. Proceeding inductively, one computes Dk-.. Do = Qk Rk... R0. The inductive step is to compute the QR factorization Dk+l Qk = Qk+l Rk+l. Dieci et al. [41] have proposed a continuous analog for this factorization of Ju based upon solving Riccati equations. The goal is produce a frame, i.e., a smoothly varying set of orthogonal coordinate systems along the trajectory, so that the variational equations become triangular when transformed to this frame. The differential equations satisfied by the frame are a Riccati system for which there are special methods of numerical integration [39,111 ]. The most direct approaches to error estimation for numerical integration are based upon interval arithmetic. Numbers are replaced by intervals and operations are replaced by enclosures. For example, the sum of two intervals is an interval that contains the sum of any numbers contained in the two summands. For calculations involving a moderate number of
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operations, interval arithmetic is often an effective means of obtaining rigorous estimates for calculations. As an example, interval implementations of Newton's method often work well to give precise estimates on the location of all the zeros of a function, including proofs of their existence. Within the context of dynamical systems, Lanford's computer based proof for the existence of a fixed point for the period doubling renormalization operator on unimodal functions [ 108,109] exploits the application of interval arithmetic to Newton's method. Application of interval arithmetic to numerical integration is an old idea [33], but the results are frequently disappointing, leading to poor bounds on the computation of a trajectory compared to the apparent accuracy of the calculation. Unless the flow is uniformly contracting in the phase space, each step of an interval based numerical integration tends to produce larger enclosing intervals. The continued growth of enclosing intervals limits severely the number of time steps that can be taken before the bounds become useless. The work on shadowing described above and three additional examples illustrate ways of circumventing these limitations of interval arithmetic. The first example comes from the work of L6hner. He replaces intervals by Taylor series (or jets) augmented by bounds that enclose function values. These Taylor series with bounds are the fundamental objects with which computations are performed. Thus, a function f :R" -+ R is approximated on a domain D C R" by a polynomial P and ~ > 0 with the property that [P (x) - f (x)l < e for all x ~ D. This is a much richer class of objects than intervals, and it is possible to construct a precise numerical calculus for these objects in the context of floating point arithmetic. Lanford's numerical proof of the Feigenbaum conjectures uses these data structures [108]. Berz has implemented this calculus in his COSY software, using the term differential remainder algebra [ 15], and applied it to normal form calculations of Hamiltonian systems to achieve strong estimates of the stability properties of accelerator designs [ 115]. L6hner uses algorithms that compute Taylor series approximations of trajectories for differential equations with automatic differentiation, and then obtains error estimates for these approximations. The error estimates come from an adaptation of the contraction operator used in the Picard proof of existence of solutions to ordinary differential equations. If f : R " --+ R" is Lipschitz continuous and T is small enough, then the operator I-l(g)(t) -- xo +
f0 t f
(g(s)) ds
acting on continuous functions g:[0, T] --+ R" with g(0) = x0 has a contracting fixed point at the solution to the differential equation 2 = f (x) with initial condition x0. Bounds on T are readily computed in terms of the domain, magnitude and Lipschitz constant of f . L6hner applies this operator in the context of Taylor series with remainders to obtain good bounds on Taylor series approximations to the solution of the differential equation. This work complements the use of automatic differentiation to produce high order numerical integration algorithms based upon Taylor series. The second example of error estimates for solutions of differential equations exploits transversality in the context of planar dynamical systems. Guckenheimer and Malo [80] observed that numerical integration of rotated vector fields [54] can be used to compute curves that are transverse to the trajectories of a planar vector field. Using the terminology
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of Hubbard and West [94], we expect the numerical trajectories of the rotated vector fields to be fences that provide barriers the trajectories of the original vector field can cross only once. Interval arithmetic can be used to verify that the trajectories of the rotated vector fields are indeed transverse to the trajectories of the original vector field. The advantage of this method compared to direct error estimation of trajectories is that the transversality computations are all local to individual time steps of the rotated vector field. The interval estimates for each time step are independent of one another, so the growth of the estimates does not propagate from one time step to the next. Using these ideas, Guckenheimer [75] described an algorithm to rigorously verify the correctness of phase portraits of structurally stable planar vector fields defined by functions for which interval evaluations have been implemented. The third example of rigorous results based upon numerical error estimates is the recent analysis of the Lorenz system by Tucker [ 146]. In 1963, Lorenz described the first strange attractor that was observed via numerical computation [ 113]. A more complete geometric model of the Lorenz attractor was formulated fifteen [84] years later. Verification that the assumptions underlying the geometric model are satisfied by the Lorenz system has been a benchmark problem in the numerical analysis of dynamical systems. The key assumption can be expressed as the statement that there exists a family of cones in the tangent spaces of points in the attractor that are forward invariant. Tucker has solved this problem. His work is based upon a careful dissection of a neighborhood of the attractor into regions on which the behavior of the variational equations can be described with interval computations. The interval analysis itself does not need to be precise once a suitable covering of the attractor has been constructed.
3. Computation of invariant sets Invariant sets with complex geometry are common in dynamical systems. The complexity comes both in the local structure of the sets and in convoluted shapes of smooth objects. Smale's horseshoe [ 141] is an important example of a fractal invariant set for a discrete time dynamical system. The analog of the horseshoe in continuous time dynamical systems is the solenoid [140]. The stable and unstable manifolds of periodic orbits in the horseshoe and solenoid are folded, with regions of arbitrarily large curvature. The Lorenz system [ 113] has been a rich source of complex geometric objects, including the convoluted, two dimensional stable manifold of the origin and its fractal, chaotic attractor. The algorithmic aspects of computing invariant sets is a relatively new subject compared to numerical integration. The list of successful methods for directly computing invariant sets has been growing. This section surveys research on computing four types of invariant sets: periodic orbits, invariant tori, stable and unstable manifolds of equilibria and chaotic invariant sets.
3.1. Periodic orbits
Aperiodic orbit of a flow ~bis a non-equilibrium trajectory x(t) with x(0) = x(T) for some T > 0. The minimal value of T is the period of the orbit. Perhaps the most common way of
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finding stable periodic orbits is to identify them as the limit sets of trajectories computed by numerical integration. However, there are circumstances in which numerical integration does not yield an accurate representation of a stable periodic orbit. An example is given below. There are also circumstances in which theoretical considerations suggest that the computation of periodic orbits with numerical integration may fail. A one-step numerical integration method with fixed steps is a diffeomorphism, and periodic orbits of flows become invariant curves of the time h map ~bh of the flow. The theory of normal hyperbolicity implies that if the periodic orbit is hyperbolic, then the map defined by a numerical integration algorithm will have an invariant curve near the periodic orbit if it is a sufficiently accurate approximation of ~b/,. On this invariant curve there may be resonance, with numerical trajectories converging to a stable, discrete periodic orbit with a finite number of points rather than filling the periodic orbit densely. This cannot happen for ~b/, if h is incommensurate with the period. When T is a multiple of h, the periodic orbit is a continuous family of periodic orbits of 4~/1.This qualitative discrepancy between the generic behavior of a numerical integration algorithm and the flow map seldom impedes the identification of stable periodic orbits as the limits of numerical trajectories. Nonetheless, when trying to compute periodic orbits whose stability is weak enough, normal hyperbolicity breaks down and the numerical algorithm may acquire more complex limit sets close to the periodic orbit. In particular, using numerical integration to accurately identify the location of saddle-node bifurcations of periodic orbits is problematic. It is desirable to have direct methods for locating periodic orbits for at least three reasons: 9 Numerical integration may fail as described above. 9 Direct methods may be more efficient than numerical integration for computing periodic orbits. 9 Unstable periodic orbits are not readily obtainable as limit sets of trajectories. The equations defining periodic orbits are boundary value problems for ordinary differential equations. Most of the extensive literature and software dealing with boundary value problems applies to two point boundary value problems with separated boundary conditions [9]. While the equations for periodic orbits can be recast in this form by enlarging the dimension of the phase space, this approach has not been successfully applied to many problems. Instead, most studies of periodic orbits use algorithms that are specifically designed for the solution of boundary value problems with periodic boundary conditions. There are two methods that dominate these studies: simple shooting methods and the collocation method implemented in the computer code AUTO [50]. For these methods to work well, the periodic orbit should itself be robust with respect to perturbation: the periodic orbit should vary continuously with deformations of the vector field. A sufficient condition for this robustness can be formulated in terms of the monodromy matrix of the orbit y. The monodromy matrix A of a point x ~ y is the Jacobian of the time T flow map at x. The monodromy matrix A always has 1 as an eigenvalue (with f ( x ) as eigenvector), but if 1 is a simple eigenvalue of A, then the periodic orbit perturbs smoothly with perturbations of f . A periodic orbit for which 1 is a simple eigenvalue of A is called regular. The periodic orbit equations of a vector field do not have isolated solutions. If Y(t) is a periodic orbit and c 6 R, then 9/(t + c) is also a periodic orbit. To obtain a regular system of equations for points approximating a periodic orbit, this degeneracy coming from
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translation in time must be removed. In s i m p l e s h o o t i n g methods, this is typically done by restricting initial conditions to lie in a cross section to the periodic orbit. The return m a p for the cross section is defined by mapping each point to the next point on its trajectory lying in the cross section. Simple shooting methods compute the return map using a numerical integration algorithm. This is augmented by using a root finding algorithm such as Newton's method to compute a fixed point for the return map. Implementing simple shooting methods is straightforward, but their performance is subject to numerical limitations that are explained below. There have been many implementations of simple shooting methods, for example in the code LOCBIF [ 102]. There is one detail worth noting. Computing the return map requires that the intersection of a trajectory with the cross-section be computed. Many numerical integration algorithms do not yield values at intermediate points of a trajectory between time steps with the same accuracy as those at the time steps. In this case, interpolation using several computed points of the trajectory or a change of time step that yields a point on the trajectory can be used to complete the calculation of the return map. If the cross-section is given by the equation xk = c, E. Lorenz observed that one can rescale the vector field by dividing by its kth component to obtain a vector field near the cross-section in which the kth component evolves with unit speed. Choosing the time step for the rescaled vector field to be c - xk gives a time step that ends on the cross-section. The numerical difficulties with simple shooting come from two sources. The first source of difficulty is the accuracy of the return map. Newton's method requires the Jacobian J of the return map. If J is computed with finite differences, this can make application of Newton's method to the return map erratic. The second source of difficulty with simple shooting is due to the potential ill-conditioning of the problem. The return map may have a Jacobian with very large norm at the intersection of a periodic orbit with a cross section. This norm can readily become large enough that changes in initial condition of unit precision produce changes in the value of a return map that are larger than a desired error tolerance in the fixed point procedure used to locate the periodic orbit. In the canard example discussed below, integration over part of the cycle produces an extremely ill-conditioned flow map. While simple shooting works with many problems despite these potential difficulties, more elaborate methods for computing periodic orbits are often required. Multiple shooting algorithms address the difficulties associated with ill-conditioning of the return map. Instead of solving the single equation q~(x, t) = x for a fixed point of the return map, one seeks a set of points (xk, tk), k = 0 . . . . . N, with to = 0, x u = xo and 9 (xk, tk+l -- tk) = xk+l for k = 0 . . . . . N - 1. This forms a system of n N equations in (n + 1) N variables. For a regular periodic orbit, there is a smooth N dimensional manifold of solutions coming from different choices of xk on the periodic orbit. Multiple shooting algorithms either constrain the (xk, tk) to lie in a set that yields a unique point on the periodic orbit, or they augment the system of equations to yield a regular system of (n + 1)N equations. Conceptually, multiple shooting is simple. By judicious choice of the length of segments along the periodic orbit, the condition number of the system of equations q~(xk, tk+l - tk) -- xk+l can be reduced to manageable levels on problems where simple shooting fails completely. We illustrate this with an example. Problems with multiple time scales are especially prone to ill-conditioned return maps since the periods of the periodic orbits may be very long when measured in the faster time scales of the problem. The following example exhibits stable periodic orbits that cannot be
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computed readily with numerical integration or a simple shooting method. These orbits are examples of trajectories called canards due to their visual appearance in a family of vector fields that generalizes the van der Pol equation [47,48]. The vector field 2 - x 3),
1
y~a~x is a translate of the van der Pol equation when a = - 1 / 3 and has two time scales when s > 0 is small. The single equilibrium point at (a, a 2 -Jr-a 3) undergoes supercritical Hopf bifurcation at a = 0 with decreasing a. Figure 1 shows periodic orbits from this family with s = 0.001, computed with a multiple shooting algorithm based upon automatic differentiation. The periodic orbits that emerge grow very quickly in a to a relaxation oscillation approximated by a pair of segments that follow the nullcline y -- x 2 + x 3 and a pair of horizontal segments that are tangent to the nullcline. The growth of the periodic orbits occurs during an interval of a whose width shrinks to zero proportionally to e x p ( - 1/e) [47]. The cycles of intermediate size in the family have a segment that follows the unstable middle portion of the nullcline with x e ( - 2 / 3 , 0). In this region, trajectories of the vector field diverge rapidly from one another at a rate comparable to e x p ( - 1/e). This divergence makes it essentially impossible to compute a trajectory that follows the nullcline using numerical integration forwards in time with double-precision (64 bit) floating point arithmetic. Starting near the local minimum of the nullcline, an error in the computation of one step comparable to the unit precision of the floating point arithmetic produces a point whose tra0.18 0.16 0.14 0.12 0.1 0.08 0.06
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jectory will leave a moderate sized neighborhood of the nullcline in a time comparable to 50e, the factor 50 being approximately the logarithm of the unit precision. For e = 0.001, this time is approximately 0.05, only long enough for y to travel a short distance up the nullcline at the rate I~1 - a - x. Consequently, forward numerical integrations of trajectories appear to "peel off" from the unstable portion of the nullcline without traveling very far along it. Varying a appears to produce almost a discontinuity in the numerical co-limit sets. Attempts to find the canard solutions with Runge-Kutta algorithms find a narrow interval of a of width comparable to 10-14 in which the integration algorithm becomes chaotic, erratically producing a combination of small and large loops along the same numerical trajectory. Such a trajectory is illustrated in Figure 2. Thus, despite the fact that the canard cycles are stable, they cannot be computed readily with numerical integration. Guckenheimer et al. [77,81 ] have been developing new algorithms for the computation of periodic orbits based upon the use of Taylor series and automatic differentiation. Automatic differentiation is used to achieve high orders of accuracy while maintaining coarse meshes and relatively small systems of equations to solve. We describe here the multiple shooting 0.2
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Fig. 2. Fourth order Runge-Kutta integration of the canard vector field. Due to the massive instability of the unstable branch of the slow manifold, the integration is unable to compute trajectories that follow this branch. Compare with the family of canards shown in Figure 1.
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algorithm used to produce Figure 1. Approximations to the periodic orbit are parametrized by a discrete mesh of points (xk, tk) near the periodic orbit, including the times associated to these points. Numerical integration from one mesh point to the next is performed using Taylor polynomial approximations to the trajectories, with step sizes chosen adaptively so that estimated errors of the trajectory lie below a predetermined threshold. The Taylor series and the Jacobians of the Taylor series coefficients with respect to variation of the mesh point are computed using procedures that are part of the program ADOL-C developed at Argonne National Laboratory [73]. The variational equations for the orbit are solved along with the original differential equation. The system of equations q~(xk, tk+l -- tk) = xk+l, k - - 0 . . . . . N, is augmented with N additional equations. These additional equations express constraints that force updates of (xk, tk) to be orthogonal to the trajectory of (xk, tk) in IK''+l . The tangent vector to this trajectory is given by ( f (xk), 1). (In the case of k = 0 where we fix to = 0, the times tl . . . . . tN -- 1 are varied instead of to.) For a regular periodic orbit, the augmented system of equations is regular and Newton's method is used to solve it. The algorithm requires an initial approximation to a periodic orbit as starting conditions, say a set of points obtained from a numerical integration. New mesh points are created adaptively when the norm of the Jacobian from the previous mesh point exceeds a specified bound. Mesh points are removed when the remaining points adequately represent the periodic orbit. Continuation, described in the next section, is easily implemented in this algorithm. Automatic differentiation is used to compute derivatives of the Taylor series coefficients with respect to parameters as well as with respect to the phase space variables. A parameter is regarded as an additional independent variable in the augmented system of equations. Continuation steps are taken by computing the tangent vector to the curve of solutions of the augmented equations and adding an increment of this tangent vector v 6 ]K('7+l)x+l to the current solution. Yet one more equation is added to the augmented system, constraining Newton updates to be orthogonal to the tangent vector v. Global boundary value methods for computing periodic orbits project the equation ~(t) = ( 1 / T ) f ( v ( t ) ) onto finite dimensional approximations of the space f" of smooth functions v : S l ~ •". Alternatively, we can try to solve ~(t) -- f ( v ( t ) ) for curves with ~, (0) = V (T). In both alternatives, the period T is an unknown that is part of the equations to be solved. The global method used in the code AUTO [52,53] is collocation. The approximations to/-" consist of continuous, piecewise polynomial functions. A discretization of V is determined by a mesh of N points (xl . . . . . xu) and N time intervals t l . . . . . tx. Inside each of the N mesh intervals k collocation points are chosen. In AUTO, as well as the boundary value solver colsys and its descendants [9,52,53], the collocation times for each mesh interval are at Gauss points in order to produce superconvergence of the method. The polynomial function on each mesh interval is required to satisfy the differential equation at the collocation points and to be continuous at the endpoints of the mesh intervals. For an n dimensional system, this yields a total of n x N x (k + 1) independent equations on the same number of variables. The orbit period is one additional independent variable, and it is balanced by an equation that removes the degeneracy associated with time translation. This can be done by restricting a point of the orbit to a cross section or it can be given by an integral phase condition in the form f g ( v ( s ) ) d s = 0 for a function g :]K'7 --+ R. The system of n x N x (k + 1) + 1 equations defined by the collocation algorithm is quite large. Fortunately, the Jacobian of this system is sparse. AUTO exploits the sparsity using
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a special Gaussian elimination procedure in its Newton iterations to obtain solutions of the periodic orbit equations. The accuracy and efficiency of the Taylor series methods are demonstrated in the following example. The planar vector field .~ __ y _ y 2 _ X (X 2 -- y 2 +
2y3/3 + c),
5~ -- x + (y - y2)(x 2 _ y2 + 2y3/3 + c) has a periodic orbit that lies in the zero set of the polynomial h - x 2 - y2 _+_2y3/3 + c when c E (0, 1/3). The value of h along a computed curve measures the distance of the curve from the periodic orbit. Figure 3 displays values of h along approximate periodic orbits with c = 0.07 and period approximately 7.7 computed in three ways. The top panel presents values of h at the 60 mesh points of an AUTO calculation of the periodic orbit. The data is representative of the most accurate approximations to the orbit produced by x 10 -1~ 1
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Fig. 3. Three different methods have been used to compute a periodic orbit that lies along the level curve h = 0.07 of the polynomial h = x 2 - y2 + 2y3/3. Values of h are plotted as a function of time during one traversal of the periodic orbit. The numerical periodic orbit in the top panel was computed with AUTO, the three in the middle panel with a fourth order Runge-Kutta method, and the bottom one with a multiple shooting algorithm employing automatic differentiation.
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AUTO while varying the number of mesh points, the number of collocation points and the error tolerances allowed by the algorithm. The middle panel shows three numerical trajectories computed with a fourth order Runge-Kutta algorithm using step sizes of fixed lengths 0.00125, 0.001 and 0.0001. The step size 0.001 appears to provide close to optimal accuracy for this method, with round-off errors apparently dominating truncation errors for smaller step sizes. The lower panel displays the results of a calculation using a multiple shooting algorithm based on automatic differentiation. There are five mesh intervals with the solution approximated on each half of a mesh interval by the degree 16 Taylor series polynomial at the boundary mesh points. Convergence was obtained in 6 Newton steps starting with mesh points that are far from the computed solution. The maximum value of Ihl in the three calculations is approximately 6 x 10 - l l , 8 x 10 -15 and 6 x 10 -16 The Taylor series methods achieve the best accuracy, comparable to the inherent precision of the floating point arithmetic, with surprisingly coarse meshes. Further development of methods based upon Taylor series appears to be very promising.
3.2. lnvariant tori Invariant tori are a prominent feature of symplectic flows and also arise through Hopf bifurcation of periodic orbits in dissipative systems. For flows with a global cross-section on a two dimensional torus, a fundamental invariant is the winding number, or equivalently the rotation number of a return map [90]. This invariant is rational if and only if the flow has periodic orbits. If the flow is C 2 and the winding number is irrational, then all trajectories of the flow are dense [38]. KAM theory [90] and the theory of normal hyperbolicity [93] provide theoretical tools for the analysis of invariant tori. There is a modest body of research on algorithms for computing these objects, much of it framed in the context of invariant curves of discrete time systems. Three different approaches, quite different from one another, are discussed here. The first approach to computing invariant tori of discrete maps has been to represent one dimensional tori as graphs of functions and to formulate a system of equations that gives a finite dimensional approximation to the invariance of these curves. This approach has been pursued in different ways. KAM Theory restricts attention to invariant tori on which the motion is conjugate to irrational rotation and solves for the Fourier series of the conjugacy. In the case of invariant curves, piecewise polynomial approximations of invariant curves lead to general algorithms that apply to invariant curves that contain periodic orbits as well as tori that have irrational rotation numbers. Implementations of such methods have been described by Kevrekidis et al. [ 101 ] and van Veldhuizen [ 147]. Their results indicate that it is difficult numerically to follow a family of invariant tori to the point at which they begin to lose smoothness and disappear. Aronson et al. [8] give a comprehensive description of ways in which tori with rational winding numbers can lose smoothness. The second approach to computing invariant tori was pioneered by Greene [72]. This method seeks to compute invariant tori in symplectic systems by approximation with periodic orbits. Most of the research has concentrated on area preserving diffeomorphisms of the plane. KAM theory proves that each periodic orbit of elliptic type is surrounded by a family of invariant curves with irrational rotation numbers satisfying Diophantine inequalities [72,155]. Each of these invariant curves is the limit of periodic orbits whose
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rotation numbers are obtained by truncating the continued fraction expansion of the irrational rotation number. Periodic orbits of high period are computed with root finding algorithms analogous to either shooting or global boundary value methods. Estimates of the convergence of the approximating periodic orbits to the (unique) invariant measure of the invariant curve gives information about the structure of the invariant curve. If the diffeomorphism depends upon a parameter, some of the invariant curves may evolve into Cantor sets. Renormalization methods have been applied to study this transition, especially for invariant curves for the golden mean and other rotation numbers with periodic continued fraction expansions [ 110]. The numerical computations of these "last" invariant curves have been based upon computations of approximating periodic orbits. Dieci et al. [42-45] have investigated the computation of invariant tori for vector fields. Their starting point has been the formulation of a partial differential equation that implies the invariance of the torus. This partial differential equation states that the vector field is tangent to the torus. The innovative aspects of these studies lie in using algorithms for solving PDE's to address this problem. The torus is represented as the image of a mapping on a discrete grid, and then partial differential equations are approximated to yield a set of equations for this mapping analogous to global methods for computing periodic orbits. Implementations of the algorithms have been tested on a few examples like the forced van der Pol equation, but experience as to the domain of problems for which these algorithms work well remains limited.
3.3. Stable and unstable manifolds Stable and unstable manifolds of equilibrium points and periodic orbits are important objects in phase portraits. In physical systems subject to disturbances, the distance of a stable equilibrium point to the boundary of its stable manifold provides an estimate for the robustness of the equilibrium point. The closer the boundary, the more likely disturbances will kick the system out of the basin of attraction of the equilibrium. In the simplest situations, these boundaries are formed by stable and unstable manifolds of saddles. In more complex situations, the basin boundaries are fractal, chaotic invariant sets containing large numbers of periodic orbits and their stable manifolds. Thus, there is great interest from both theoretical and practical perspectives in computations of stable and unstable manifolds. From a naive perspective, it would appear that the computation of stable and unstable manifolds of equilibria is no more difficult than numerical integration. For one dimensional manifolds this is true. One dimensional stable and unstable manifolds of equilibria of flows consist of pairs of trajectories, so their computation can be implemented by applying an initial value solver to a well chosen initial condition. Higher dimensional stable and unstable manifolds are harder to compute. The two dimensional stable manifold of the origin for the Lorenz system [113] 2-
lO(y-x), - 28x-y-xz, 8 -- - - ~ z + x y
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has served as a benchmark problem. There are two difficulties in computing this manifold. First, the stable eigenvalues at the origin of this system are approximately - 2 . 6 7 and - 2 2 . 8 with a ratio that is approximately 8.56. As a result, backwards trajectories in the manifold tend to flow parallel to the strong stable direction. Numerical integration of initial conditions in the stable manifold uniformly clustered near the origin produces only a strip along the strong stable direction. The second difficulty in computing this stable manifold is that it becomes highly convoluted far from the origin. Part of the manifold spirals around the z-axis while part of it curls around the stable manifolds of the equilibria located at (-+-6 x/2, -+-6x/2, 27) [126]. Symbolic methods can be used to compute high order approximations to the Taylor series of stable and unstable manifolds at equilibrium points. One approach to these algebraic calculations is to subsume the computation of stable and unstable manifolds of equilibria into the linearization problem: finding a smooth coordinate transformation that transforms the system 2 = f (x) into a linear system of equations near an equilibrium. In the transformed coordinates, the stable and unstable manifolds are linear subspaces. Formally, the linearization problem can be reduced to a sequence of systems of linear equations for the Taylor series of the coordinate transformation [78]. These linear systems degenerate if the eigenvalues )~i at the equilibrium satisfy resonance conditions of the form il
)~i -- ~
aj)~j
j=l
with non-negative integer coefficients aj. The order of the resonance condition is ~ aj. When resonance conditions are satisfied, transformation to normal forms containing only nonlinear terms associated with the resonance conditions can still be accomplished but the system can only be linearized with finite smoothness related to the order of the resonance conditions [96]. Transformation to simpler nonlinear systems, called normal forms, is used extensively in the analysis of bifurcations [78,131]. Algebraic computation of linearizations and normal forms are readily implemented in symbolic systems for vector fields of moderate size [133]. Nonetheless, the complexity of these computations grows quickly with problem size. For large problems, instead of computing a full linearization, one would like to extract more limited information. Problems are common for which almost all modes are highly damped and a low dimensional submanifold in the phase is attracting. These problems often arise from investigations of instability when a system is driven by external forces until its attractors are time dependent, but not highly disordered. Beyn and Kless [ 17] have examined the computation of low dimensional invariant manifolds within this context. They study the use of iterative methods in linear algebra to compute the location and normal forms of invariant manifolds while avoiding such operations as the factorization or inversion of the full system Jacobian at an equilibrium. The most complete work on computing stable and unstable manifolds has been done in the context of one dimensional stable and unstable manifolds of fixed points of discrete time dynamical systems. These methods have been applied to the return maps of periodically forced continuous systems and to computation of two dimensional stable and unstable
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manifolds of periodic orbits [24]. One dimensional stable and unstable manifolds of fixed points for maps have fundamental domains: if the eigenvalue of the manifold is positive, each half of the manifold is the union of iterates of a segment joining a point to its image. Moreover, the manifold lies close to its tangent near the fixed point. Thus an initial approximation of the manifold can be obtained by iterating points that lie in a small fundamental domain of the linearized map of the fixed point. However, this procedure does not always give a well resolved approximation to the manifold because the points may separate from one another as they iterate away from the fixed point. Algorithms that avoid this difficulty have been implemented [ 104]. Yorke et al. [ 114] have used a divide and conquer algorithm to compute one dimensional stable and unstable manifolds of saddles in two dimensional maps. These straddle algorithms locate a stable manifold by finding segments whose endpoints iterate towards the saddle point and then proceed in opposite directions along the unstable manifold of the saddle. Continuity implies that a point of the segment lies in the stable manifold. Iteratively following the midpoint of the segment and selecting the half that straddles the stable manifold, the intersection of the stable manifold with the segment can be located precisely. The method is inherently very robust, but it does not emphasize computational efficiency. Recently, Osinga and Krauskopf [ 104] have described a different procedure to compute one dimensional stable and unstable manifolds. Some research has been done on the global computation of two dimensional stable and unstable manifolds of equilibria for flows. Several different strategies have been used with reasonable success on such problems, all tested with the stable manifold of the origin in the Lorenz system. Johnson et al. [97] rescaled the vector field so that it had constant length. This approach makes trajectories advance at uniform speed, but their direction continues to follow the strong unstable manifold. Guckenheimer [85] experimented with computation of the geodesic rays in the induced metric of the stable manifold. This procedure appeared to work well, but developed numerical instabilities far from the equilibrium. Osinga [ 125] and Osinga and Krauskopf [ 104] have developed methods based upon the graph transform. The graph transform is an operator that is used to prove the stable manifold theorem, and Osinga in her thesis implements methods that follow closely the proof. As the Lorenz system stable manifold grows, it acquires complex folds and twists [126]. Tracking the manifold through these folds and twists has been difficult. The graph transform methods are based upon a decomposition of the phase space into a product of linear stable and unstable manifolds near the equilibrium, but the manifold does not remain transverse to the unstable manifold of the equilibrium. Therefore, an adaptive set of coordinate systems is required to track the manifold as it turns. In the methods of Guckenheimer and Johnson, the manifold is computed as a set of curves that bound a growing disk in the manifold. These curves grow in length quickly enough that an interpolation procedure that places new points on the curves as they grow is required to resolve the stable manifold adequately. In places where the manifold develops sharp folds, it becomes difficult to perform this interpolation accurately. A complementary method for computing the stable and unstable manifolds of low dimensional systems similar to the straddle algorithms of Yorke et al. was implemented by Dellnitz and Hohmann [37]. To compute a compact portion of the manifold, a region in phase space is partitioned and each partition element is marked as to whether it might intersect the desired manifold. Starting with a coarse partition, many partition elements can
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usually be marked as not containing an intersection point. These are discarded, and the remaining elements of the partition are refined and then tested to see whether they intersect the manifold. The number of rectangles in successive refinements that must be tested depends on the dimension of the manifold being computed rather than on the dimension of the phase space, so the methods appear feasible for two dimensional manifolds of rather large systems. Doedel [51 ] has suggested yet another procedure for computing stable and unstable manifolds based upon the solution of boundary value problems. The idea advanced by Doedel is to formulate an iterative procedure in which each step is the solution of a two point boundary value problem. If W is the invariant manifold and U C W is a neighborhood of the equilibrium point that has been determined, then one wants to compute a larger neighborhood of the equilibrium in W. The boundary value solver end point conditions for one end of the interval will be chosen so that the end point of the desired trajectory lies on the boundary of U. If W has dimension d, then these boundary conditions have dimension d - 1. The other end point is required to lie on a specified manifold V transverse to W. If V has complementary dimension to W, one more boundary condition is needed. This can be obtained either by enlarging the dimension of V or by allowing the transit time from one end point to the other to vary. The latter strategy is similar to that used by a boundary value solver to obtain the period of a periodic orbit by fixing its length in time while rescaling the vector field with a free parameter. Robust implementations of algorithms to compute two dimensional stable and unstable manifolds of equilibria have not yet been achieved. The work described above reveals some of the obstacles that have been discovered. These obstacles appear surmountable. Recent improvements in computers should make methods feasible that previously required too much floating point computation or memory use. Better adaptive methods to discretize the intricate geometry of two dimensional stable and unstable manifolds are needed before we will have general purpose codes that reliably compute two dimensional stable and unstable manifolds of equilibria.
3.4. Chaotic invariant sets Chaotic invariant sets have been the focus of a large amount of dynamical systems research. Chaos is a term that has come to mean any type of asymptotic dynamics more irregular than quasiperiodicity. Numerous papers have made the claim that chaos occurs in a particular system, but most of these claims are based only upon visual observation of numerical trajectories. Infrequently, the claims are substantiated with arguments demonstrating that the system has a property that implies the existence of chaos. The strongest criterion for the existence of chaos is the existence of horseshoes in discrete systems or solenoids in flows [ 140]. These are invariant sets which are topologically equivalent to subshifts offinite type in the case of discrete time and their suspensions in the case of flows. All of these objects have been extensively studied from measure theoretic and statistical viewpoints. They carry invariant measures which are ergodic and have positive entropy and Liapunov exponents [59,158].
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Horseshoes and solenoids vary continuously with C l perturbations of a map or flow, lending credence to numerical observations of chaotic structure. The Smale-Birkhoff Homoclinic Theorem [140] gives a necessary and sufficient criterion for the existence of horseshoes for diffeomorphisms, namely that there are transversal intersections of stable and unstable manifolds of a periodic orbit. Application of this theorem to return maps of a flow gives the same result there. When stable and unstable manifolds of periodic orbits can be calculated, this result gives a procedure for verifying the existence of chaotic dynamics in a system. The Melnikov method [78,132] gives criteria for perturbations of nontransversal homoclinic orbits of periodic orbits to become transversal as a system is deformed. Note, however, that many examples have intersections of stable and unstable manifolds in which the angles of intersection are small, making numerical verification of chaotic dynamics difficult [78]. This is especially true in Hamiltonian systems [ 123] where Melnikov theory applied to resonant layers of nearly integrable systems fails. Asymptotic analysis of these systems reveals that the angles between stable and unstable manifolds in these layers is "beyond all orders" of the perturbation theory [ 135]. Simo and his collaborators have investigated carefully several Hamiltonian systems arising in celestial mechanics, including the restricted three body problem [139]. They have made significant strides in demonstrating the existence of very small transversal intersections between stable and unstable manifolds of periodic orbits. Although numerical evidence is often used to substantiate claims of chaotic behavior, this evidence can be unreliable. One step numerical integration algorithms with fixed time step h define maps that approximate the time h maps of flows. There is a notable qualitative difference between these objects, namely that the trajectories of flows are one dimensional curves while the trajectories of the numerical integrators are sequences of points. Homoclinic orbits of an equilibrium point for a flow cannot be transverse because the stable and unstable manifolds have complementary dimension and any intersection has dimension at least one. The numerical method will have a fixed point near the equilibrium with stable and unstable manifolds of the same dimensions as those of the flow. However, their intersection can be zero dimensional since the trajectories of the numerical method are sequences of points rather than curves. Indeed, the Kupka-Smale Theorem states that for a generic set of maps, homoclinic intersections of periodic points will be transverse [ 140]. Thus numerical integration can be expected to introduce chaotic behavior to simulations of dynamical systems that cannot have chaotic behavior. The canard example in the previous section displays this property in a slightly different setting. The scale on which such chaos occurs is frequently small, but claims for chaos in a dynamical system based upon observations of a numerical simulation should be bolstered by additional analysis. Conversely, chaotic dynamics is sometimes difficult to observe in simulations of systems that are indeed believed to be chaotic. Guckenheimer et al. [ 103] studied an example of this kind in a family of diffeomorphisms of the two dimensional torus. Investigating resonances in these maps, they discovered the presence of codimension two Takens-Bogdanov bifurcations. Generic two parameter families of maps that undergo Takens-Bogdanov bifurcation have nearby parameters at which a saddle has transversal intersections of its stable and unstable manifolds. In the example investigated by Guckenheimer et al., the region in which this behavior was found was very, very small - a strip of width less than 10-l0 in a problem for which the parameter space is naturally the unit square. Moreover, the angle be-
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tween the manifolds became large only in very small neighborhoods of the periodic orbit. Without a systematic search, the chaotic behavior in this family is difficult to find. Similar phenomena occur in the analysis of unfoldings of bifurcations of flows. Two parameter families of flows near codimension two bifurcations of equilibria with a zero eigenvalue and a pair of pure-imaginary eigenvalues have chaotic dynamics in a persistent manner. However, truncated normal forms of these bifurcations do not have chaotic dynamics and once again the angles of transverse intersections of stable and unstable manifolds are initially very small. Thus the failure to detect chaos in numerical simulations does not always mean that it is not present. Compare the study of toral maps by Yorke et al. [71 ] with the torus maps described above as an example where chaos is almost certainly present but hard to stumble across. The existence of chaotic attractors has been a subject of intense theoretical investigation. Structurally stable chaotic attractors have uniform hyperbolic structures [140], but it is apparent that many examples which appear to have chaotic attractors cannot have uniform hyperbolic structures on these attractors. The most studied discrete system of this kind is the Henon attractor [89]. Beginning with the theory of iterations of one dimensional mappings [36], an understanding of the properties of chaotic attractors that do not have uniform hyperbolic structures has begun to emerge. Benedicks and Carleson [ 11 ] proved that there are families in which non-uniformly hyperbolic attractors occur on parameter sets of positive measure. Their theory and its extensions [ 12] lend credibility to the belief that chaotic dynamics observed in numerical simulations does indicate that the underlying system has a chaotic attractor. One of the principal theoretical tools for investigating uniformly hyperbolic invariant sets has been the concept of Markov partitions [20]. These partitions lead immediately to representations of these invariant sets as images of subshifts of finite type by maps for which most points have a single preimage. From a statistical perspective, the invariant sets behave like subshifts of finite type. Anosov diffeomorphisms are defined to be diffeomorphisms with dense trajectories and uniform hyperbolic structures on a compact manifold. For two dimensional Anosov diffeomorphisms, the elements of Markov partitions are rectangles whose boundaries are smooth segments of stable and unstable manifolds [3]. In odd dimensions, the boundaries of Markov partitions of Anosov diffeomorphisms are always fractal [21,26]. Algorithms to compute these partitions have only recently been studied [100,137]. The work thus far has been restricted to linear Anosov diffeomorphism of the torus and is heavily dependent on algebraic constructions. There are several phenomena that occur in other examples of chaotic attractors that highlight the bewildering complexity of dynamical systems. One such phenomenon is partial hyperbolicity. Higher dimensional attractors may exhibit partial hyperbolicity in which the dimensions of the unstable manifolds of points are always positive, but vary from point to point. Abraham and Smale [2] described an early example of this phenomenon. More recently, Pugh and Shub [ 127] and others have devoted renewed attention to the analysis of partial hyperbolicity. Numerical investigations of partial hyperbolicity have hardly begun. A second complex phenomenon is riddled basins of attraction [4] in which two or more invariant sets have positive measure "domains" of attraction that are densely intertwined. Every open set that contains points tending to one of these invariant sets also contains points tending to another invariant set. In these circumstances, there appears to be an in-
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herent unpredictability about the limit behavior of initial conditions in large regions of phase space.
3.5. Statistical analysis of time series The numerical analysis of chaotic dynamics has dealt with the statistical properties of invariant sets as well as with algorithms for locating the sets and describing their basins of attraction. Most of the statistical methods are based upon ergodic theory [99] and formulated in terms of invariant measures. From this perspective, the analog of topological transitivity for invariant sets is ergodicity of invariant measures. Hyperbolic invariant sets support many ergodic invariant measures, including the measure theoretic limits N - 1 6 ( F i (x)) of atomic measures along trajectories tending to the invariant set. lim ~1 ~ i-0 These limit measures are frequently called the time averages of the trajectories. They do not always exist, even for almost all initial conditions, as has been demonstrated for one dimensional mappings [79]. For attractors, special emphasis has been given to natural measures, defined as the limits attained from sets of initial conditions having positive Lebesgue measure. For hyperbolic attractors, these natural measures are the SBR (Sinai-Bowen-Ruelle) measures characterized by a variational principle [58]. The convergence of trajectories to measure theoretic limits has been investigated for various examples [13,14]. Three important statistics of ergodic attractors are their entropy, Liapunov exponents and dimension [ 156,157]. Computation of entropy has received relatively little attention compared to computation of Liapunov exponents and dimension. Most algorithms to compute these quantities use data from trajectories, and the methods have been applied to observational data as well as simulations. Nonlinear time series analysis based on these methods provides tools that help assess whether a system might be modeled effectively by one with a low dimensional chaotic attractor. Reconstruction of attractors and construction of models from a scalar time series is a topic that has been extensively studied. The theoretical basis for methods of recovering attractors from one dimensional data was studied by Takens [ 145] who formulated adaptations of the Whitney Embedding Theorem [92]. This theorem states that generic mappings of an n dimensional manifold into a manifold of dimension 2n + 1 are embeddings. Extensions of the theorem have been used to justify the view that the method of time delays can be used to embed a chaotic attractor of dimension d into IKk when k > 2d. The method begins with a scalar time series of observations yi that are assumed to be values of the function y at points x(i A) sampled along a trajectory of the attractor. Vectors of the form (Yi, Yi+l . . . . . Yi+lk) are used as observations of the map Ek(x) = (y(x), y ( x ( l A ) . . . . . y ( x ( l k A ) ) . Takens [145] demonstrates that for a generic observable y, attractor of dimension d and k > 2d, Ek is a 1 - 1 map of the attractor into •k. Procedures for choosing l and k to obtain reliable estimates of the dimension of an observed attractor have been extensively investigated. The accuracy of the methods tends to degrade rapidly with the dimension of the attractor [74]. Liapunov exponents measure the exponential rates of growth of solutions of the variational equations of a vector field. Oseledec [124] proves their existence as measurable functions with respect to any invariant measure. Consequently, the Liapunov exponents of an ergodic measure are invariants of the measure. If xt is a trajectory of the vector
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field ~ = f ( x ) in Euclidean space R", then the variational equations of f along xt are -- Df~., (~), a nonautonomous system of linear differential equations. Its fundamental solution 3 (t) is the matrix solution with initial condition ,T,(0) = I, the identity matrix. Denoting cri (t) the singular values of ,E (t) in decreasing order, the Liapunov exponents of xt are defined to be liminf(1/tlog(cri(t))). Positive Liapunov exponents indicate that there are nearby trajectories that diverge from xt at an exponential rate. Computation of the largest Liapunov exponent is straightforward: cri (t) is comparable to 113 (t)ll. Determining smaller Liapunov exponents is more difficult because, when the largest Liapunov exponent is simple, Z (t)/ll ,T,(t)ll tends to a rank one matrix and round-off errors interfere with the calculation of the smaller Liapunov exponents. This problem has been addressed by reorthogonalization of the solutions of the variational equation. The time interval [0, T] is subdivided into k segments of length 6j, and ,T,(t) is written as a product ,~k""" ,F,l of the fundamental solutions for each of these segments. A series of QR factorizations is then calculated so that 3 j . . . ,El = Qj Rj. At step j of this iteration, the QR factorization of the matrix 3j+l Qj is needed. The matrices Rj a r e products of right triangular matrices of moderate size, so their singular values are expected to be more reliable estimates of ~ri than those obtained from a singular value decomposition of the matrix obtained for ,~ (t) by numerical integration. The continuous methods of Dieci et al. [46] provide an alternate approach to this decomposition. There are several distinct definitions of the dimension of an attractor. The two that have been used the most in analysis of numerical and observational data are the pointwise dimension and the correlation dimension [58,59]. Both of these concepts implicitly rely upon an invariant measure of the attractor. The starting part for their computation is a time series of a trajectory x(t) sampled at N discrete times ti. Appealing to the ergodic theorem, one assumes that the discrete measure ~1 y~Uol (~(x(ti)) approximates an invariant measure # of the attractor. For any measurable set U, the proportion of the points x(ti) that lie in U is then approximately # ( U ) . The calculation of the pointwise dimension of/Z and of the correlation dimension of the attractor use interpoint distances di,j = d(x(ti),x(tj)) with respect to a metric d on the phase space. The pointwise dimension gives the rate at which the volumes of balls shrink as their radius tends to zero. Denoting Bx (r) the ball of radius r centered at x, the point-wise dimension of the measure/Z is lim(log(#(Bx(r)/log(r)) as r -+ 0 for/z-almost all x in the attractor [ 156]. To estimate the pointwise dimension, x is chosen to be one of the points x(ti) in the time series and the sequence di.j, j ~ i, is sorted to produce an increasing sequence r,. A proportion s / ( N - 1) of the points x(t~) lie in the (closed) ball Bx (rs), so/z (Bx (rs)) is estimated to be s / (N - 1). Extrapolating the slope of log(s) vs log(rL,) as r, --+ 0 yields an estimate of the pointwise dimension. There are statistical fluctuations in this estimate that depend on N and the choice of x = xi [74]. Noise in data affects these calculations, so judgment must be exercised about the range of scales above which deterministic dynamics dominates the location of the observed points and below which noise dominates. If the points on a trajectory are regarded as a random sample of points drawn from the measure/z, variance due to sampling can be reduced by averaging the results for several choices of initial points xi. Estimating the volume of small balls with feasible amounts of data is problematic for attractors of large dimension c~, since the volume decreases like r ~. The correlation dimension is computed with a similar calcu-
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lation to the pointwise dimension, but instead of sorting the sequence of N - 1 numbers j :/: i, all of the N ( N - 1)/2 interpoint distances di,j, j < i, are used. "Nonlinear" methods for the analysis of time series data have been extensively investigated since the early 1980s [ 149]. This research is an inverse problem to the numerical analysis of dynamical systems models, in that it seeks to identify models that fit data. The research began with the observation that linear time series analysis methods did not readily distinguish characteristics of data produced from low dimensional attractors from data produced by systems with large random fluctuations or from systems with high dimensional attractors. The Ruelle-Takens theory of transition to turbulence [ 130] motivated this research, prompting careful scrutiny of time series data from fluid systems as flows evolved from steady states to turbulent flow [22,76]. A rich set of methods has been developed using many of the ideas described above, as well as others such as multi-fractal analysis [6] There have been attempts to reconstruct dynamical models directly from data [ 1]. The mathematical foundations for most of this work is poor compared to the remainder of the material reviewed in this survey. Nonetheless, from a practical standpoint, this area of research has great potential to enhance industrial design and scientific study of systems that can be adequately represented by dynamical systems with low dimensional attractors. The methods are less appropriate for systems that have high dimensional attractors because the amount of data required to reconstruct attractors grows very rapidly with the dimension of the attractors.
di,j,
4. Bifurcations
Bifurcation theory is the study of how phase portraits of families of dynamical systems change qualitatively as parameters of the family vary. It is a subject filled with complex detail. Singularity theory [67] is an analog to bifurcation theory, providing a framework that has been partially transplanted to the setting of dynamical systems. These efforts have produced a wealth of valuable information, but some of the mathematical completeness and elegance of singularity theory does not carry over to bifurcation theory. The intricacy of dynamical phenomena act as a barrier to the formulation of a theory that classifies all bifurcations that occur in generic families of dynamical systems. Nonetheless, the mathematical concepts adapted from differential topology and singularity theory provide the foundations for successful algorithms. The focus here is upon describing those concepts that are used in numerical methods. Less attention is devoted to results concerning structural stability or genericity.
4.1. Bifurcation theory Let f :R n x R k ---> ]R'1 be a k-parameter family of vector fields on R '1. Equilibrium points (x, ~,) of f are the solutions of f ( x , ~,) = 0. The goal of local bifurcation theory is to analyze the set of equilibrium points and their stability, taking into account the dependence upon the parameters [68]. We discuss the location of equilibrium points first and then consider their stability. Near equilibrium points (x, ~) where Dx f is regular or, equivalently,
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has full rank n, the Implicit Function Theorem states that the solutions of the equilibrium equations form a k-dimensional submanifold of R 'Z x R k that can be parametrized as the graph of a function Xe:R k --+ IR" from the parameter space to the phase space. Continuation methods implement the computation of xe. By changing coordinate systems to mix parameters and phase space variables, equilibrium point manifolds that are not graphs from phase space to parameter space can be computed. Local bifurcations include all points where Dx f is singular. To use equation solvers that rely upon the regularity of the system being solved, we require reformulation of the problem at bifurcation points. A fundamental example, saddle-node bifurcation, introduces the methods used to do so. Saddle-node bifurcation occurs at equilibrium points (x,)~) where Dx f has a simple eigenvalue zero. Thus (n + 1) defining equations in R n x IRk for saddle-node bifurcations are given by zeros of the map
f(x,)~) F(x, )~) =
)
det(Dxf)(x,)~)
"
If the system of equations F = 0 is regular, then Newton's method can be used to locate the points of saddle-node bifurcation. Consider the case in which there is a single parameter: k = 1. Assume that (x0,)~0) is an equilibrium point at which the defining equations are satisfied. The derivative of DF is then a square (n + 1) x (n + 1) matrix with block structure Dx f
DF =
Dx det(Dx f )
D)~ det(Dx f )
and Dx f singular. In order for this matrix to have full rank, Dx f must have rank at least n - 1 since the addition of a single row or column to a matrix increases its rank by at most 1. Here, DF can be obtained from Dx f by the successive addition of one column and one row, so the difference between the ranks of DF and Dx f is at most 2. If Dx f has rank (n - 1), then it has unique left and right eigenvectors w T, v ~ R 'Z up to scalar multiples. The regularity of DF implies that the products (w v O)DF and DF(o ) are nonzero, yielding that w D)~f :/: 0 and Dx(det(Dr f ) ) v 7/=O. The second of these equations is satisfied if wDxx f (V, v) :/: O. The inequalities w D ~ f :/: 0 and w D x x f (v, v) :/: 0 are nondegeneracy conditions for saddle-node bifurcation. Together with the assumption that Dx f has rank (n - 1), they give sufficient conditions that the defining equations F -- 0 for the saddle-node bifurcation are regular. Regularity of the defining equations for saddle-node bifurcation are not quite enough to characterize the dynamics of a family in the neighborhood of the bifurcation point. However, if the nondegeneracy conditions are strengthened to the requirement that the only eigenvalue on the imaginary axis at the bifurcation point is a simple zero eigenvalue, then the local dynamics near the bifurcation are determined up to topological equivalence and perturbations of the family will have topologically equivalent phase portraits in the neighborhood of the bifurcation. The following theorem summarizes this discussion.
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THEOREM 2. Let Yc = f (x, )~) be a smooth n-dimensional vector field depending upon a scalar parameter )~. Let (xo, )~o) be a solution of the system of equations
f (x,)~) = 0 , det(Dx f ) ( x , )~) = O.
This system of equations is regular at (x0,)~0) if (1) Dx f has rank n - 1. Denote the left and right zero eigenvectors of zero by v and w. (2) w Dxx f (v, v) =/:O. (3) w D z f =fi O. If properties (1)-(3)are satisfied, the curve y of equilibrium points f (x, )~) = 0 is smooth. Furthermore, if zero is a simple eigenvalue of Dx f and Dx f has no pure imaginary eigenvalues, then there is a neighborhood U of (xo, )~o) such that all trajectories that remain in U for all time are equilibrium points o n / . Singularity theory [7,67] provides a set of tools for analyzing the variation of equilibrium points with respect to parameters in generic families of vector fields. Before tackling the general theory, consider one more example, the family of scalar vector fields )c - - )~1 -+- ~.2X -~- X 3 9 For fixed ~-2 and varying ~.l, this family fails to satisfy the condition wDxx f ( v , v) ~ 0 when ~.2 = 0. However, as a two parameter family, the system of equations
f(x,~) =0, Dx f (x, ~.) = 0 , Dxx f (x, )~) = 0 is regular. The curve on which saddle-node bifurcation occurs in this family is obtained by eliminating x from the pair of equations ~.1 -q- ~.2X -Jr- X 3 - - 0
and
Df(x)
- - ~2 -+-
3x 2 - - 0 .
Parametrically, the curve is given by )~2 - - 3 x 2 and )~l - 2x3. This implies that ()~1/2) 2 -k(~,2/3) 3 - 0. Solutions of this equation form a cusp in the ()~1,)~2) parameter plane. It is not smooth at the origin. The example illustrates that the locus of local bifurcation in a generic multi-parameter family may not be a smooth manifold. Nonetheless, singularity theory gives a set of geometric tools that can be used to formulate regular systems of defining equations for local bifurcations. The theory is typically applied at the level of germs [67], but the description here avoids this language. Jets are objects that give coordinate-free expressions for the Taylor series of smooth maps between manifolds. The r-jet extension of a map f associates to each point of the domain of f the Taylor expansion of degree r, viewed as an object in a suitable jet space. Thom's transversality theorem [92] states that if P is a submanifold of a jet space, then any smooth map can be perturbed so that its jet extension is transverse to P. In the example of the cusp, the two-jet extension of the family of maps g(x, )~l, )~2) - )~l + ~.2x + x 3 is
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given by j2g(x, AI, A2) -- (g, g', g") = (s + A2X -q- X 3, A2 Jr- 3X 2, 6X) which vanishes at the origin. The Jacobian of J 2g at the origin is the non-singular matrix
(il 0) 0 0
1 0
.
Therefore, the cusp gives a family of maps whose two jet extension is transverse to the zero dimensional manifold consisting of the origin. Local bifurcations determined by smooth submanifolds of the jet spaces have regular systems of defining equations in those jet spaces. The transversality theorem implies that solutions of these defining equations yield smooth submanifolds of the product of parameter and phase spaces in generic families of vector fields. The Thom-Boardman stratification in singularity theory illustrates concretely how these procedures work. The T h o m - B o a r d m a n decomposition of a map g" R m ~ R'; is constructed as follows. Partition R m into the sets 27 / on which Dg has rank min(m, n) - i or corank i. For generic maps g, the sets 27i are submanifolds. The defining equations for 27i c a n be expressed (locally) in the space of 1-jets in terms of minors of Dg. Partition each of the 27i by restricting g to 27i and repeating the construction. This produces sets 2 7 i , j on which gl s; has rank j. For non-increasing sequences of integers (il, i2 . . . . . ik), Thorn defined 2 7 i l , i 2 ..... ik inductively as the set on which the derivative of g restricted to r i l ' i 2 ..... ik-I has corank i~. Boardman proved that, for generic maps g, these sets are submanifolds of R m . The saddlenodes and cusps described above correspond to the singularities 271 and 27 l, 1. Interest in the T h o m - B o a r d m a n stratification was motivated by its relationship to the stability of mappings. The groups of diffeomorphisms of R'" and R" act on C ~ (Rm, R';) by composition on the left and right: (h, k) 6 Diff ~ (R m) • Diff ~ (R") send g to hgk. If g is an interior point of its orbit with respect to this action, then it is stable. The action clearly preserves the T h o m - B o a r d m a n stratification, so transversality with respect to this stratification is necessary for stability. In a seminal series of papers [ 116-121], Mather formulated necessary and sufficient conditions for stability of a mapping. In some cases, transversality with respect to the T h o m - B o a r d m a n stratification is sufficient, but in other cases it is not. Local bifurcation theory seeks stratifications of the jet spaces of families of dynamical systems that are analogous to the T h o m - B o a r d m a n stratification. These stratifications are expected to give necessary conditions for the structural stability of a family, but they will give sufficient conditions in only a limited number of cases. The definition of a stratification used here is naive: a stratification of a closed set V is a sequence of closed subsets V - V1 D Vt-l D ... D V0 D V-i -- 0 such that each difference S i - - V i - Vi-1 is a smooth manifold of dimension i, called a stratum, or empty. The codimension of the stratum Si is k - i. Locally, there are regular systems of k - i defining equations that define Si as a subset of R k. These are defining equations for the bifurcations in Si. Ideally, Si has a finite number of components, each consisting of vector fields with similar properties near their bifurcation points. A k - i parameter family that is transverse to Si is an unfolding. A particular choice of a point in Si and an unfolding is a normal form. In the "best" circumstances, the normal forms are structurally stable k - i parameter families. Even when this is true, it can be difficult to prove and each case requires a separate analysis.
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The analyses of local bifurcations have tended to follow a common pattern. The first step is to identify submanifolds of the space of vector fields that fit into a stratification. These submanifolds should be preserved by topological equivalence or other equivalence relations that are used to describe when vector fields are qualitatively similar to one another. Once the submanifolds are identified, the next step is to choose normal forms for each submanifold. The choice of normal form is usually based upon polynomial coordinate transformations that simplify the analytic expression of the vector field near the bifurcation. The third task in the analysis of local bifurcations is to study the dynamics of the normal form families, seeking to establish their structural stability. The unfoldings of a bifurcation of codimension j will contain in their parameter spaces submanifolds of bifurcations of codimension smaller than j. Many of these lower codimension bifurcations are global bifurcations, making it awkward to maintain a separation between the theories of local and global bifurcation. The primary distinction from a computational perspective is that the defining equations for local bifurcations are formulated directly in terms of the Taylor series of the vector field rather than in terms of the flow of the vector field. Part of the bifurcation analysis is to identify geometric properties of how strata of smaller codimension limit on the codimension j bifurcation. When the normal forms do not produce structurally stable families, there are two possible scenarios. The first possibility is that a more refined analysis with normal forms of higher degree and additional nondegeneracy conditions on the normal form leads to a structurally stable family. The second possibility is that normal families defined by finite Taylor expansions never produce structurally stable families. As in some cases of double Hopf bifurcation, there may be an infinite number of families of bifurcations that intersect the neighborhood of a bifurcation of codimension j and no simple decomposition of the bifurcation set as a stratified set is possible. Takens-Bogdanov bifurcation [19,144] provides a good illustration of the analysis of a local bifurcation of codimension two. This bifurcation occurs at equilibrium points of a vector field for which zero is an eigenvalue of (algebraic) multiplicity two and no eigenvalues are pure imaginary. The defining equations can be expressed easily in terms of the characteristic polynomial of the Jacobian at an equilibrium. Near a point of Takens-Bogdanov bifurcation, the Taylor expansion of degree two for an unfolding can be transformed to .~ ~ y , - - ~1 +
ax 2 -+- Y()~2 -+- bx)
in the plane corresponding to the generalized eigenspace of zero. This is a normal form for Takens-Bogdanov bifurcation. The normal form is a structurally stable family and the phase portraits near the bifurcation are determined if neither a or b is zero. In the two dimensional parameter plane there are three bifurcation curves that meet at the TakensBogdanov point: a curve of saddle-node bifurcations that passes through the TB point, a curve of Hopf bifurcations that terminates at the TB point and a curve of homoclinic bifurcations that terminates at the TB point. These three curves meet with a quadratic tangency, and in the region of parameters between the Hopf and homoclinic bifurcation curves, the vector field has a periodic orbit. This picture illustrates that global bifurcations can appear in the neighborhood of local bifurcations.
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The classification of local bifurcations up to topological equivalence of their unfoldings is hardly complete, even for relatively low codimension. There are examples beginning with codimension two bifurcations of three dimensional vector fields in which chaotic dynamics appear in the unfoldings. These examples have an infinite number of bifurcation curves that terminate at the codimension two point in the parameter space, and the families are never structurally stable. Kuznetsov [ 106] gives a comprehensive summary of information about codimension one and two local bifurcations. Dumortier, Roussarie and Sotomayor [55,56] have analyzed codimension three local bifurcations of planar vector fields. Their work is the current frontier in attempts to systematically classify local bifurcations of increasing codimension. Similar principles to the ones discussed above apply to global bifurcations, but the defining equations are expressed in terms of the flow maps instead of directly in terms of the vector field. Chapter 4 of this volume contains more specific information about global codimension one and two bifurcations, including discussion of bifurcations of homoclinic and heteroclinic orbits and the numerical methods implemented in the HomCont package [27] that is part of the 1997 version of AUTO [50]. If flow maps and their derivatives can be computed accurately with numerical integration, then similar numerical methods can be used to compute bifurcations of periodic orbits. There are aspects of global bifurcations that have no counterparts in the theory of local bifurcation. One example is the breakdown of invariant tori. In generic two parameters of vector fields, invariant two dimensional tori with fixed irrational winding number may be present along curves in the parameter space. These parameter space curves corresponding to invariant tori with an irrational winding number may have endpoints beyond which the invariant torus "breaks down" into a Cantor set or a chaotic invariant set. The singularity theory based methods described above are inadequate to analyze the break down process. Renormalization methods that search for self-similar patterns in these phenomena have been used [60].
4.2. Continuation methods Continuation methods solve underdetermined systems of equations F = c,
F : R m -+ R '~,
with m > n. They usually assume F is smooth and and regular; i.e., DF has rank n on the level set of c. In these circumstances, the Implicit Function Theorem implies that the level set is a smooth manifold of dimension m - n whose tangent spaces are given by the null spaces of DF. Sard's Theorem [92] implies that for almost all c (with respect to Lebesgue measure in ~"), the level set is a manifold. Continuation algorithms are best developed when m = n + 1 and the solution manifolds are curves. Multiparameter continuation with m > n + 1 is not yet in widespread use, but remains an active research area [5,23]. The topological complexity of higher dimensional level sets has not been fully incorporated into robust algorithms. Single parameter continuation can be formulated as a combination of numerical integration and root finding. The goal is to compute the level curve F --c. On the set of regular
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points of F, one can define the line field that assigns to x the null space of DF(x). This line field can be represented by vector fields in a variety of ways: for example, as a unit vector field determined by an orientation of the null space or via a parametrization of the level curve in the form y ( x ) where (x, y) are coordinates on IR''+l with D y F a regular n • n minor of DF and Dxy -- - ( D y F ) - l Dx F. The integral curves of this vector field are level curves of F. Continuation methods exploit this fact to choose predicted steps along a level curve, but they then utilize root finding methods to refine these steps so that they once again satisfy F = c. Without this cycle of prediction and correction, numerically integrated curves will likely drift away from the level curve on which they start. The use of the initial prediction step (typically an Euler step that gives a tangent approximation to the level curve) helps pick seeds for iterative root finders that are close to the desired solutions. This is important when using a root finder like Newton's method that is not globally convergent. As with numerical integration, the choice of step length in a continuation method is important. Large step lengths tend to make the root finding less reliable or slower. Small step lengths take more steps to traverse a level set. Choosing unit vectors to parametrize the level curves leads to pseudo-arclength continuation. Fixed step sizes yield points along the curve that are approximately equidistant. If the level curve has tight folds with areas of large curvature, then pseudo-arclength continuation is likely to require very small steps. Therefore, adaptive strategies typically monitor the curvature of the level curve and adjust the step length to control the estimated error from each prediction step. The final choice that needs to be made in implementing a continuation method is the choice of subspace in which to perform the root finding. To obtain a "square" system, the original system of equations is constrained to a hypersurface on which F is regular. Common choices are to fix one coordinate; i.e., use a subspace parallel to a coordinate subspace, or to use the hypersurface that is orthogonal to the continuation step. Continuation methods have been extremely useful in the study of dynamical systems. Here, we examine their use in computing information about local bifurcations. Consider the system of differential equations .;c = f (x, ~.)
with x 9 R n and )~ 9 IRk. Local bifurcations locate parameter values ~, at which equilibria of this dynamical system have qualitative changes. The Transversality Theorem [92] implies that the equilibrium set of f is a smooth manifold for generic f . In this case, continuation methods can be used to compute the equilibrium manifold. As we compute the set of equilibria f = 0 with continuation, we expect to occasionally see bifurcations along the branch. These occur when Dx f is singular or has eigenvalues along the imaginary axis. Thus the problem of computing local bifurcations consists of a continuation problem together with solving additional equations that explicitly depend upon the derivatives of f . Bifurcations of high codimension serve as "organizing centers" where multiple types of lower codimension bifurcations meet. The branching patterns of many bifurcations of codimension 2, 3 and 4 have been analyzed by first computing their normal forms and then studying the dynamics exhibited by the normal form families. Single parameter continuation has been used to locate and identify high codimension bifurcations with the following strategy, implemented in CONTENT [107]. In computing branches of equilibria for a
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generic dynamical system with one active parameter, one expects to meet saddle-node and Hopf bifurcations. These are detected by evaluating a function that changes sign at the bifurcation. When a bifurcation point is located, a new continuation can be started to follow the bifurcation curves. The defining equation is added to the equilibrium equations and a second parameter is made active, producing n + 1 equations in n + 2 variables. At selected points along these bifurcation curves, codimension two bifurcations may be encountered and detected by evaluation of suitable functions. When this happens, a new continuation is established with a pair of defining equations for the codimension two bifurcation and three active parameters. This process can bootstrap from codimension j to codimension j + 1 bifurcations as long as explicit defining equations for the bifurcations have been formulated and the root finding converges. The package CONTENT implements computations of all local codimension two bifurcations of vector fields and discrete maps (and much more as well). There are important cases in which we want to study systems whose equilibrium sets are not manifolds. For example, systems that are equivariant with respect to a symmetry group are common in varied applications. Equivariance can force the zero level set of a vector field or family of vector fields to have singularities. This complicates the computation of bifurcations substantially. The analysis of these systems is framed in terms of group theoretic concepts. Dynamical analysis of the normal forms of even moderately complex normal forms of symmetric systems is incomplete [61 ]. Computation of the equilibria and local bifurcations in these systems can require substantial amounts of algebra [ 153]. Gatermann [65] and Sanders [133] have made initial steps towards the construction of general software for the computation and analysis of normal forms of symmetric systems. Continuation methods have been used to track curves of periodic orbits as well as equilibria. AUTO [49] implements continuation methods superimposed on collocation algorithms for periodic orbits. The basic advantages of using a continuation method to compute periodic orbits is that initial conditions close to the desired orbit are used for each point along the continuation path after the first. Thus, convergence of the method to the desired orbit is much more likely than with random or fixed data to start each periodic orbit calculation, and fewer calculations are required at each step along the continuation path. There are circumstances in which the use of a global boundary value solver like that employed in AUTO offers additional advantages when coupled with continuation. First, unstable periodic orbits can be computed. As an iterative method, the algorithm has stable fixed points corresponding to all approximate periodic orbits, not only those that are attracting in the flow. With suitable procedures for choosing dependent and independent parameters in the root finding, curves can be followed around folds in which the periodic orbits do not vary smoothly with the parameters. Second, in problems with multiple time scales, one finds families of stable periodic orbits that cannot be computed readily with numerical integration. The canard example in Section 3.1 shows periodic orbits that can be computed with boundary value solvers but not with numerical integration.
4.3. Numerical methods for computing bifurcations Bifurcation theory provides a mathematical foundation for algorithms that locate bifurcations in specific families. Implementation of methods based upon singularity theory en-
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counters three types of numerical issues: (1) Formulation of regular systems of defining equations. (2) Accurate evaluation of defining equations that depend upon derivatives of a vector field. (3) Numerical condition number of the defining equations, especially for large systems and systems with multiple time scales. These issues can be viewed from both theoretical and practical perspectives. Practically, the most desirable numerical methods are those that give accurate answers for large classes of interesting systems. There are several important choices that enter the construction of software for computing bifurcations, so the potential number of distinct methods is large. Yet, different methods are seldom compared carefully with one another. There have been few attempts to gather suites of test problems in this domain as there has been for numerical integration [95]. Picking parameters in bifurcation algorithms that make methods work well remains an art. Thus, opinions of different methods tend to be very subjective, based upon the experience of users and the skill they develop in adjusting algorithmic parameters when a method initially fails. Regular systems of defining equations for saddle-node bifurcations are presented above in terms of the determinant of the Jacobian of a vector field. This choice of defining function is natural from a theoretical perspective, but may lead to numerical problems. If the Jacobian has eigenvalues of large magnitude, then these eigenvalues contribute to the condition number of the determinant and may make it difficult to satisfy the defining equations to a desired tolerance. The larger the system, the worse this problem becomes. Thus there are circumstances in which it is desirable to seek alternate defining equations for saddlenode bifurcation that avoid calculation of the determinant. The singular matrices do not form a smooth submanifold of the space IR'1x,l of n x n matrices, so there is no regular function whose values measure the distance of a matrix from being singular. However, the corank one matrices are a smooth hypersurface in It~n xn. Saddle-node bifurcations occur at matrices of corank one, and this fact can be used in the formulation of defining equations. The following result about bordered matrices is the basis for one method. THEOREM 3 [70]. Let A be an n x n matrix that has a single eigenvalue zero. For most choices o f n vectors B and C and scalar D the (n + 1) x (n + 1) block matrix
M--
Ct
D
is nonsingular. There are constants cl > 0 and C2 and a neighborhood U o f A so that if A ~ U with smallest singular value or, then the unique solution (u t, v) o f the system o f equations m
.4
satisfies c l o <
B
Ivl <
u
C2 O'.
(1) 0
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Applying the theorem with A the Jacobian of the vector field gives the quantity v as a measure of the distance of the Jacobian from the set of singular matrices. The system of linear equations can be solved using Gaussian elimination with partial pivoting, an algorithm that is efficient and reliable for most systems. Moreover, when v is zero, u is the right zero eigenvector of A, an object needed to compute the normal form of the bifurcation. High dimensional vector fields often have sparse Jacobians. For these, iterative methods can be used to compute the solution of the system of linear equations, avoiding the need to calculate a full factorization of the matrix M. Thus, this method is feasible for discretized systems of partial differential equations for which computation of the determinant of the Jacobian can hardly be done. A second approach to computing saddle-node bifurcations is to rely upon numerical methods for computing low dimensional invariant subspaces of a matrix. Subspace iteration and Arnoldi methods [ 142] are effective techniques for identifying invariant subspaces that are associated with the eigenvalues of largest magnitude for a matrix. Inverse iterations can be used in this framework to identify invariant subspaces associated with eigenvalues close to the origin. Cayley transforms [64,122] extend these methods to compute invariant subspaces for any cluster of eigenvalues on the Riemann sphere. This can be especially useful in finding Hopf bifurcations, but subspaces associated to eigenvalues of large magnitude on the imaginary axis cannot be readily separated from subspaces associated with negative eigenvalues of large magnitude. If appropriate invariant subspaces are computed, then the bifurcation calculations can be reduced to these subspaces. On the remaining small problems, the choice of function that vanishes on singular matrices matters less than it does for large problems. Deriving explicit defining equations for bifurcations other than saddle-nodes requires additional effort. For example, Hopf bifurcation occurs when the Jacobian at an equilibrium has a pair of pure imaginary eigenvalues. There is no familiar function that vanishes when a matrix has pure imaginary eigenvalues analogous to the determinant for zero eigenvalues. Guckenheimer et al. [83] described algebraic procedures that produce single augmenting equations analogous to the determinant and the bordered matrix equation for saddle-node bifurcation in Section 4.1. The algebraic equation can be derived from the characteristic polynomial of the Jacobian. A determinant, the Sylvester resultant of two polynomials constructed from the characteristic polynomial, vanishes if and only if the Jacobian matrix has a pair of eigenvalues whose sum is zero. There are two ways in which a real matrix can have a pair of eigenvalues whose sum is zero: they can be real or they can be pure imaginary. There is an explicit algebraic inequality in the coefficients of the characteristic polynomial that distinguishes these two cases. While the formulas that arise from this analysis are suitable for computations with low dimensional systems, they rapidly become unwieldy as the dimension of a vector field grows. They suffer from all of the problems associated with the use of the determinant as a defining equation for saddle-node bifurcations as well as the additional difficulty that computations of the characteristic polynomial tend to suffer from numerical instability [ 150]. Tensor products yield a procedure for computing Hopf bifurcations without forming the characteristic polynomial of a matrix. Given n x n matrices A and B, their tensor product is an n 2 x n 2 matrix A | B whose eigenvalues are the products of the eigenvalues of A and B. Therefore, the eigenvalues of the matrix C = A | I + I @ A are sums of pairs of the
384
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eigenvalues of A. Moreover, C can be decomposed into a symmetric part that commutes with the involution u | v ~ v | u and a skewsymmetric part that anticommutes with this involution. The skewsymmetric part is an n ( n - 1)/2 x n ( n - 1)/2 matrix (called the b i p r o d u c t of A) whose eigenvalues are the sums of distinct eigenvalues of A. Therefore, A has a single pair of eigenvalues whose sum is zero if and only if its biproduct has corank one. Applying the bordered matrix construction described above to the biproduct gives a defining function for A to have a single pair of eigenvalues whose sum is zero. Govaerts et al. [69] studied the Jordan decomposition of the biproduct of matrices with multiple pairs of eigenvalues whose sum was zero and used a bordering construction to implement a system of defining equations for double Hopf bifurcation. The methods described above for computing saddle-node and Hopf bifurcations construct minimal augmentations of the defining equations. There are alternative methods that introduce additional independent variables and utilize larger systems of defining equations. For example, in the case of Hopf bifurcation, many methods solve for the pure imaginary Hopf eigenvalues and eigenvectors associated with these. In addition to the equilibrium equations, one method solves the equations D f v = cow and D f w - - - c o y for vectors v and w as well as the eigenvalue ico [ 128]. To make this system of equations regular, additional equations that normalize v and w are required. The complexity of the expressions appearing in these defining equations is reduced compared to that of minimal augmentation methods. This advantage is offset by the expense of having larger systems to solve with root finding and the necessity of finding initial seeds for the auxiliary variables. Guckenheimer and Myers [82] give a list of methods for computing Hopf bifurcations and a comparison between their method and the one of Roose and Hvalacek [ 128]. The defining equations of local bifurcations include derivatives of f . In the cases of some bifurcations of codimension two and larger, the expressions for these defining equations are very complex and involve higher derivatives of f . Consequently, accuracy and efficient evaluation of the defining equations is important. Automatic differentiation [ 15] provides methods for the accurate evaluation of the derivatives themselves that avoids truncation errors inherent in finite difference formulas. General expressions for defining equations of some types of bifurcations have been derived only recently, so only a small amount of testing has been done with computation of these bifurcations [82]. The description of high codimension singularities of maps has proceeded farther than the description of high codimension bifurcations of dynamical systems. The thesis of Xiang [ 154] contains results that surmount a technical difficulty in implementing the computation of Thom-Boardman singularities [ 18]. The problem is that the singularities are defined by equations on submanifolds of a domain: ~ i l ,i2 ..... ik is the set on which the map restricted to ~__il,i2 ..... ik-l has corank ik. The corank conditions can be expressed in terms of minors of the derivative of the restricted map, but numerical computations only yield approximations to ~__~il ,i2 ..... ik-I. These approximations do not automatically produce good approximations of tangent spaces and regular systems of defining equations. Xiang [ 154] altered the construction of defining equations to produce a regular systems of equations for ~ i l , i 2 ..... ik defined in neighborhoods of ~ i l , i 2 ..... i k - I . These methods were tested with seven dimensional stable maps containing ,~v,2,1 singularities, the smallest example of stable maps in which this difficulty arises.
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[141] S. Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, A Symposium in Honor of Marston Morse, Princeton Univ. Press, Princeton, NJ (1965), 63-80. [142] D.C. Sorensen, The k-step Arnoldi process, Large-Scale Numerical Optimization (Ithaca, NY, 1989), SIAM, Philadelphia, PA (1990), 228-237. [143] A.M. Stuart and A.R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge Univ. Press, Cambridge (1996). [1441 E Takens, Forced oscillations and bifurcations, Comm. Math. Inst. Rijkuniversiteit Utrecht 3 (1974), 1-59. [1451 E Takens, Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), Springer, Berlin (1981), 366-381. [1461 W. Tucker, The Lorenz attractor exists, Thesis, Uppsala (1998). [1471 M. van Veldhuizen, A new algorithm for the numerical approximation of an invariant curve, SIAM J. Sci. Statist. Comput. 8 (1987), 951-962. [148] E Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott Foresman, Glenview, IL (1971). [149] A. Weigend and N. Gershenfeld, Time Series Prediction, Addison-Wesley, Readings, MA (1994). [150] J. Wilkinson, The Algebraic Eigenvalue Problem, Oxford Univ. Press, Oxford (1965). [151] J. Wisdom, Two lectures on chaotic dynamics in the solar system, Nonlinear Evolution and Chaotic Phenomena (Noto, 1987), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 176, Plenum, New York (1988), 185-187. [152] A. Wolf et al., Determining Lyapunov exponents from a time series, Phys. D 16 (1985), 285-317. [1531 R Worfolk, Zeros of equivariant vector fields: algorithms for an invariant approach, J. Symbolic Comput. 17 (1994), 487-511. [154] Y. Xiang, Computing Thom-Boardman singularities, Thesis, Cornell University (1998). [155] J.-C. Yoccoz, Recent developments in dynamics, Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (ZUrich, 1994), Birkh~iuser, Basel (1995), 246-265. [156] L.S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems 2 (1982), 109-124. [157] L.S. Young, Entropy, Lyapunov exponents, and Hausdorff dimension in differentiable dynamical systems, IEEE Trans. Circuits and Systems 30 (1983), 599-607. [158] L.-S. Young, Ergodic theory of attractors, Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (ZUrich, 1994), Birkh~iuser, Basel (1995), 1230-1237.
CHAPTER
9
Conley Index Konstantin Mischaikow* Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, GA 30332, USA E-mail: mischaik@ math.gatech, edu
Marian Mrozek t lnstytut lnforma~ki, Uniwersytet Jagielloriski, 30-072 Krak6w, Poland E-mail: mrozek@ ii. uj. edu.pl
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Decompositions of invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Attractor-repeller pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Morse decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Conley's decomposition theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
395 399 399 402 405 407
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3.1. Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. 3.3. 4. The 4.1. 4.2. 4.3. 4.4. 4.5.
Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . index and the structure of invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The structure of Morse decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The topology of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symbolic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fast slow systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Multivalued dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
407 413 420 420 420 437 442 442 445 447 447
5.1. Multivalued maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
447
5.2. Representable multivalued maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Approximation and inheritance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
448 449
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5.5. Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *Research supported in part by NSF grant DMS-9805584. +Research supported in part by KBN, Grant 2 P03A 029 12 and 2 P03A 011 18. H A N D B O O K OF D Y N A M I C A L S Y S T E M S , VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 393
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5.6. C o m p u t a b i l i t y of the index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
453
5.7. R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
455
6. E x a m p l e s of c o m p u t e r assisted proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. L o r e n z equations 6.2. T h e H 6 n o n m a p
455
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
457
6.3. K u r a m o t o - S i v a s h i n s k y equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4. R e f e r e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References
455
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
457 458 458
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1. Introduction Given the goals and constraints of this Handbook a complete self contained introduction to the Conley index theory is not possible. Therefore, these notes will focus on that part of the theory which is most closely related to the study of the dynamics of differential equations. We will work in the context of flows on locally compact metric spaces. In this setting most of the essential ideas can be explained without too much technical difficulty. Of course the disadvantage to this approach is that the results are not directly applicable to the dynamics generated by partial and functional differential equations. There are two responses to this. First, having read this survey the reader is encouraged to turn to Rybakowski's book [74] where the theory is generalized to semi-flows on infinite dimensional spaces. Second, many physically interesting infinite dimensional systems are dissipative and contain global compact attractors. Furthermore, upon restricting the dynamics to these attractors one typically has a flow. This, of course, is precisely the setting of the index theory described herein. With this in mind we adopt the convention that throughout this paper X will denote a locally compact metric space with metric #. Furthermore, q9 :R x X --~ X will represent a flow; that is a continuous map satisfying: ~0(0, x) = x,
~0(t, ~0(~, x)) - ~(t + s, x). A set S C X is an invariant set for the flow q9 if qg(•, S ) ' - - U qg(t, S) - S. t6•
Much of the qualitative theory of differential equations involves the study of the existence and structure of invariant sets. There are at least four difficulties associated with the investigation of nonlinear systems: (1) Invariant sets are global objects, and therefore, their study involves obtaining global estimates of a nonlinear problem. (2) Invariant sets possess an infinite variety of different structures some of which can be extremely complicated (chaotic dynamics and fractal structures). (3) The structure of invariant sets can change dramatically with respect to perturbations (bifurcation theory, normal forms, catastrophe theory); (4) The bifurcation points need not be isolated (structurally stable systems are not dense). The fact that for a fixed family of differential equations all of these problems must be considered simultaneously makes the analysis of nonlinear systems difficult. Consider for instance the simplest example of an invariant set, that of a fixed point or equilibrium, i.e., a point x E X such that qg(R, x) = x. For many differential equations proving the existence of fixed points either analytically or numerically is a nontrivial task. Degree theory is by now a standard tool for this type of problem. The degree of a map can be thought of as a function from a set of continuous maps to the integers with three important properties:
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(1) It is defined in terms of a region for which there are no fixed points on the boundary. (2) If the degree of a map is nonzero, then the map possesses a fixed point. (3) Degree is a continuous function. To successfully apply this theory requires a few essential steps. Direct computation of the degree of a map is often quite difficult. Therefore, the first step is to choose a continuous family of maps going from the map of interest to a simpler map for which the fixed points are explicitly known. One then chooses a region containing one or more equilibria of the simple map for which the resulting degree is nonzero. The third step typically involves analysis; one must show that each function in the continuous family of maps does not possess a fixed point on the boundary of the region. Having done this one observes that since the degree is integer valued and continuous, it cannot change throughout the continuous family of maps. Therefore, the degree of the original map of interest is the same as that of the simpler map. In particular the degree is nonzero, and hence, the original map has a fixed point. The power of this approach arises from the fact that one does not need to study the actual behavior of the map on the interior of the region. This short digression into degree theory was included since the application of the Conley index typically involves a similar process. The regions of interest are called isolating neighborhoods. A compact set N C X is an isolating neighborhood if Inv(N, 9)"-- {x e N I ~o(IR,x) C N} C intN, where int N denotes the interior of N. S is an isolated invariant set if S = Inv(N) for some isolating neighborhood N. The most important property of an isolating neighborhood is that it is robust with respect to perturbation. To state this more precisely consider a continuous family of dynamical systems ~oz:IRxX--+X,
)~E[-1,1].
(1)
PROPOSITION 1.1. Let N be an isolating neighborhood for the flow 99o. Then, for sufficiently small 3 > O, N is an isolating neighborhood for all qgz, [)~1< 3. The proof of this proposition is fairly straight forward; the essential ideas being as follows. Since N is an isolating neighborhood, x 6 ON implies that x r Inv(N, qg0). Equivalently, there exists a time tx > 0 such that qg([-tx, tx], x) ~ N. The compactness of N implies that T :-- maxxcaN tx < c~. By continuity of the flows, one may approximate qg0 over the time interval [ - T , T] arbitrarily closely by choosing 6 sufficiently small. This robustness of isolating neighborhoods with respect to perturbations leads to the following definition. DEFINITION 1.2. Let N C X be a compact set. Let Sz = Inv(N, ~oz). Two isolated invariant sets Szo and Sz~ are related by continuation or Sz o continues to Sz~ if N is an isolating neighborhood for all ~oz, )~ c [)~o, )~l]. Observe that in itself the definition of continuation is about isolating neighborhoods. It says nothing about the associated isolated invariant set. For example if Sz0 is a degenerate
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fixed point, then it typically is related by continuation to the isolated invariant set consisting of the empty set. It is in this sense that the inherent difficulties of bifurcation theory are avoided. Of course what we are interested in is the structure and properties of invariant sets not isolating neighborhoods. Therefore we need a means by which we can pass from knowledge of isolating neighborhoods to an understanding of its associated invariant set. This is the purpose of the Conley index. The index will be defined in Section 3. However, for the moment, consider its important characteristics: (1) The Conley index is an index of isolating neighborhoods. Furthermore, if N and N' are isolating neighborhoods for the flow q9 and Inv(N, 99) = Inv(N', qg), then the Conley index of N is the same as the Conley index of N'. Observe, that this implies that we can also consider the Conley index as an index of isolated invariant sets. We shall make use of both interpretations depending on which is most convenient. (2) (Was Property) If the Conley index of N is not trivial, then Inv N r 0. (3) (Continuation) If N is an isolating neighborhood for a continuous parameterized family of flows qgz, )~ 6 [0, 1], that is Inv(N, ~p~) C int N
for k 6 [0, 1],
then the Conley index of N under ~P0 is the same as the Conley index of N under 991. The applicability of the Conley index depends essentially on these three properties. The first gives great freedom in the choice of regions in phase space on which one will perform the analysis. The second allows one to pass from the isolating neighborhood to an understanding of the dynamics of the isolated invariant set. The Wa2ewski property, while the most fundamental, is the simplest result of this type. Section 4 contains a variety of more sophisticated theorems which can been used to prove the existence of connecting orbits, periodic orbits, and even chaotic dynamics in the sense of symbolic dynamics. The third property is important for the following reason. The Conley index is a purely topological index and as such it is a very coarse measure of the dynamics. Typically, if it can be computed directly at a particular parameter value, then one's knowledge of the dynamics at that parameter value is reasonably complete. The power of the index (as in degree theory) comes from being able to continue it to a parameter value where ones understanding of the dynamics is much less complete. So far the discussion of the Conley index has been restricted to dynamical systems generated by flows. However, the index theory can also be extended to the setting of maps. Even in the context of differential equations this extension is important for at least two reasons. First, there are many situations in which the flow admits a Poincar6 section for which one can study the dynamics of the associated Poincar6 map. As will be made clear in Section 4.2, the Conley index for the Poincar6 map carries essentially more information than the index for the flow. The second reason arises from the fact that numerical approximations to the flow can take the form of maps. Again, to avoid as many technical details as possible we will consider discrete dynamical systems that are generated by homeomorphisms f : X --+ X. If one begins with a flow qg:IK x X --+ X, then natural way to generate a discrete dynamical system is to take the time r map of the flow f : X --+ X defined by f (x) = qg(r, x).
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Several ideas from the Conley index for flows carry over directly to the discrete case. The most fundamental is that of an isolating neighborhood. Given N C X the maximal invariant set of N is defined by Inv(N, f ) : = {x ~ N l f n ( x ) ~ N for alln E Z}. A compact set N is called an isolating neighborhood if Inv(N, f ) C int N and a set S is an isolated invariant set if there exists an isolating neighborhood N such that S = Inv(N, f ) . Another set of ideas that are identical in the continuous and discrete case involves the notion of decompositions of isolated invariant sets. These issues are discussed in Section 2 where the results are all stated in the context of flows. The reader, however, should feel confident in replacing flow by homeomorphism throughout that section. The real difference between the setting of flows and homeomorphisms comes in the definition of the index. This will be explained in greater detail in Section 3.2. However, the essential difference is that if f : X --+ X is the time r map of a flow as defined above, then f is homotopic to the identity map on X. This is not the case for an arbitrary homeomorphism. Never the less, the Conley index for maps as described in these notes is a true generalization of the Conley index for flows. If S is an isolated invariant set for a flow ~0, then S will be an isolated invariant set for the associated time r map, and furthermore, the Conley indices of these two objects will agree. In Section 5, yet another, framework for the index theory will be developed; that of multivalued maps. While the theory is relatively new, its inspiration is not. One of the fundamental theorems in dynamical systems is the decomposition theorem of Conley which states that any compact invariant set can be divided into its chain recurrent part and the rest. Furthermore, on the latter part one can define a strictly decreasing Lyapunov function. These two portions are therefore, precisely the sets on which one has recurrent dynamics and gradient-like dynamics. This result appears as Theorem 2.19 of these notes. However, in the definition of the chain recurrent set one moves forward by the flow for a minimal time and then is allowed to make an error of size e > 0. This is the simplest form of a multivalued map; points get sent to sets, in this case the e neighborhood around the true image. A slightly more complicated method for generating multivalued maps is to use a numerical scheme to approximate the trajectories of a differential equation and to keep track of error estimates or, better yet, error bounds. Observe that this produces a map that provides an a priori outer estimate for the dynamics of the differential equation. Furthermore, if the numerical estimates are good, then the set of possible single valued maps that are approximated by the multivalued map will be close to one another in the C o norm. However, this is precisely the setting in which the Conley index continues. Therefore, one might expect that the Conley theory for multivalued maps might provide a means by which numerical computations could be used to obtain index information for differential equations. This is in fact the case as will be made clear in Section 5. The final section contains a sample of the types of numerical results that have been obtain. We conclude this introduction by returning to the comments in the opening paragraph. The goal of these notes is to indicate how the Conley index theory can be used to understand the structure of dynamics arising from applications. This precludes a discussion of the important influence that Conley's ideas had in geometry and topology, in particular in
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the development of Floer homology and the proof of the Arnold conjecture. For this the reader is referred to Conley and Zehnder's original work [15,16], the series of papers by Floer [ 19-23] or some recent books on the subject [38,66].
2. Decompositions of invariant sets As was indicated earlier the purpose of the Conley index theory is to understand the structure of isolated invariant sets. To describe this structure requires the ability to decompose invariant sets into subinvariant sets. Since one of the properties of the index theory is its stability under perturbation, it is important that the method of decomposition also be stable with respect to perturbation.
2.1. Attractor-repeller pairs The coarsest decomposition of an invariant set is that of an attractor-repeller pair. Our discussion begins with some standard definitions from the theory of dynamical systems. Let Y C X. The omega limit set of Y is
n d(+(t,,
+> .=
r))
t>O
while the alpha limit set of Y is
, ( r ) - . ( r . +):= N
r)).
t>O
DEFINITION 2.1. Let S be a compact invariant set. A C S is an attractor in S if there exists a neighborhood U of A such that
co(U n S) = A. The dual repeller of A in S is R "= {x 6 S lw(x) n A - - 0}. The pair (A, R) is called an attractor-repeller pair decomposition of S. The following properties of attractors and repellers are easy to check. PROPOSITION 2.2. Let S be an isolated invariant set. Let (A, R) be an attractor-repeller pair decomposition for S. Then (1) A and R are isolated invariant sets; (2) if A' is an attractor in A, then A' is an attractor in S.
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EXAMPLE 2.3. Consider the ordinary differential equation m y,
= cy + (X
-x2),
c6R,
)~ 6 [ - 1 , 1].
Let N = [ - x , x] x [ - x , x] C R 2 9 It is fairly easy to show that for sufficiently large x, N is an isolating neighborhood for all c 6 R and )~ E [ - 1, 1]. Let Sc,Z = Inv(N) for the given parameter values. Define H(x, y) := y2/2 + x3/3 - )~x, then d H / dt = cy 2. So for c # 0, H acts as a Lyapunov function. For the moment let us restrict our attention to c > 0 and 0 < )~ ~< 1. Then there are two equilibria in N, (-+-~, 0). Since H(Vc~, 0) > H(-~/-~, 0) we can conclude that ((~/~, 0), ( - ~ / ~ , 0)) is an attractor repeller pair decomposition of Sc,x. From the definition of an attractor-repeller pair decomposition it is clear that if x 6 S then it is impossible for co(x) C R and c~(x) C A, i.e. there can be no connecting orbits from the attractor to the repeller. Of course, connections in the other direction may exist. The set of connecting orbits from R to A in S is denoted by
C(R, A" S)"-- {x ~ S Iw(x) C A, or(x) C R}. The following result, though trivial, is important enough to be designated a theorem. THEOREM 2.4. Let (A, R) be an attractor-repeller pair decomposition of S. Then
S = A U R U C(R, A; S). In Example 2.3 an attractor-repeller pair decomposition was constructed. As will be demonstrated in Section 4.1, in appropriate situations the Conley index can be used to show that C (R, A; S) # 0, i.e., that connecting orbits exist. THEOREM
2.5. Attractor-repeller pair decompositions continue.
PROOF. Assume that So is an isolated invariant set with isolating neighborhood N. Then, by Proposition 1.1 there exists )~s > 0 such that N is an isolating neighborhood for all qgz, )~ 6 [-)~s, )~s]. Let (A0, R0) denote an attractor repeller pair decomposition of So. Since A0 and R0 are isolated invariant sets, they have isolating neighborhoods NA C N and NR C N. Furthermore, there exists ~,A > 0 and )~R > 0 such that NA is an isolating neighborhood for )~ E [--)~Z, )~A] and NR is an isolating neighborhood for )~ 6 [-)~R, )~R]. Let )~0 = min{)~s, )~A, )~S}. Thus, for all )~ E [-)~0, )~0], the Ax := Inv(NA, ~0x) are related by continuation, the Rz := Inv(NR, qgx) are related by continuation, and the Sz are related by continuation. The final point that needs to be checked is that there exists 0 < )~1 ~< )~0 such that for all ~. E [--~,1, ~-1], (Ax, Rz) is an attractor repeller pair for Sz. Assume not. Then there exists a sequence of parameter values )~n --+ 0 and points Xn ~ Sz,, such that xn ~ Az,, W Rz,, and co(xn) ~ Az,,. Invoking the continuity of the family of flows and the fact that (A0, R0) is
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an attractor repeller pair decomposition of So gives a contradiction. Thus, there exists an interval [--~-1, ~1] over which the attractor repeller pair decomposition continues. U] EXAMPLE 2.6. Let us return now to Example 2.3 and examine the question of continuation. To begin with, since N is an isolating neighborhood for all values of c 6 • and )~ 6 [ - 1 , 1] the corresponding isolated invariant sets Sc,~ are related by continuation. Furthermore, if we restrict as above to c > 0 and 0 < )~ ~< 1, then the attractor repeller pair decomposition ((V~, 0), ( - x / ~ , 0)) also continues. However, it does not continue over the full parameter space of c c • and )~ 6 [ - 1 , 1]. In particular, the attractor repeller pair decomposition breaks down in two ways in this example. The first is that at )~ = 0 the two equilibria merge and for )~ < 0, Sc.~ -- 0. The second way occurs for )~ > 0 when c -- 0. In this case the system is Hamiltonian and the integral curves consisting of elements of Sc,~ extend from (x/~, 0) to ( - v / ~ , 0). In the above examples a Lyapunov function was used to obtain the existence of an attractor-repeller pair decomposition. In fact, as the following theorem indicates they are in some sense equivalent concepts. THEOREM 2.7. Let S be a compact invariant setwith attractor repellerpair (A, R). Then, there exists a continuous function V : S ----~ [0, 1]
such that: (i) V - ' (1) = R, (ii) V-~(O) = A, (iii) for x ~ C(R, A) and t > O, V(x) > V(~p(t,x)). PROOF. Observe that if A or R equals the empty set, then the result is trivially true. So assume that A :/: 0 # R. The proof is now broken down into three steps. The first is to define a function f :S --+ [0, 1] by
f(x) =
#(x,A) #(x,A)+#(x,R)
Clearly, (since A A R = 0) f is continuous, f - l (0) = A, and f - l ( 1 ) -- R. Second, define g : S --+ [0, 1] by
g(x)t/>0
x))}.
Again, it is obvious that g-1 (0) = A, and g - I (1) = R and g(~o(t,x)) <<,g(x) for all t ~> 0. It is left to the reader to show that g is continuous. The third and final step is to define
V(x) -
f
o~
e-~ g(~o(~,x))d~.
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Clearly (i) and (ii) are satisfied. If x ~ C(R, A), then
V(x)-
V(x,t) =
f0
e-~g(qg(~,x))d~-
-
x)) -
f0
e-~g(qg(~ + t , x ) ) d ~ + t, x ) ) ]
o.
Now V(x) - V(qg(t,x)) - - 0 if and only if g(qg(~,x)) - g(qg(~ + t , x ) ) = 0 for all ~ ~> 0, i.e., g(qg(~, x)) = ~ a constant for all ~ ~> 0. This implies that co(x) n (A U R) -- 0, a contradiction. Thus V(x) - V({p(t, x)) > 0, and condition (iii) is satisfied. D
2.2. Morse decompositions EXAMPLE 2.8. Consider the following modification of Example 2.3. m
y,
= cy +
(x 2 -
1)(X -Jr 1/2),
C > 0.
(2)
As before, N = [ - x , x] • [ - x , K] C ]R 2 is an isolating neighborhood for large x. This system still possesses a Lyapunov function H ( x , y) "= y 2 / 2 - ( x 4 / 4 + x 3 / 6 - x Z / 2 - x / 2 ) . However, in this case there are three equilibria ( - 1 , 0 ) , ( - 1 / 2 , 0), (1, 0) and it would be nice to decompose the isolated invariant set Sc into three isolated subsets. This requires a generalization of the concept of an attractor repeller pair. In an attractor repeller pair (A, R) decomposition of S there is a natural total ordering that can be imposed on these two isolated invariant subsets given by R > A, implying that there is no connecting orbit from A to R. If one has a multitude of isolated invariant subsets, then it is more natural to use partial orders. A partial order I on a set 79 is a relation > which satisfies: (i) p > p never holds for p 6 79. (ii) I f p > q a n d q > r t h e n p > r . If, in addition, the partial order satisfies (iii) for all p, q E 79, either p > q or q > p. Then > is called a total order. From now on (79, >) will be used to denote afinite indexing set 79 with a partial order >. DEFINITION 2.9. A finite collection
,A4(S) - { M ( p ) I p ~ 79} 1 Actually w e are defining a use the above definition.
strict partial order, however, to save the use of an unnecessary adjective we shall
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403
of disjoint compact invariant subsets of S is a Morse decomposition if there exists a partial order > on the indexing set 7J such that for every
x6S\
UM(p) pc T~
there exists p, q E 7) such that p > q and
co(x) C M(q)
and
oe(x) C M(p).
The sets M(p) are called Morse sets. Observe that it is not assumed that there is unique order on 7~. In fact, any ordering on 7) with the above property is called admissible. Having chosen an admissible order > we shall often write
M(S)-
{M(p) I p ~ (79 , >)}.
Strictly speaking this is an abuse of notation since the Morse decomposition only refers to the collection of Morse sets. In applications, however, the construction of a Morse decomposition typically involves knowledge of a Lyapunov function, which can be used to define an admissible order. EXAMPLE 2.10. Returning to Example 2.8, let M(3) := ( - 1 / 2 , 0), M(2) = ( - 1 , 0) and M (1) := (1, 0). Then, for c > 0 we have the Morse decomposition
A d ( S c ) - {M(i) l i -
1 , 2 , 3 , 3 > 2 > 1},
where the ordering is determined by the Lyapunov function H. Observe that > is an admissible ordering for all c > 0. This is not, however, the only partial ordering that is admissible. Consider equation (2) at the two extreme parameter values c = 0 and c -- oo. At c -- 0 we have a Hamiltonian system where the set of bounded solutions has two components: one consisting of the point (1, 0); and the other containing ( - 1,0), ( - 1/2, 0), a variety of periodic orbits, and an orbit homoclinic to ( - 1, 0). From this one can conclude that for 0 < c << 1 there can be no connecting orbit between (1, 0) and the other two equilibria. Therefore, an admissible order is 3 >0 2 with 1 unrelated to either 2 or 3. A simple rescaling of time allows one to write down ~ = - ( x 2 - 1)(x + 1/2) as the limiting equation as c --+ oc. For this equation it is easy to check that the set of connecting orbits consist of heteroclinic orbits from ( - 1/2, 0) to ( + 1, 0). Thus, for c sufficiently large one can use the ordering 3 > oc 2 and 3 > oc 1. In both these examples the order obtained by an explicit study of the flow at a given parameter value is different from that obtained using the Lyapunov function. In particular, > contains more relations that either >0 and > oc, and therefore, less information about the structure of the dynamics. This example leads to the observation that while there may be many admissible flows for a given Morse decomposition, there is, a unique minimal (in the sense of the number
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of order relations) admissible order which is called the flow defined order. Recall that an order > ' on 7~ is an extension of > if p > q implies p > ' q. Thus, any admissible order must be an extension of the flow defined order. In applications one typically does not know the flow defined order, and therefore, must work with an extension. As in the case of attractor repeller pair decompositions, Morse decompositions are strongly related to the existence of Lyapunov functions. The following result is a direct generalization of Theorem 2.7. THEOREM 2.1 1. Let S be an isolated invariant set. Let
M ( S ) = {M(p) I p ~ 7~} be a finite collection of disjoint invariant compact subsets of S. Then, ,All(S) is a Morse decomposition of S if and only if there exists a continuous function V" S --+ [0, 1] such that: V x, y ~ M(p)
implies
V ( x ) - - V(y)
x ~ S \ U M(p)
implies
V(x) > V(~p(t,x))
Vt>O.
pEP
Observe that given a Morse decomposition WI(S) = {M(p) ] p 6 (7J, >)} one can always coarsen it. To do this systematically we introduce the following notation. A subset I C 7~ is called an interval if p, q 6 I and p > r > q implies that r 6 I. The set of intervals on (7~, >) will be denoted by 2-(7J, >). PROPOSITION 2.12. Let I ~ 2"(7~, >) and define
u
(u
pEI
p,qEl
Then, M ( I ) is an isolated invariant set. The proof is fairly straight forward and follows from the compactness of S. PROPOSITION 2.13. All(S) - {M(p) [ p E "P\l} U {M(I)} defines a Morse decomposition of S. Furthermore, an admissible partial order >' is given by
p>tq p>t I I>~p
~,
p>qif if if
p, q ~ 7 ~ \ l ,
3 q E I such that p > q, 3 q 6 I such thatq > p.
This section began with the statement that Morse decompositions are generalizations of attractor-repeller pair decompositions. To see this observe that an attractor-repeller pair decomposition (A, R) of S defines a Morse decomposition
M(S)-
{M(p)]p-
1,2, 2 > 1}
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405
where M (1) = A and M (2) = R. In the converse direction, within a Morse decomposition there are many attractor repeller pairs. To make this precise define an adjacent pair of intervals in (79, >) to be an ordered pair (I, J ) of mutually disjoint intervals satisfying: (i) I U J E 2-(79, >), (ii) p E I, q E J implies that p ~ q. The collection of adjacent pairs is denoted by 2-2(79, >). If (I, J ) is an adjacent pair, then setlJ ::lUJ. LEMMA 2.14. If (I, J ) E I2('P, >), then ( M ( 1 ) , M ( J ) ) is an attractor-repeller pair in M(IJ). EXAMPLE 2.15. Returning to Example 2.10, consider c ~ cx~. The flow defined order is 3 > ~ 2 and 3 > ~ 1. Thus (2, 3), (3, 1) E 2-2(79, > ~ ) . Therefore, (M(2), M(3)) and (M(1), M(3)) form attractor-repeller pair decompositions of M(23) and M(13), respectively. Knowing that Morse decompositions can be coarsened it is natural to ask whether given an isolated invariant set S, there exists a finest Morse decomposition. The following example shows that the answer is no. EXAMPLE 2.16. Consider the equation .~ -- X2 sin
zr X
Let N = [ - 2 , 2]. Then, Inv N -- S = [ - 1 , 1] is a compact isolated invariant set. We can define Morse decompositions as follows. Let
.A4,,(S) "= { + l / k l k - - 1 . . . . ,n - 1} U { [ - I / n , l / n ] } . For each n this is a Morse decomposition with 2n - 1 Morse sets and as n increases the Morse decomposition becomes more refined. Since, by definition, a Morse decomposition can only consist of a finite number of Morse sets there is no finest Morse decomposition.
2.3. Conley's decomposition theorem As was indicated by Example 2.16 there need not be a finest Morse decomposition. However, one can still ask the following question. Given a compact invariant set S is there a finest collection of subsets of S off of which one can define a Lyapunov function? One of the fundamental theorems of dynamical systems states that the answer is yes. To present it the notion of chain recurrence is essential. An (e, r) chain from x to y is a finite sequence
{(xi,ti)} C X X ['t', CX3),
i -- 1 . . . . . n,
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such that x = xl, ti >/"c, #(qg(ti,xi),Xi+l) <~ e and #(qg(tn,Xn), y) <~ e. If there exists an (e, r) chain from x to y, then we write x ___(e,r) Y. If x ___(e,r) Y for all (e, r), then x >- y. DEFINITION 2.17. The chain recurrent set of X under the flow ~0 is defined by ~ ( x , ~o) - {x ~ X lx • x}.
The chain recurrent set is fundamental for two reasons. First, it is minimal in the sense of the following theorem. THEOREM 2.18 (Conley [7]). I f S is a compact invariant set, then
~ ( ~ ( s , ~o), ~ol~r
= ~ ( s , ~o).
Second, it captures all the recurrent dynamics as is indicated by the following theorem. THEOREM 2.19 (Conley [7]). Let ~.i (S), i = 1,2 . . . . . denote the connected components o f ~ ( S ) . Let S be a compact invariant set. Then there exists a continuous function V" S--+ [0, 1] such that: (i) if x q~ ~ ( S ) and t > O, then V ( x ) > V(~p(t,x)); (ii) f o r each i = 1, 2 . . . . . there exists cri ~ [0, 1] such that ~i more, the {oi} can be chosen such that ai =fi crj if i =/: j.
C V-1 (tyi) and, further-
Unfortunately, for direct applications the components of the chain recurrent set are difficult to work with since they are not robust with respect to perturbation. Furthermore, while the individual components are invariant sets they need not be isolated invariant sets. EXAMPLE 2.20. Returning to Example 2.16
~(S)-
{1
-4--In~Z
/
U{0},
n
and hence, has an infinite number of components. In addition, observe that 0 is a component of 7~(S), but is not an isolated invariant set. Now consider the parameterized system ~"
--
x 2
sin --+)~(x rc 2-1),
~[-1,1],
x
with corresponding flow qgz. Again, choose N - [ - 2 , 2] and let Sz " - Inv(~0z). Then, for ~, --fi0, 7~(Sz) has only finitely many components.
Conley index
407
2.4. References The material of this section is by now classical. The most complete reference is Conley's monograph [7], though Salamon's article [75] is also recommended for its clarity of proof. A proof due to Robinson of Theorem 2.19 for compact invariant sets in the setting of maps can be found in [72]. See also the exposition in [73]. In the non compact case there is work by Hurley [39,40].
3. Conley index In the previous section we discussed the decomposition of isolated invariant sets into Morse decompositions. Furthermore, we provided a series of simple examples in which we were able to determine the existence of an attractor-repeller pair or of a Morse decomposition, but we were unable to say anything about the existence or structure of connecting orbits between these invariant sets. To do this we will need to use the Conley index.
3.1. Flows Before providing a definition of the index we recall a few elementary notations from topology. DEFINITION 3.1. A pointed space (Y, Yo) is a topological space Y with a distinguished point y0 E Y. Given a pair (N, L) of spaces with L C N,
N / L : = ( N \ L ) U[L] where [L] denotes the equivalence class of points in L in the equivalence relation: x -~ y if and only if x = y or x, y 6 L. In the sequel we will usually use N/L to denote the pointed space (N/L, [L]). The topology on (N/L, [L]) is defined as follows: a set U C N \ L is open if U is open in N and U n L = 0, or the set (U n (N \ L)) U L is open in N. If L = 0, then
(N/L, [L])"-- (N U {.}, {.}) where {.} denotes the equivalence class consisting of the empty set. Let (X, x0) and (Y, y0) be pointed topological spaces and let f, g: (X, x0) --+ (Y, Y0) be continuous functions. Implicit in this notation is the assumption that f (x0) = g(xo) = y0.
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DEFINITION 3.2. f is homotopic to g, denoted by f ~ g, if there exists a continuous function F :X • [0, 1] --+ Y such that F (x, 0) = f (x),
F(x, 1) = g(x), F(xo, s) = Yo,
0~<s~
Obviously ~ is an equivalence relation. The equivalence class of f in this relation is called the homotopy class of f and denoted [f]. DEFINITION 3.3. Two pointed topological spaces (X, x0) and (Y, y0) are homotopic (X, xo) ~ (Y, Yo) if there exists f : (X, x0) ~ (Y, Y0) and g : (Y, Y0) --+ (X, x0) such that fog'~idy
and
gof~idx.
Observe that homotopy defines an equivalence class on the set of topological spaces. The reader for whom homotopy is a new concept may wish to check that ~ 2 \ {0} '~ S 1 . We now turn to the definition of the Conley index. Recall that r is a flow on X a locally compact metric space. As was indicated in the introduction, even though the Conley index is an index for isolating neighborhoods we shall define it in terms of the isolated invariant set. DEFINITION 3.4. Let S be an isolated invariant set. A pair of compact sets (N, L) where L C N is called an index pair for S if: (1) S = Inv(cl(N \ L)) and N \ L is a neighborhood of S. (2) L is positively invariant in N; that is given x ~ L and 99([0, t],x) C N, then r t], x) C L. (3) L is an exit set for N; that is given x ~ N and tl > 0 such that r x) ~ N, then there exists to 6 [0, tl] for which r to], x) C N and r x) 6 L. The following theorem gives the existence of index pairs. THEOREM 3.5. Given an isolated invariant set S, there exists an index pair. Isolating neighborhoods and index pairs are very general objects; and as such useful for computational purposes since they are relatively easy to find. A much stricter notion is that of an isolating block. This is a compact set B such that
B-:=
B I
is closed and for all T > 0 InvT(B, ~o) C int B,
T),
r B, VT > 0},
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409
where InvT(B, ~o):= {x 6 B I ~o([-T, T], x) C B }. THEOREM 3.6. Given an isolated invariant set S and a smooth flow, there exists an isolating block B such that S -- Inv B. Furthermore, (B, B - ) is an index pair for S. DEFINITION 3.7. The homotopy Conley index of S is
h(S) = h(S, ~o) ~ ( N / L , [L]). Observe that the Conley index of S has been defined in terms of any index pair. Furthermore, typically an isolated invariant set possesses a multitude of isolating neighborhoods. Therefore one needs the following theorem. THEOREM 3.8 (The Conley index is well defined). Let (N, L) and (N', L') be indexpairs for an isolated invariant set S. Then
( N / L , [L]) ~ (N'/L', [L']). Observe from the definition that the homotopy Conley index is the homotopy type of a topological space. Unfortunately, working with homotopy classes of spaces is extremely difficult. To get around this, it is useful to consider the homological Conley index defined by
CH,(S) := H , ( N / L , [L]) ~ H,(N, L). Since the homology of two homotopic spaces is the same, the homology index is well defined. REMARK 3.9. It is not true that given any index pair (N, L), H , ( N / L , [L]) ~ H , ( N , L). However, one can prove that one can always find index pairs for which this isomorphism holds. Since for some of the algebra it is more convenient to work with the pair (N, L) directly, rather than the quotient space we will hence forth assume that we are working with these types of index pairs. Finally, we state the continuation theorem for the Conley index. THEOREM 3.10 (Continuation Property). Let Sz o and Sz o be isolated invariant sets that are related by continuation. Then,
CH,(Szo) ~ CH,(Sz, ). Let us now consider some examples of the Conley index.
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K. Mischaikow and M. Mrozek
EXAMPLE 3.1 1. Observe that the empty set 0 is vacuously an isolated invariant set. Furthermore, (0, 0) is an index pair for the empty set. Thus CH, (0) ~ O.
The contrapositive of this example, though trivial, has significant enough implications to be designated as a theorem. THEOREM 3.12 (Wa2ewski Property). Let N be an isolating neighborhood and assume that C H , (Inv N) ~ O. Then, Inv N ~ 0. This result provides the simplest example of an existence result which can be obtained via the Conley index. It also demonstrates an important point concerning the way one wishes to view the Conley index. The computation of the index was done with knowledge of the invariant set. The theorem of interest was stated in terms of the isolating neighborhood. After the empty set, the simplest isolated invariant sets are hyperbolic fixed points. THEOREM 3.13. Let S be a hyperbolic fixed point with an unstable manifold of dimension n. Then CHk ( S)
|Z I0
i f k = n, otherwise.
PROOF. Let 2 = f ( x ) , x E R n+m, be the ordinary differential equation for which S is a hyperbolic fixed point. By the Hartman-Grobman theorem, the flow in a neighborhood of S is topologically equivalent to the flow in a neighborhood of the origin of -- D f (S)y.
(3)
Thus it suffices to compute the Conley index of the origin under this linear dynamics. A linear change of variables transforms (3) to
[z 0 z, 0 lEz2] where A is an m • m matrix for which the real parts of all its eigenvalues are less that zero and B is an n x n matrix for which the real parts of all its eigenvalues are greater that zero. An isolating neighborhood of the origin is given by [ - 1 , 1]m x [ - 1 , 1],7. The exit set is given by [ - 1 , 1]m x 0 ( [ - 1 , 1]n). The result now follows from computing H. ( [ - 1 , 1] m x [ - 1 , 1]", [ - 1 , 1] m • 0 ( [ - 1 , 1In)). D EXAMPLE 3.14. Consider Example 2.3. Linearizing about the equilibria and counting the number of eigenvalues with positive real part one can check that for c > 0, k = 1, ((-z, 0)) {0z ifotherwise,
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411
and CHk ( ()~
,0))
/ Z
if k -- 2,
!0
otherwise.
EXAMPLE 3.15. The same type of argument applied to Example 2.8 gives CHk ((--I-1,0)) ~ { OZ
if k - 1, otherwise,
CHk((1/2, 0)) ~ { Z 0
if k = 2, otherwise.
and
The Thom isomorphism theorem [4,49,77] provides a general means of computing the Conley index for normally hyperbolic invariant sets. THEOREM 3.16. Assume that a manifold S is a normally hyperbolic invariant set. Let E be the vector bundle over S defined by the local unstable manifold of S. If E is a rank n orientable bundle and one uses homology with field coefficients, then CHk(S) ~ Hk+n(S).
As a corollary we obtain the following result. COROLLARY 3.17. Let S be a hyperbolic invariant set that is diffeomorphic to a circle. Assume that S has an oriented unstable manifold of dimension n + 1. Then CHk ( S)
[Z I0
i f k = n , n + l, otherwise.
EXAMPLE 3.18. Observe that from Corollary 3.17 the dynamics on the isolated invariant set S plays no role with respect to C H , ( S ) . In particular, it implies that the index of S is the same whether S is a hyperbolic periodic orbit; a circle on which there are two hyperbolic fixed points connected by two heteroclinic orbits; or even a circle consisting entirely of fixed points. One can obtain the same conclusion by a continuation argument. However, the point is that given an isolating neighborhood N with maximal invariant set S -- Inv N and
CHk ( S) ~
Z
ifk-n,n
0
otherwise,
+ l,
one may not conclude that S contains a periodic orbit. This is an important issue. Because the Conley index remains invariant under continuation, in general, the index alone does not provide sufficient information to draw conclusions about the structure of the invariant set. The results described in Section 4 are devoted to
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this problem. What will be shown is that index information plus some knowledge about the dynamics on N can, in many cases, be used to obtain reasonably strong conclusions concerning the dynamics of S. In particular, Theorem 4.20 states that if S has the Conley index of a periodic orbit and that the isolating neighborhood possesses a Poincar6 section, then S must contain a periodic orbit. To apply Theorem 3.16 one needs to be able to determine the orientability of the unstable manifold of S. For a general nonlinear differential equation, doing this analytically is extremely difficult. For this reason, the index theory is typically computed using Z2 coefficients. In this case the Thorn isomorphism theorem implies the following result. THEOREM 3.19. Assume that a manifold S is a normally hyperbolic invariant set. Let E be the vector bundle over S defined by the local unstable manifold o f S. I f E is a rank n bundle, then CHk(S; Z2) ~'
Hk+n(S, Z2).
While these theorems provide nice abstract means of determining the Conley index, in applications the most common approach is to perform a continuation to a simple system. EXAMPLE 3.20. Returning to Example 2.3, recall that N is an isolating neighborhood for all ,k E [ - 1, 1]. At X = - 1, Sc,- 1 = 0 and therefore, CH, (Sc,z) ~, O. EXAMPLE 3.21. For Example 2.8 we wish to compute the index of Sc. Consider the one parameter family of equations
.~
m
y,
(4) ) -- cy 4- (1 - ~ . ) ( x 2 - 1)(x 4- 1/2)4- X ( x -
1),
c>0.
N is an isolating neighborhood for all ~, 6 [0, 1]. Furthermore, at ~, = 0 we have Equation (2) while at ~. = 1 we have a system for which Inv N consists of a single hyperbolic fixed point with a one dimensional unstable manifold. Therefore, CHk(Sc)
,~,[Z I0
if k - l , otherwise.
The following theorem is fundamental to many of the most significant applications of the index theory to date. THEOREM 3.22 (Summation Property). Assume S = So U S1 is an isolated invariant set where So and $1 are disjoint invariant sets. Then C H , ( S ) ~ CH,(So) EDCH,(S1).
Conley index
413
PROOF. Since So and S1 are disjoint invariant sets, there exist disjoint isolating neighborhoods No and Nl such that (No, L0) and (Nl, L1) are index pairs for So and Sl respectively. Thus, C H , ( S ) -- H , ( N o U NI, L0 U LI) -- H , ( N o , LO) | H , ( N I , L I ) -- C H , ( S o ) 9 CH,(SI).
U]
EXAMPLE 3.23. Consider yet again Example 2.3. The index computations of Examples 3.14 and 3.20 allow us to apply Theorem 3.22 to conclude that S 7~ {(iX, 0)}. Therefore, Theorem 2.4 implies that there exists a heteroclinic orbit from (-X, 0) to 0v, 0).
3.2. Maps As was indicated in the introduction the Conley index can be defined for discrete dynamical systems, i.e., for continuous maps f ' X --+ X where X is a locally compact metric space. As in the case of flows one begins with the notion of an index pair. DEFINITION 3.24. A pair (N, L) of compact subsets of X is called an index pair for an isolated invariant set S if L C N and (1) cl(NXL) is an isolating neighborhood isolating S, (2) f (L) N N C L (positive invariance), (3) f (NXL) C N (exit set). THEOREM 3.25. For any neighborhood V of an isolated invariant set S there exists an indexpair (N, L) for S such that NXL C V. So far the Conley index theory in the discrete case mimics the flow case. To construct the index we need to take two index pairs (N, L) and (N', L') for the same isolated invariant set S and show that they carry some common information. In the flow case the most general common information is the homotopy type of the quotient space [N/L] and the required homotopies are built along the trajectories of the flow. The difficulty in generalizing the Conley index to maps lies in the lack of such homotopies. Actually things are even worse: the homotopies needed in the construction of the Conley index for flows may not exists at all in the discrete case. Consider the following two examples. EXAMPLE 3.26. Let f : R --+ R be given by f ( x ) := x + 1. For every n E N put L,, := [n, n + 1/2] and N,, := U{Li I i = 0, l, 2 . . . . . n}. One can easily verify that for every n c N the pair (N,, L,) is an index pair for f such that N,, \L,, isolates the empty set. On the other hand [N,,/L,,, [L,, ]] is the homotopy type of a pointed set of n elements. EXAMPLE 3.27 (Smale's Horseshoe). Let N := [0, 1] • [0, 1] C R 2. Assume f : R 2 --+ R 2 is a continuous map such that f maps two rectangles R0 and R l linearly onto rectangles So and Sl, as indicated in Figure 1. Assume also that f maps NX(Ro U Rl) into ]R2XN and 1~2\N into It~2\(80 U S1).
K. Mischaikow and M. Mrozek
414
D
C
R1
>
so
s~
R0
A
B A'
B'
C'
D'
Fig. 1. The U horseshoe.
Put
C n ' - - { o t l / 5 + ' " + O t n / 5 n + o t / 5 n+l Ioti e {1,3}, i - - 1 , 2 . . . . . n, a 6 [0, 1]}, Ln := [0, 1] • cl([O, I]\C,,). Then S "-- Inv N C ["]{Cn • Cn In -- 1, 2 . . . . }. One can easily verify that (N, Ln) is an index pair for S for all n e N and [N/Ln, [Ln ]] is the homotopy type of the wedge sum of 2 n copies of a pointed circle. The examples show that unlike the flow case, the homotopy type of the quotient space [N/L] of an index pair (N, L) is not an invariant of S. This can raise the question if the Conley index in the discrete case makes sense at all. Surprisingly it does, and the theory incorporates the index for flows as a special case. To see how this can happen we need to introduce the index map. Given an index pair (N, L), the index map is the map fN,L : N / L ~ N / L defined by
fN,L([x]) " - - { f L ( ] )
if f ( x ) e N , otherwise.
(5)
The following proposition follows easily from the definition of the index pair. PROPOSITION 3.28. For any index pair (N, L) the index map f N,L is continuous. Index maps turn out to be essential in extracting common information from index pairs of the same isolated invariant set. This may be done in several different ways depending on how general the information we want to extract is. Recently Franks and Richeson [24] proposed to use shift equivalence. Assume c : C --+ C and d : D --+ D are continuous maps. They are shift equivalent if there exist continuous maps r : C --+ D, s : D ~ C and a natural number m such that rc --
Conley index
415
= c m and sr = d m . We say that the homotopy classes of c and d are shift equivalent if there exist continuous maps r : C --~ D, s: D --+ C and a natural number m such that rc "~ dr, s d ~ cs, rs ~ c m and sr ~ d m . It is straightforward to verify that shift equivalence is an equivalence relation.
dr, s d = cs, rs
THEOREM 3.29. I f (N, L) and (N', L') are index pairs f o r an isolated invariant set S, then the homotopy classes o f index maps [fN,L] and [ f N',L'] are shift equivalent.
We define h(S, f ) , the homotopy Conley index of S as the shift equivalence class of [ f u , c ]. The correctness of this definition is justified by Theorem 3.29. One can prove that the Conley index for discrete dynamical systems has the same basic properties as the Conley index for flows. In particular it has the Wa2ewski Property, Continuation Property and Summation Property. The homotopy Conley index it is the most difficult to compute, because we do not know much about shift equivalence in the homotopy category of compact metric spaces. Therefore in applications we usually work with algebraic Conley indices, which may be obtained from the homotopy index by applying algebraic functors like the homology or cohomology functor. In the sequel we will study the simplest case when the homology coefficients are in Q and the phase space X is a compact ANR (for instance a manifold or a polyhedron). Under such assumptions one can show that in every neighborhood of an isolated invariant set there exist index pairs (N, L) such that H , ( N , L) is a finite dimensional vector space. Two endomorphisms of finite dimensional vector spaces e: E --+ E and f : F --+ F are shift equivalent if there exist linear maps r: E ~ E, s: F --+ F and a natural number m such that re -- f r, s f = es, rs -- e m and sr = f m . Applying the homology functor with rational coefficients to the homotopy Conley index we obtain the homology Conley index Con,(S, f ) as the shift equivalence of H , ( f x , c ) . Such shift equivalences may be found easily by means of the Leray reduction. Let e : E --+ E be an endomorphism of a finitely dimensional vector space. The generalized kernel of e is defined by gker(e)
" - -
U le-,, (O) ln ~ 1~}.
Since e(gker(e)) C gker(e), we have an induced isomorphism e"E/gker(e)
9 [x] --+ [e(x)] E E l gker(e).
We call e' the Leray reduction of e. PROPOSITION 3.30. I f e' is the Leray reduction o f an endomorphism e: E --+ E o f a finite dimensional vector space then e and e' are shift equivalent. PROPOSITION 3.31. A s s u m e e: E --+ E and f : F --+ F are two automorphisms o f finite dimensional vector spaces. They are shift equivalent if and only if they are conjugate. Since the Leray reduction of a finite dimensional endomorphism is obviously an automorphism, we have the following theorem.
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K. Mischaikow and M. Mrozek
THEOREM 3.32. Two endomorphisms e and f of finite dimensional vector spaces are shift equivalent if and only if their Leray reductions are conjugate. By the above theorem we may consider the homological Conley index for maps on compact ANR's as a pair Con(S, f ) = (CH,(S, f ) , x , ( S , f ) ) , where CH,(S, f ) = {CHn (S, f)} is a graded finite dimensional space equal to
H , ( N / L , [L])/gker(fN,L), for any index pair (N, L) such that H , ( N / L , [L]) is finite dimensional and x,(S, f ) = {Xn(S, f)} is a graded automorphism on CH,(S, f ) equal to the Leray reduction of
(fN,L),. Consider a hyperbolic fixed point x0 6 ]1~n of a C l-diffeomorphism f :]Kn ~ ]Kn. Let k denote the number of eigenvalues of D f (xo) out of the unit circle (counted with multiplicity). Let 1 denote the number of real eigenvalues of Df(x0) which are less than - 1 . The pair (k, l) will be referred to as the Morse index of x0. THEOREM 3.33. Assume xo is a hyperbolic fixedpoint of a Cl-diffeomorphism R n --+ R n. Then {x0} is an isolated invariant set and
Coni({xo})- { 0(Q, ( -
1) lid)
for i~: k, for i -- k.
(6)
PROOF. Without loss of generality we may assume that x0 = 0. We will first consider the case when f is linear, i.e., f ( x ) = Ax for some A 6 ]1~nxn . Let ]1~n = U ~) V be the decomposition of ]1~n into eigenspaces of A corresponding to eigenvalues with modulus greater than one and less than one, respectively. Let
A
c~
be the corresponding decomposition of A. From the decomposition it follows that {0} is the only compact trajectory and consequently it is an isolated invariant set. In order to prove (6) in the linear case first assume that A has n different eigenvalues. Choose a basis in which A has the block-diagonal matrix with one-dimensional blocks [~j ] corresponding to the real eigenvalues )~j and two-dimensional blocks
I rj c o s ~oj --rj sinqgj
rj sin qgj ] rj cos ~0j
corresponding to the pairs of complex eigenvalues rj e x p ( - i 99j) and rj exp(i qgj). For t [0, 1] let At be the matrix of the same block-diagonal structure and with corresponding blocks of the form
[h(Xj, t)]
and
h(rj, t)
[ cost~oj - sint~oj
sin t~oj ] cost~oj '
Conley index
417
where
h(u, t)"-- tu + (1 - t ) s g n ( u ) e x p ( s g n ( l u l - 1)ln2). Then the origin is the only non-trivial isolated invariant set with respect to each At and the homotopy property of the Conley index shows that Con({0}, A l) = Con({0}, A0). Thus we may assume that A = A0, i.e., A has a diagonal matrix in which - 2 appears 1 times, 2 appears k - l times and the remaining non-zero entries are 1/2 or - 1/2. Put N "-- {x + y e U 9 V • ]R"-k I Ilxll ~ 2, IlYll ~ 1}, L "-- { x + y 6 N I I I x l l > l } , B "-- {x E g I Ilxll ~< 1},
S'-
{xEglllxll-1},
It is easy to verify that (N, L) is an index pair of {0}. Let the mappings
d : (B, S) --> (B, S), o t : ( B , S ) --+ (N,L) be given by d(u, v) := (u, - v ) , a ( x ) := (x, 0). One can easily verify that the following diagram
N/L
fN,L> N/L
d
B/S
l
> B/S
is commutative up to a homotopy. Thus we get the following diagram in homology
H,(N/L) (fN,L~ H,(N/L)
H,(B/S)
d,
> H,(8/S).
It is straightforward to verify that or. and d. are isomorphisms. Therefore Con({O} f ) = ,
(H,(B/S) d,) - { 0 ,
(Q, ( _ 1 ) / i d )
for/#k, for/=k.
418
K. Mischaikow and M. Mrozek
In order to obtain formula (6) in case of a linear map with multiple eigenvalues it suffices to construct a homotopyjoining the map A with a nearby map A' which has pairwise different eigenvalues and then apply the homotopy property. Now we shall consider the case of a general f. The fact that also in this case {0} is an isolated invariant set follows directly from the linear case and the H a r t m a n - G r o b m a n theorem. Let A := D f (0). We have
f (x) = A x + r(x), where r(x) --
o(llxll). For ,k E [0, 1] define
fz (x) := A x + ~r (x ) . For each fixed ~. 6 [0, 1] we can apply the Hartman-Grobman theorem to find 6()~) > 0 such that B(6()~)) := {x 6 IRn: Ilxll ~< ~()0} is an isolating neighborhood with respect to f)~, isolating {0}. A compactness argument shows that there exists a 6 > 0 such that B(6) is an isolating neighborhood with respect to fz for each ~. 6 [0, 1]. Obviously we can make small enough to ensure that Inv(B(6), f ) = {0}. Since fl = f, f0 = A, the thesis follows now from the homotopy invariance of the Conley index and the proved linear case. [2 THEOREM 3.34. Let xo ~ ]l~n be a hyperbolic periodic orbit of f , i.e., a hyperbolic fixed point of f d for some d E N. Let (k, l) be the Morse index of xo with respect to f d . Assume d is the minimal period of xo, i.e., f i (xo) 7~ xo for i = 1, 2 . . . . . d - 1. Then S := {x0, f (xo) . . . . . f d - l ( x o ) } is an isolated invariant set and Coni (S)
: {0
(Qd, D)
for i 7~ k,
(7)
for i -- k,
where D : Q d --+ Qd is given on the canonical basis {ei } i - - 1 , d of Q d by D(ei)--ei+l
fori=
l,2 ..... d-l,
D(ed)=(--1)lel. PROOF. Choose (N, L), an index pair of {x0} with respect to g := f d . Put x i :-- f i (xo), N i :__ f i (N), L i := f i (L) for i = 0, 1, . . . , d - 1. Taking N smaller if necessary, we may assume that N i 71N j -- 0 for i r j and f ( N d - l ) A N = 0. Obviously x i are hyperbolic fixed points of g with the same Morse index (k, 1). One can easily verify that (N i , L i) for i = 0, 1 . . . . . d - 1 is an index pair for g. It follows that N t := ~ i d--01 Ni is an isolating neighborhood with respect to f and (N', L') with
L' "-- U di =-0 1 L i is an index pair for f which isolates {x ~ x 1' H , ( N ' / L ' ) -- H , ( N ~
"'"
x d - l }" We have
O) G "" 9 H , ( N d - 1 / L d - 1 ) .
Hence it follows from Theorem 3.33 applied to g that H i ( N ' / L ' ) is zero for i ~ k and Hk (N ~/ L') is a d-dimensional vector space.
Conley index
419
Choose arbitrarily a generator a ~ 6 H k ( N ~ ~ and define recursively a sequence {oti}i=l,d_l of generators in Hk(N i / L i) by a i + l := Hk(f)(oti) for i = 0, 1 . . . . . d - 2. Then {ei}i=O,d_l with 8.i : : (0 . . . . . oti . . . . . 0) is a basis of H k ( N ' / U ) . It is straightforward to verify that (fN, L,)k(S i) = S i+l for i = 0, 1,2 . . . . . d - 2. We have also by (6) that -
_
0 .....
0)
= ((-1)lot ~ 0 . . . . . 0) -- ( - 1 ) l s ~ This shows that (fu',L'). is an isomorphism and formula (7) holds. EXAMPLE 3.35. Consider again the horseshoe map from Example 3.27. Take 1
3
4
Then (N, L) is an index pair for f and ( f u , L ) . has the matrix 1 1
-1 11 "
It follows that (fu,L)2 _ 0 and consequently the homological Conley index of the classical Smale's horseshoe is zero. EXAMPLE 3.36. This is not the case for the isolated invariant set S of the horseshoe in Figure 2. (N, L) as above is again an index pair but ( f u , c ) . has the matrix 1
I'
1
1]"
D
C R1 SO
S1
R0 A
B A~
Fig. 2. The G horseshoe.
B~
K. Mischaikow and M. Mrozek
420 Since in this
case
( f N , L ) 2 -- 2 ( f N , L ) , , it follows that
Conk(S) - / 0 / (Q, 2id)
fork#l, f o r k = 1.
3.3. References The original development of the Conley index for flows took place for the most part in the seventies. Again [7] and [75] are excellent sources. Early work includes that of Conley and Easton on isolating blocks [9]. Since that time the theory has undergone several developments. One important direction is the generalization to semi-flows on arbitrary metric spaces. For this one should consult the work of Rybakowski [74], Benci [2] and Benci and Degiovanni [3]. The first version of the Conley index for maps was presented by Robbin and Salamon [70]. Their definition of an index pair is more general than that presented here in that they only require that the compact pair (N, L) possess the property that the induced index map f u , c : N / L --+ N / L be continuous. The index based on the Leray reduction was proposed in [62]. In this paper it was also observed that the index map gives rise to an index automorphism which is also an invariant of the isolated invariant set. Szymczak [78] constructed the homotopy Conley index as a functor into an abstract category and showed that any other Conley index can be factorized through his homotopy Conley index. Recently Franks and Richeson [24] presented a version of Szymczak's approach based on shift equivalences.
4. The index and the structure of invariant sets
4.1. The structure of Morse decompositions In order to fully describe the dynamics of an isolated invariant set for which we have a Morse decomposition we need to understand the connecting orbits between the Morse sets. This will be done by relating the Conley indices of Morse sets to the Conley index of the total isolated invariant set. The construction of the algebraic machinery needed to do this in the general setting of Morse decompositions is rather formidable and will not be presented in these notes. However, to give the reader a flavor of the issues involved we shall indicate in some detail what happens in the setting of an attractor repeller pair decomposition. For the sake of simplicity we will restrict our discussion to the setting of flows. 4.1.1. Attractor repeller pairs. The goal of this section is to show that it is possible to reduce the answer to the question of the existence of a connecting orbit in Example 2.6 to a problem in linear algebra. More precisely we will show that given an isolated invariant set S with an attractor-repeller pair decomposition (A, R), the relationship between their Conley indices can be expressed in terms of a matrix called the connection matrix. The first step is to be able to determine the indices in a consistent manner.
Conley index
421
DEFINITION 4.1. Let S be an isolated invariant set and let (A, R) be an attractor-repeller pair decomposition. An index triple for (A, R) is a collection of compact sets (N2, Nl, No) where No C N1 G N2 such that: (1) (N2, No) is an index pair for S; (2) (N2, N1) is an index pair for R; (3) (Nl, No) is an index pair for A. THEOREM 4.2. Let (A, R) be an attractor-repeller p a i r decomposition o f an isolated invariant set S. Then, there exists an index triple (N2, Nl, No). PROOF. By Theorem 3.5 there exists an index pair (N2, No) for S. It is left to the reader to check that there exists a closed neighborhood U of A and a constant e > 0 such that if x 9 U and ~0([0, t], x) C N2 then #(~0([0, t], x), R) > e. Define Z "-- {y c N2 13 t > 0 x 9 U such that ~0([0, t l , x ) C N2 and ~o(t,x) - y}. Let Nl := No U Z. Then (N2, N1, No) is the desired index triple.
V]
Since No C N1 C N2 there exists on the chain level a short exact sequence 0 --+ C , ( N l , No) --+ C,(N2, No) --+ C , ( N 1 , No) --+ 0
which gives rise to the long exact sequence
9" --+ H n ( N I , NO) --+ H~,(N2, NO) --+ HI,(N1, NO) 2_~ Hn-1 (N2, N1) --+ " " .
(8) But this is equivalent to 9.. --+ CH,, (A) --+ CH,, (S) --+ CH,, (R) -~ CH,,_ 1(A) --+ . . . .
(9)
The following theorem relates this exact sequence to the underlying dynamics. THEOREM 4.3. Let (A, R) be an attractor-repeller p a i r decomposition o f an isolated invariant set S. I f S = A U R, then On = O.
PROOF. Let U and V be disjoint neighborhoods of A and R, respectively. Since S = A U R there exists an index triple of the form (N a U Nff, N a U NOR, N~ U N~), where N a C U and N~ C V. Equation (8) now takes the form
9.. + H,, (N.~ v N:, No~ . N:) + t+,, ( N # . N~, N 2 . N : ) -+ H~7(NA U Nff , NIA U N~) -~ Hn-i (N # U N~, N~ U N~) - + . . .
K. Mischaikow and M. Mrozek
422
which by excision is equivalent to
9"" -+ Hn (N~, N~) --+ Hn (N1A, N ~ ) + Hn (NR , NR) --+ Hn(Nff , N~) -~ Hn-1 (N R, N A) - + ' " . Since the sequence splits,
0n =
0.
[2
It is the contrapositive of this theorem which is typically of use in applications. COROLLARY 4.4. If On(A, R) 7~ O, then S 7~ A L) R, i.e., C(R, A; S) ~ 0. REMARK 4.5. Example 3.18 shows that the converse of this corollary is not true. In preparation for the rather complicated algebra which will be introduced in 4.1.2, we will now try to present the information that is carried by 0, in the form of a matrix. To simplify the presentation we shall now only consider homology with field coefficients. The information which we have consists of three parts. (1) A collection of graded vector spaces arising from the indices of the attractor and repeller,
CH,(A) ~) CH,(R). (2) The index of the total invariant set S,
CH,(S). (3) The connecting homomorphism or boundary map On which is a degree - 1 operator, i.e., it sends n-level homology to n - l-level homology. We shall, from now on, denote this homomorphism by
O,(A, R) "CH,(R) --+ CH,(A). We now pose the following question. Let us view CH,(A) @ CH,(R) as a chain complex with a boundary operator A. Can we choose A in such a way that the resulting homology H A , is isomorphic to CH,(S)? To be more precise, let
An "CHn(A) ~) CHn(R) --+ CHn-I (A) G CHn-1 (R), denote the obvious restriction of A to each homology level of the indices. Then, by definition ker(An) H A n := . Image(An+l)
Conley index
423
Thus we are asking whether it is possible to choose A in such a way that
HA,;~CHn(S),
n=0,1,2
.....
Observe that if S = A U R, then C H , ( S ) ~, C H , ( A ) G C H , ( R ) . In this case we can choose A = 0. This is a special case of the following proposition. PROPOSITION 4.6. If we define
A=[
~0
O,(A, R) ] I " C H , ( A ) 9 C H , ( R ) --+ C H , ( A ) G C H , ( R ) 0 J
then H A , ~ CH,(S). PROOF. We begin by checking that A is a boundary operator, i.e., that A is degree - 1 and A o A = 0. The first is obvious since On is of degree - 1 . The second condition is equally obvious. We now need to show that
H A,; ,~ CH,, (S),
n = 0, l, 2 . . . . .
Observe that ker A,, = CH,;(R) G ker O,, (A, R) and image A,,_ 1 = i m a g e 0,;- l (A, R). Thus, we need to show that
CH,; ( S) ~
CH,,(A) 9 kerOn(A, R) image 0,,+l (A, R)
~
CHn(A) image 0,,+1 (A, R)
9 ker Ot~(A, R).
Now consider the sequence
9. . - + CHn(A) -~ C H , ( S ) -~ CHn(R) -~ CHn-I (A) - + . . . which, since we are working with field coefficients, can be written as
9.. --> CH,,(A) ---->image/,, • L,, --+ CH,,(R) -~ CH,,-I (A) ---> . . . .
K. Mischaikow and M. Mrozek
424 By exactness Ln ~ ker3n(A, R) and
CH,,(A) image in ~ . image 0n+l (A, R)
IS]
The matrix in Proposition 4.6 is the simplest example of a connection matrix which will be described further in the next section. EXAMPLE 4.7. We now return to Example 2.3 and show that for all c > 0 and 0 < )~ ~< 1, there exists a connecting orbit from ( - ~ / ~ , 0) to (~/~, 0). For the index computations we shall use the field coefficients Z2. In Example 3.20 it was shown that CH,(Sc.x) = O. Linearizing about the equilibria shows that
CHk ( A ) ~
Z2 0
if k = 1, otherwise,
CHk(R) ~'~ { Z2 0
if k - 2, otherwise.
Thus, the portion of the exact sequence (9) which is of interest is
9..--+ CH2(S) --+ CH2(R) -~ CHI (A) --+ CH1 (S) --+ ... which of course reduces to 999 ----~ 0 ----~ Z2 - ~ Z2 ~
0---~ . . ' .
Since this sequence is exact 02(A, R) ~ O. By Corollary 4.4 there exists a heteroclinic orbit from R to A. We can abstract this example into the following proposition. P R O P O S I T I O N 4.8. Assume that S is an isolated invariant set with an attractor-repeller pair decomposition (A, R) consisting of twofixed points. Furthermore, assume that
CH,,(R) ~ { Z 0
if n - k + l, otherwise,
CHn(A) ~ { Z 0
and C H , ( S ) ~ O. Then, there exists a heteroclinic orbit from R to A.
if n = k, otherwise,
Conley index PROOF. By Theorem 3.22, S -r A U R. Thus, by Theorem 2.4, C (R, A; S) :/= 0.
425 F-1
Consider Proposition 4.8. This is a pure existence result; the conclusion is that there exists an orbit q)(R, x) C S such that lim (p(t, x) = A, t-----~ r
lim ~0(t, x) = R. t -----~ - - r
What it does not do is describe the dynamics on S as a whole. Clearly, in this simple example it is obvious that the only type of orbits which can occur are the fixed points A and R and the heteroclinic orbits from R to A. However, for the sake of clarity it is worthwhile trying to recast this example in the context of semi-conjugacies. More complicated examples will be presented in later sections, but there the technical difficulties can overwhelm the simple idea. With this in mind, therefore, we shall prove the following theorem. THEOREM 4.9. Let q9 be a flow f o r which S is an isolated invariant set with an attractorrepeller pair decomposition (A, R). Assume that CH,(S) ~ CH,(A) 9 CH,(R). Let ~ : IK x [ - 1, 1] --+ [ - 1, 1] be the flow on the unit interval generated by 2 = ( x 2 - 1). Then, there exists a continuous surjective function p : S --> [ - 1 , 1] such that A = p-l(-1),
R = p-J(1)
and the following diagram commutes ~p
RxS
>~S g,
Rx [-1,1]
> [ - 1 , 1]
PROOF. Let V :S ---> [0, 1] be a Lyapunov function for ~0 as defined by Theorem 2.7. Because V is strictly decreasing off of A U R, given x 6 C ( R , A; S) there exists a unique tx ~ R such that 1
5.
K. Mischaikow and M. Mrozek
426
Observe that the function r :C(R, A; S) ---> IR given by r (x) = tx is continuous. Now, define p : S --+ [0, 1] by -1
p(x) --
1
2 ?tan- 1 tx
i f x ~ A, i f x ~ R, if x 6 C(R, A; S).
Since CH,(S) ~ CH,(A) @ CH,(R), C(R, A; S) 7~ 13, and hence, p is surjective.
D
There are several points that should be remarked concerning the proof of this theorem. The first is that one uses the decomposition and the Lyapunov function to define the semiconjugacy. The second is that it is the Conley index information that allows one to conclude that p is a surjection. On a more general level, the reader should note that a theorem of this form provides a global description of the dynamics. The existence of a semi-conjugacy provides a lower bound on the complexity of the dynamics. Since p is surjective, every trajectory of 7r is lifted to 4~. Of course this lifting need not be one to one, and hence, the dynamics of r may be more complicated than that of gr. Just as one can decompose isolated invariant sets, one can decompose sets of connecting orbits. Let (A, R) be an attractor-repeller pair decomposition of S with associated boundary operator 0 (A, R). A separation of C (R, A) is a collection {Cj (R, A) I j = 1. . . . . J } of open subsets of C (R, A) such that J
C(R,A)--UCj(R,A). j=l
If N is an isolating neighborhood for S, then there exists Nj C N such that Nj is an isolating neighborhood for Sj := A U R U C j ( R , A). Observe that (A, R) is an attractorrepeller pair for Sj, and hence, there is an associated boundary operator O(A, R; j). THEOREM 4.10 (McCord [45]). For any separation of C(R, A) J
O(A,R)--~O(A,R;j). j=l
4.1.2. Connection matrices. In the previous section we constructed a connection matrix for the case of an attractor repeller pair. We now wish to do the same for a Morse decomposition. As was indicated earlier, the construction is not trivial. For this reason we will only describe the properties that connection matrices possess, referring the reader to [2528,69,71 ] for further details. We remind the reader that throughout this section we will be using homology with field coefficients. In practice one usually uses Z2 coefficients since this avoids one having to deal with orientations.
427
Conley index
Let A,4(S) = {M(p) [ p 6 (79, >)} be a Morse decomposition. Each Morse set is an isolated invariant set, and therefore, has a Conley index of the form CH,(M(p)). Let
A "@ CH,(M(p)) ---> ( ~ CH,(M(p)) pET)
pc7)
be a linear operator which will be written as a matrix
A=[A(p,q)] where A(p, q) : CH,(M(q)) --+ CH,(M(p)). A is upper triangular with respect to the partial order > if p ;r q implies A (q, p) -- 0. A is a boundary operator if A o A = 0 and it is a degree - 1 map, i.e., it maps nth level homology to (n - 1)st level homology. Finally, for each interval I E 2-(79, >) define A ( I ) -- [ A ( p , q)]p,qEI "~
CH,(M(p)) --> ~
pEI
CH,(M(p)).
pEI
If A is a connection matrix, then it satisfies the following four properties: (1) A is upper triangular. (2) A is a boundary operator. (3) For every interval I E 2-(7~, >) kerA(1)
H , A ( I ) "-- image A(I)
~ CH,(M(I)).
(4) Consider a pair of adjacent intervals (I, J) E I2 (~, >). Observe that there are three associated chain complexes: (1) ( ~ p E 1 C H , ( M ( p ) ) with boundary operator A(I), (2) CH,(M(p)) with boundary operator A ( J ) , and (3) CH,(M(p)) with boundary operator A ( I J ) . Furthermore, under inclusion and projection the following is a short exact sequence,
(~pEJ
(~pEZJ
0--> ~CH,(M(p)) --> 0 pEI
pEIJ
CH,(M(p)) --->@CH,(M(p)) --+ O. pEJ
Applying the boundary operators one obtains the following long exact sequence
9..-+ H,,A(I) --+ H n A ( I J ) ~ H,,A(J) --+ H,,_IA(/) --+ . . . . Then, this sequence is isomorphic to the long exact sequence for the attractor repeller pair (M(I), M(J)) decomposition of M ( I J)
9..--+ CHn(M(I))--+ CHn(M(IJ))--+ CH,,(M(J)) --+ CHn-, ( M ( I ) ) - - + . . . .
K. Mischaikow and M. Mrozek
428
From property (4) one can deduce that given adjacent Morse sets M (p) and M (q), i.e., a pair of Morse sets such that (M(q), M(p)) ~ 2-2(79, >), then
,4(qp) _ [ O O,(M(q), M(p)) ] 0 0 " It is important to mention that the four properties listed above do not define the connection matrix. For the proper definition the reader is referred to [26]. The fundamental theorem is the following. THEOREM 4.1 1 (Franzosa [26]). For any isolated invariant set S with Morse decomposi-
tion M (S) the set of connection matrices are not empty. As is implied in the theorem, connection matrices need not be unique. EXAMPLE 4.12. Consider once again Example 2.8. We shall u s e Z2 coefficients for the index computations. The associated connection matrices must be linear operators of the form 3
3
,4"~CH,(M(p))--~ OCH,(M(p)). p=l
p=l
From Theorem 3.13 each of these is a one-dimensional vector space implying that A is a 3 x 3 matrix. Using the upper triangularity with respect to > and the fact that ,4 is a degree - 1 map one obtains:
A--
00 0
00 0
??1 . 0
Now consider the case of 0 < c << 1. As was indicated in Example 2.10 the flow defined order is 3 >0 2 and 1 is unrelated to 2 or 3. Thus upper triangularity with respect to >0 gives
,4 0
O 0 0 0 0 0
I
O] ? . 0
In Example 3.21 it was shown that CH, (S) is one dimensional. Thus applying property 3 of the connection matrices to the interval {3, 2, 1} forces
Ao--
0 0 0
0 0 0
01 1 . 0
Conley index
429
For the case c ~ ec the unknown entries are computed directly using property 4. In particular, one considers the attractor-repeller pairs (M(2), M(3)) and (M(1), M(3)) and shows that C H , ( M ( 2 3 ) ) ,~ 0 and C H , ( M ( 1 3 ) ) ~ O. This implies that
Aoc--
i
0 0 0
0 0 0
1] 1 . 0
4.1.3. E x a m p l e - bistable gradient like systems9 In Section 4.1.1 the Conley index was used to obtain a semi-conjugacy describing the dynamics of an attractor repeller pair decomposition for which a connecting orbit exists9 One can ask whether similar results can be obtain in the context of Morse decompositions 9 The answer is yes as the following example indicates9 Consider the following assumptions: (A 1) A is a global compact attractor for a semi-flow q) on a Banach space9 Furthermore, if q9 denotes the restriction of q) to ,4 then ~0 defines a flow on A. (A2) Under the flow q) :IR x .,4 --> .,4
.Ad(A) = { M ( p + ) l p = O . . . . . P - 1} U {M(P)} with ordering P > P - 1+ > . . . > 1+ > 0 + is a Morse decomposition of A. (A3) The Conley indices of the Morse sets are
CHk(M(P))
/ Z2 [0
i f k = P, otherwise,
and for p - 0 . . . . . P - 1
CHk (M ( p + ) ) ~ { Z2 0
if k -- p, otherwise.
(A4) The connection matrix for A4 (A) is given by -0 0 A
m
D1
0
0
D2
0
m
0
where for p - 0 , . . . ,
9 9 9
"-
...
0 0
O
m
0
Dp 0
m
P - 1
cH (M(p )) | cH,(M(p+)) --+ CHp_! ( M ( p - 1-)) (9 CHp_! ( M ( p
-
1+))
K. Mischaikow and M. Mrozek
430
is given by 1
DP--['
1
'1
and Dp "CHp(M(P))
--+ C H p - I ( M ( P
-
1-)) ~ C H p _ 1( M ( P - 1+))
is given by
De--
['] 1
"
These assumptions may seem somewhat strange at first, but in fact are satisfied by a wide variety of systems [52]. (A1) is typical of many dissipative systems [37,36,81]. The simplest interpretation of assumptions (A2) and (A3) is that the Morse sets consist of hyperbolic fixed points, with the equilibria on the pth level having a p dimensional unstable manifold. Finally, (A4) is a statement that each Morse set on the pth level connects to every Morse set on the (p - 1)st level. The simplest ordinary differential equation that possess these properties is the following. Let D P "-- { z -
ze-1) I Ilzll ~< 1} C R P
(zo . . . . .
be the closed unit ball in R P and S P - l = OD P be the unit sphere. Let 7z P : ~ • D P --+ D P denote the flow generated by the following system of ordinary differential equations -
Q~ -
( Q ~ , ~)~,
E S P-I,
(10)
r ~ [0, 1],
(11)
~
? -- r(1 - r), where
O
__
1
0
0
1
o
0
...
0
,
1
T
The dynamics of 7z e is easily understood if one realizes that (10) is obtained by projecting the linear system ~ = Q z onto the unit sphere. Let ep+ - ( 0 . . . . , +1 , ... , 0) be the unit vectors in the pth direction. Then one has the following result.
Conley index
431
THEOREM 4.13. Given assumptions (A1)-(A4) there exists a continuous surjective map f " A ~ D P and ~9 a flow obtained by an order preserving time reparameterization o f r such that the following diagram commutes
RxA
idxf I
(o
>A
lPP I f
IR x D e
> De
where M ( p +) -- f - I (eip) f o r 0 <~ p <~ P - 1, and M ( P ) -- f - I (0). The proof of this theorem can be found in [52]. However, the essential idea mimics that of Theorem 4.9. One defines a continuous map from ,4 to D P by describing each point in A in terms of the amount of time its trajectory spends in the neighborhood of the pth Morse set and its value under the abstract Lyapunov function of Theorem 2.11. One then uses the index theory to prove that the map must be surjective. To apply this theorem the connection matrix must be known. The direct computation of a connection matrix is typically a difficult task and often done through a continuation argument. For additional examples of theorems of this nature and their applications to differential equations the reader is referred to [17,31,33,47,54,43]. 4.1.4. Transition matrices. In Examples 2.10 and 4.12 it was shown that the flow defined orders and connection matrices differ for different values of c. This implies the existence of bifurcations. Since the Morse decomposition continues over the parameter range c > 0, the bifurcation must be of a global nature. In this section we will introduce the notion of a transition matrix and show that it can be used to detect global bifurcations. To discuss transition matrices one must begin with a continuous family of flows. We shall take them to be generated by ordinary differential equations defined on N", 2-f(x,X),
X6R.
(12)
For the sake of clarity assume that N is an isolating neighborhood for all values of A and let Sz -- Inv N under the dynamics induced at the parameter value A. Also, assume that the Morse decomposition AA(Sz) := {Mz(p) I P E (79, >)} also continues. Then choosing A = 4-1 one has two connection matrices A+ which are boundary operators on the complexes @ C H , ( M + I (p)) respectively. Transition matrices are chain maps between these chain complexes. There are several ways in which transition matrices have been defined. The simplest and most general is based on a singular perturbation. Referred to as singular transition matrices, the idea was due to Conley and was carried out by Reineck [68]. Consider the following augmented family of differential equations 2 -- f ( x , A ) , (~3) -
8>0.
K. Mischaikow and M. Mrozek
432
For each e > 0 this generates a flow for which N x [ - 2 , 2] is an isolating neighborhood. Let Ke = Inv(N x [ - 2 , 2]) under the e flow. Define
M(p+) "-- Ml (p),
M(p-) "--M-l(p)
then
A/l(Ke)= lM(p+) l p e 79} is a Morse decomposition. This follows from the fact that e > 0, and hence, X < 0, if X e ( - 1, 1). Furthermore, an admissible order is given by q+ >> p - , q
_
>>p
m
q+ >> p+
~, ,,. ~, ,~
q>p, q>p.
Let Ae denote a connection matrix for .Ad(Ke). Since the dynamics on the subspaces defined by X = + 1 are given exactly by the flows generated by 2 = f (x, + 1),
A" 0
CH,(M(p-)) ( ~ CH,(M(p+))
pep
pep
--->0 CH,(M(p-)) ( ~ CH,(M(p+)) p e t)
p e t)
takes the form
where
cm+, (M(p+)) pep
(9 CH,(M(p-)) p e t)
and
Te(q-, p+) "CH.+I (M(p+)) --+ CH.(M(q-)). Two comments need to be made at this point. The first is that technically the entries A+ in A~ cannot be the connection matrices A• for A//(S+), since they are defined on different spaces. One of the contributions of [68] was to overcome this technical obstacle and justify the expression in (14). Second, while in principle TE may change as a function of e, it is always possible to choose a sequence 6n ~ 0 along which Te,, is constant. The set of matrices obtained
Conley index
433
through the above mentioned limiting process are referred to as singular transition ma,]--sing trices and the set of singular transition matrices is denoted by _ 1, l" Consider T e ,7--sing -~,1 and assume p, q are adjacent elements with respect to the ordering > for which T ( q - , p+) r O. Then, there exists a sequence e,, --+ 0 for which Te,, ( q - , p+) -r 0. This in turn implies the existence of a connecting orbit Fe,, from M1 (p) to M - l (q) for the parameter value e,,. Taking the limit of Fe,, as e,, --+ 0 implies the existence of a connecting orbit Mx (p) to M)~(q) for the dynamical system defined by (12) for some value of )~ between - 1 and 1. Thus, non zero entries in the singular transition matrix can be used to conclude the existence of connecting orbits. A generalization of this argument leads to the following theorem. 9
,-/-.sing
_
THEOREM 4.14 Let T ~ ~__1, l and assume T (q , p+) 7~ O. Then there exists a finite sequence 1 > Xl > )~2 > " ' " > )~k > 0 and rk+l -- p >xk rk > . . " >)~2 r2 >x~ rl = q, elements of (T9, >) where >xi is the flow defined ordering for the Morse decomposition of (12) at parameter value Xi. This form of transition matrix is by far the most general. In fact, the only necessary assumption is the existence of Morse decompositions of isolated invariant sets which are related by continuation. There is, however, a price to be paid for this generality. Observe that the algebraic properties of Te for e small are a complete mystery. For example, it is reasonable to ask what is the relation between Te and T--eI . Unfortunately the spaces on which T, and T_e are defined are different, and hence, even formalizing the question poses some difficulty. These algebraic questions led to the development of another transition matrix called a topological transition matrix denoted by Ttop -l.l
CH,(M1 (p)) ~ ( ~ CH,(M_I (p)). pE~
"~
pET)
Using the same setting as in the discussion of the singular transition matrix, the idea behind topological transition matrices can be described as follows. Since each Morse set M (p) continues over R, there exist isomorphisms
F-l,1 (p) " C H , ( M I (p)) ~ C H , ( M _ I (p)). Similarly, since S continues over R, there is an isomorphism
F_I.I " CH,(SI) --~ CH,(S_I).
UpcPM+l (p),
Furthermore, if S+l there exists an isomorphism -
++, O peP
i.e., if the set of connecting orbits is empty, then
c m (M• (p>) --, cm(s+,>.
K. Mischaikow and M. Mrozek
434
Thus, if there are no connections at X = 4-1, then the following diagram can be constructed:
~pcP F-t,l (p) (~pcP CH,(MI (p))
~- ~ p c P CH,(M_I (p))
F-I.l CH,(SI)
~
CH,(S_I)
In general, this diagram is not commutative and it is precisely its failure to commute that gives information about connecting orbits. For the purpose of applications it is useful to be able to express this last statement in the form of a matrix. Of course this requires choices of basis on each of the spaces. This can be done as follows. Let/31 be a basis for ~]~p~p CH,(MI (p)) and let 13-1 = ~
F - l , l (p)(Bl)
p6P be a basis for E[~p~e CH,(M_I (p)). Using these bases define the topological transition matrix by T -1,1 t~ = ~ - l o
F - l , l (p) o q~-I
p6P Observe that this construction only makes sense when the Morse decomposition continues over the entire parameter space, and when there are no connecting orbits with respect to the Morse decomposition at the parameter values )~ = 4-1. In this setting it is perfectly clear what is meant by the composition of two transition matrices. It is equally obvious that Ttop
-,,,-
top (rl,-l)
- 1
Furthermore, Ttop _ l, l shares many properties with elements o f ,~sing l, l" In particular, non-zero off diagonal entries imply the existence of connecting orbits for some parameter values b e t w e e n - 1 and 1. An important fact is that when both singular and topological transition matrices are defined then they are "equal". Equal needs to be put in quotation marks since the maps are defined on different vector spaces. However, as is shown in [46] there exists a canonical isomorphism between these spaces which justifies this claim. -
EXAMPLE 4.15. To see how the transition matrices can be used in practice, consider Example 2.8. Again, we use Z2 coefficients in the calculations. From Example 2.10 we know that the Morse decomposition continues over the interval c > 0. Furthermore,
435
Conley index
A0 and Aoo were calculated in Example 4.12. Thus, augmenting (2) by the equation -- e(c - co)(c - coo) where 0 < co << 1 and 1 << coo and applying (14) one obtains 0 0 0 0 0 0
0 0 0 0 0 0
0 1 0 0 0 0
T ( 1 - , 1 +) T ( 2 - , 1 +) T ( 3 - , 1 +) 0 0 0
T(1-,2 + ) T(2-,2 + ) T(3-,2 + ) 0 0 0
T(1-,3 + ) T(2-,3 + ) T(3-,3 + ) 1 1 0
"
Since, for all c > 0 the Lyapunov function implies that 3 > 2 > 1, and using the fact that A~ must be a degree - 1 map, this reduces to: 0 0 0 0 0 0
A E
0 0 0 0 0 0
0 1 0 0 0 0
T ( 1 - , 1 +) 0 0 0 0 0
T ( 1 - , 2 +) T ( 2 - , 2 +) 0 0 0 0
0 0 T ( 3 - , 3 +) 1 1 0
"
From the simple form of the flow it is easy to check that ( M ( p - ) , M ( p + ) ) forms an attractor-repeller pair. Furthermore, C H , ( M ( p - p + ) ) -- 0 which implies that T ( p - , p + ) -- 1. Thus 0 0 0 0 0 0
z~ 8
0 0 0 0 0 0
0 1 0 0 0 0
1 0 0 0 0 0
T ( 1 - , 2 +) 1 0 0 0 0
0 0 1 1 1 0
"
However, A~ o Ae -- 0, which implies that T ( 1 - , 2 +) -- 1. Therefore, there exists c* > 0 for which Equation (2) possesses a heteroclinic orbit from ( - 1,0) to (1,0). 4.1.5. E x a m p l e - infinitely many connecting orbits. equations ~
Consider the system of differential
y,
[y -- X y - x ( x - a ) ,
x, y , X , a 6 R ,
(15)
5~ -- e(1 - X2), Observe that for e :/: 0 there are exactly four equilibria B + "-- (0, 0, + 1 ) and A + "= (a, 0, + 1). Our goal is to show that for e > 0 there exist infinitely many connecting orbits from B - to B +. (15) has the same form as (13), suggesting that the singular transition matrix can be used. Also, notice that at A - + 1 this system has the same form as that of Example 2.3.
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K. Mischaikow and M. Mrozek
We begin by studying the case e < 0. The same arguments used in Examples 4.7 and 4.15 give rise to the following result. PROPOSITION 4.16. For e < O, A4(S~) = {B • A +} is a Morse decomposition of Se with an admissible ordering A + > B + > B - > A - . Furthermore, using Z2 coefficients the corresponding connection matrix is
A~
m
o, 0 0 o 0 o 01
i
0 0
0 0
0 0
1 0
"
Proposition 4.1 6 implies that the transition matrix is T sing -- r 0 -l,l
LO
00] . CH,(B+) 9 C H , ( A +) --+ C H , ( A - ) G C H , ( B - )
sing! ( B - , B +). HowSince Ae is a degree - 1 map, the only possible non-zero entry is T -l, ever, it is easy to check that the segment B x ( - 1 , 1) is an isolated connecting orbit from B + to B - , i.e., it is an element of a separation of C (B +, B - ) . Furthermore, the associated boundary operator is O(B-, B +) -- 1. Thus, if this were the only connecting orbit, then T-1,1 sing ( B - ' B +) -- 1 By Theorem 4.10 there must be another connecting orbit in the separation which has a nontrivial boundary operator. Some simple analysis shows such an orbit must wind around the line segment A x [ - 1 , 1 ]. To simplify the rest of the argument we shall assume that C (B +, B - ) consists of exactly two orbits; the trivial orbit, B x ( - 1 , 1), and one which winds around the line segment A x [ - 1, 1] exactly once. For this particular system one can prove this fact using Melnikov methods. In a more general setting one can continue the topological argument presented here allowing for multiple orbits with multiple windings [44]. Returning to the proof of infinitely many connecting orbits, let N be an isolating neighborhood for Se. To be precise about what is meant by the winding of orbits consider the following set
D--N\({A}xR). D is homotopic to S l, and hence Jr 1 ( D ) ~ 7r1 (S 1) ~ Z where zr I denotes the fundamental group. This suggests the following definition. DEFINITION 4.17. Let ye denote a connecting orbit from B + to B~-. Then ye U (Be • [ - l , 1]) generates an element of 7/"1( D ) which defines the winding number of Fe. Let D denote the universal covering space of D with covering map
p ' D - + D.
Conley index
437
There exists a unique local flow qse on D which projects onto qge. Let {B,~ I n 6 Z} -- p-1 (B +) and let ,~ -- p - l (N). Unfortunately,/V is not an isolating neighborhood for qs,, since it is not compact. However, one can show that
K
K
U
s%(K) =
U c(8+, 87; < )
u
k=l
j,k=l
is an isolated invariant set. Using the fact that all the index information of 45e must project onto qge one can compute
~sing
the singular transition matrix -1,1 associated with Se(K). Since we are assuming that there is the trivial connection and exactly one other connection with winding number 1 from B + to B - this implies that -1
0
1
0
...
1
1
...
0
00
1 0
1
sing 1,1 0 _0
...
1_
Observe that there are no connecting orbits at )~ = -+-1. Therefore, we can apply the topological transition matrix. As was indicated in the previous section, when both topological and singular transition matrices are defined on the same Morse decomposition they agree9 Hence, modulo some canonical isomorphisms one may write
~sing
~top -1
0
_0
(~si~ g -1
~top-1 1
1
1
1
... ...
,)
1
11
1
1
0
1
But this implies that for e > 0, there exist the desired connecting orbits with arbitrary winding numbers. Under p each orbit with a different winding number projects to a different solution of (15). Therefore, there are an infinite number of distinct connecting orbits.
4.2. Periodic orbits Given an isolated invariant set S with a Morse decomposition .L4 (S), Theorem 2.11 implies that the only type of dynamical objects that lie in S, but not within the Morse sets are connecting orbits. The theme of Section 4.1 was how to use the Conley index to prove the existence of and describe the structure of these solutions. Theorem 2.19 implies that all
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K. Mischaikow and M. Mrozek
dynamics in an invariant set either consist of connecting orbits or lie in the chain recurrent set. With this in mind we now turn to the question of how can the Conley index be used to understand recurrent dynamics. The simplest form of recurrence is that of a periodic orbit. As Example 3.18 demonstrates it is not enough to know that the index of an invariant set S is that of a periodic orbit to conclude that S contains a periodic orbit. What is surprising, however, is that the only additional necessary hypothesis is that the invariant set have a minimal amount of recurrence. 4.2.1. P o i n c a r d s e c t i o n . To insure recurrence we shall make use of a Poincar6 section. Since we are adopting the point of view that it is the isolating neighborhood which is the observable object rather than the invariant set, we want to formulate all of our hypotheses in terms of N, rather than S. Therefore, we employ the following definition. DEFINITION 4.18. ,~ C X is a P o i n c a r d s e c t i o n for N under tp if Z, is a local section, ~,U " = ~, O N is closed, and for every x E N there exists tx > 0 such that (p(tx,X) 6 E.
(16)
Observe that it is not necessary to know S in order to find a Poincar6 section. Also, Z, is not required to be a subset of N. Indeed, if N has an exit set, no subset of N can be a Poincar6 section, as there will be points in N whose orbits exit N before they cross the section. The following example is included to indicate that this definition is sufficiently weak to make it possible to prove the existence of Poincar6 section in the context of a differential equation. EXAMPLE 4.19. The Nagumo equations (see [76] for a derivation of the equations) are given by, fi
m
i; -
V~
Ov - f (u) + w ,
(17)
0
The nonlinearity f is a cubic like function as indicated in Figure 3. In this problem, 0 is treated as a constant rather than as a parameter value. ~0e denotes the flow on R 3 generated by (17) for a fixed e. For e = 0 the Nagumo equation becomes a parameterized family of differential equations in the plane with parameter w. In particular, for each fixed w one has a flow ~ w ' R • R 2--+R 2 generated by the f a s t s y s t e m
(18) i; -- Ov - f (u) + w.
Conley index
439
w
q
) u
Fig. 3. The nonlinearity of the Nagumo equation.
At e = 0 the equilibria of (17) are given by the set {(u, 0, w) I f(u) = w}. Let [0, wl] be an interval such that for c 6 [0, wl], there are three branches of solutions to f(u) -- c (see Figure 3). Observe that for each fixed value, w = c, the dynamics of (18) is similar to that described in Example 2.8. In particular, the left and right roots of f (u) -- c give rise to saddle points in flow 7tc. Extending this argument to all w 6 [w0, w l] one obtains two pa'rameterized branches of equilibria q (w) and p(w), respectively. For an appropriate choice of 0 one can prove, using the same type of argument employed in Example 4.15, that there are parameter values w, and w*, with 0 < w, < w* < wl, for which there are heteroclinic connections y, from q(w,) to p(w,) and V* from p(w*) to q(w*), respectively. Let Q be the set of equilibria given by q(w) for w 6 [w0, wl] and P be the set of equilibria given by p(w) for w 6 [w0, wl ]. Now observe that for e > 0, tb < 0 along the curve Q and tb > 0 along the curve P. This suggests that for each fixed 0 < e << 1 there may exist a periodic orbit of qg~ close to the segments V, U P U y* U Q. To prove this using the index theory requires several steps. The first is to find an appropriate isolating neighborhood by choosing tubular neighborhoods around each of the segments. More precisely, let Tp and TQ be tubular neighborhoods around the segments P and Q, and let T, and T* be tubular neighborhoods around the segments V, and V*. Set N = Tp U TQ U T, U T*. It can be shown using the techniques described in Section 4.4 that N is an isolating neighborhood for 0 < e << 1. The second step is to compute the index of Sc = Inv(N, qg~) wherein one obtains
CHk(S~) ~ { Z2 0
ifk=l,2, otherwise.
The techniques for doing this are also discussed in Section 4.4. The final step is to apply Theorem 4.20 and conclude the existence of a periodic orbit. To do this it must be shown that N has a Poincar6 section which can be done as follows.
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K. Mischaikow and M. Mrozek
Since N is an isolating neighborhood, by Theorem 3.6 there is an isolating block B that lies within an arbitrarily small neighborhood of Se such that S~ - Inv(B, qg~). Furthermore, it can be shown (see [75]) that for any time T > 0 there is an isolating block E such that: (1) Inv(E, qg~) -- Inv(B, qge), (2) E C int(B), and (3) x E E implies qg~([0, T], x) C B. Observe that it is sufficient to show that E has a Poincar6 section and then apply Theorem 4.20 to Inv E. Let K -- {(u, v, w) 6 B n TQ I (w* + w , ) / 2 - 6 < w < (w* + w , ) / 2 + 6} for 6 > 0, but small and let K0 -- {(u, v, w) 6 B n TQ I w -- (w* + w,)/2}. Since tb < 0 along the curve Q, choosing B within a sufficiently small neighborhood of Se implies that K0 is a local section. Clearly if (u, v, w) stays in E for all forward time under q9e, then its forward orbit passes through the tubular neighborhoods in the periodic progression of TQ --+ T, --+ Tp --+ T* ~ TQ. This is because the flow in each tubular neighborhood is determined by the flow near the segments. Choose the time T mentioned about to be longer than the time needed for any orbit in Se to pass through all four tubular neighborhoods. Observe that if Se -- 0, then B - is a Poincar6 section for E. Thus, what needs to be dealt with are those points that remain in E for all positive time. Let E "-- ( B - \ K) U K0.
We claim that Z is a Poincar6 section for E. E U Y, is clearly closed. It is a section to the flow because both B - and K0 are. Finally, we must show that the forward orbit of every point in E intersect 3 . Clearly if (u, v, w) stays in E for all forward time under q9e its forward orbit intersects/7. Thus, we only need to worry about points (u, v, w) for which there exists a time r0 > 0 such that qg~(r0, (u, v, w)) ~ E. Since Inv(E, qgc) = Inv(B, qg~), this is equivalent to the existence of rl > 0 such that qge(rl, (u, v, w)) r B. This in turn implies that there exists r2 > 0 such that qge(rl, (u, v, w)) E B - . If qge(rl, (u, v, w)) 6 B - \ K then qge(rl, (u, v, w)) E Y,. So we have further reduced the problem to qge(rl, (u, v, w)) B - n K. Observe, however, that (u, v, w) 6 E implies that there exists a positive time 0 < s < T such that qge([0, s], (u, v, w)) C B and qge(s, (u, v, w)) 6 K0. 4.2.2. Existence ofperiodic orbits. The following theorem provides very general conditions under which an isolated invariant set must contain a periodic orbit. THEOREM 4.20. I f N is an isolating neighborhood which admits a Poincard section Z, and f o r all n E Z either dim CH2n (Inv N) - dim CHzn+l (Inv N)
(19)
dim CH2n (Inv N) -- dim CH2n-! (Inv N)
(20)
or
where not all the above dimensions are zero, then Inv N contains a periodic trajectory.
441
Conley index
COROLLARY 4.21. U n d e r the above hypothesis, iflnv N has the Conley index o f a hyperbolic p e r i o d i c orbit, then Inv N contains a p e r i o d i c orbit. A key step in the proof of Theorem 4.20 is the following exact sequence which relates the index of an isolated invariant set which admits a Poincar6 section with the index of the invariant set under the Poincar6 map. THEOREM 4.22. A s s u m e N is an isolating n e i g h b o r h o o d f o r the f l o w ~p a n d a s s u m e that N admits a Poincard section ,g,. Let H denote the c o r r e s p o n d i n g Poincard map, S = Inv(N, qg), a n d K = 3 fq S. Then, there is the f o l l o w i n g exact s e q u e n c e o f homology Conley indices:
9 . . - + C H , , ( S , qg) --+ C H n ( K , H )
i d - X,, ( K, 17) ) CH,,(K,/7)
--+ C H n - l (S, ~p) ~
.... (21)
As was indicated in the introduction and as the following simple examples will demonstrate the Poincar6 map may carry more information than the index of the flow. EXAMPLE 4.23. Let N be an isolating neighborhood for a flow q9 and let S = Inv(N, qg). Assume that N admits a Poincar6 section ~,. L e t / 7 be the induced Poincar6 map. Furthermore, assume t h a t / 7 is the G-horseshoe map of Example 3.36. Let K -- Inv(~,,/7). If we now apply Theorem 4.22, then the only nontrivial portion of (21) is
9 .. -+ CHI (S, ~o) -+ CHI ( K , / 7 )
id--xi (K,17) ) CHI
( K , / 7 ) --~ C H o ( S , qg) ~
....
(22) Observe that id - X1 ( K , / 7 ) = 1 - 2 = - 1, and hence (22) becomes 9 .. --+ 0 --+ CHI (S, 99) --+ Q -1
Q --+ C H o ( S , qg) --+ 0 --+ . . . .
Exactness of this sequence implies that C H , ( S , 99) = O. It is worth emphasizing just how much information has been lost in the flow index. Since C H , ( S , 99) - - 0 , we can draw no conclusion from this information. Since the map index is nontrivial, the Wazewski principle implies that K and hence S is nonempty. In fact, however, the map index can be used to say much more about the structure of K. In particular, the fact that x I(K, H ) = 2 allows one, by the direct application of a result of Baker [ 1], to conclude that the entropy of K is bounded from below by In 2. On a more general note, as will be made clear in Section 4.3, the spectrum of x , ( K , H ) can be used to conclude the existence of rather complicated orbit structure in K. From (21) it is clear that any information carried by spectral values different from 1 is lost in the flow defined index.
K. Mischaikow and M. Mrozek
442 4.3. The topology of S
As was discussed in the introduction, the topology of invariant sets can change dramatically under even small perturbations. Since the Conley index is stable with respect to perturbations one cannot expect that knowledge of the index will immediately give information about the structure of the invariant set. In fact, even if the Conley index of S is extremely complicated, it is possible that S consists of a single equilibrium point. In Section 4.1, we were able to obtain additional information about the structure of S by assuming knowledge of an index filtration. In Section 4.2, the existence of periodic orbits could be concluded if one knew that there was a Poincar6 section for the isolating neighborhood. Another approach, which we now describe is due to Floer [18] and is based on the idea of restricting the topology of the isolating neighborhood. To be more precise, let qgz : I1~ • X ~ X be a parameterized family of dynamical systems with )~ 6 [0, 1]. Let So be a normally hyperbolic invariant set for qg0 and let N be a tubular neighborhood for So. Then N is an isolating neighborhood for So. There are two observations to make at this point. The first is that any retract ot:N --+ S is homotopic to the canonical projection of N onto S coming from the tubular neighborhood structure. The second is that the homotopy Conley index of S is essentially the Thorn complex for the unstable disk bundle for S. Using this Floer obtains the following theorem. THEOREM 4.24. Assume that N is an isolating neighborhood for r for all )~ ~ [0, 1]. Then
(otls~)* : H*(So) ~ H*(SI) is an injection. In other words, the topology of the invariant set can only increase as the flow is continued. The simplest example of this phenomenon is to consider a hyperbolic saddle in a planer flow. This is clearly a hyperbolic invariant set with a disk as a tubular neighborhood. Now consider a continuation of the flow that preserves the hyperbolic equilibrium but introduces a degenerate fixed point on one of the trajectories that passes close to the equilibrium. This new isolated invariant set consists of two points, thus the topology of the isolated invariant set has become more complicated. This result has far reaching consequences. This lower bound on the co-homology of the invariant set can be used to estimate the cup length of S1. In turn, if one knows that ~01 is a gradient flow, then one has a lower bound on the number of critical points of the potential. It should be noted that in [ 18] Floer showed that this construction can be carried out in the setting of equivariant dynamics. Work in a similar spirit was also carried out by Poiniak [67].
4.4. Symbolic dynamics In Section 3.2 we saw that the Conley index of a U-horseshoe is trivial. This is not surprising, because one may easily continue the horseshoe to the empty set. This also means
Conley index
443
that, similar to the case of periodic orbits, the index itself is not sufficient to reconstruct the horseshoe dynamics. So far we do not know what assumptions, if any, would guarantee the existence of an embedded horseshoe. However it is possible to formulate assumptions in terms of the Conley index which imply the existence of a semiconjugacy onto shift dynamics, i.e., a continuous map onto bi-infinite sequences of prescribed set of symbols, which commutes with the shift dynamics on the sequences. To be more precise, let r k := {a : Z ~ { 1, 2 . . . . . k} } denote the space of bi-infinite sequences on k symbols with the product topology. Let o- : Sk -+ Zk be the shift map given by
(o-(a)),, "--an+l. The semiconjugacy onto the shift dynamics is a continuous surjective map p : Inv(N, f ) --+ r k such that the diagram j
.
Inv(N, f )
> Inv(N, f ) (23)
Sk
~
> Sk
commutes. Returning to Example 3.27 we easily see that replacing the isolating neighborhood N by the set R := R0 U R l we obtain another isolating neighborhood which isolates the same isolated invariant set. This suggests how the semiconjugacy map may be defined in situations when the isolating neighborhood decomposes into a number of disjoint compact components as in the case of R. Namely, if N = No U Nl U - - - U Nk is a union of k disjoint compact sets then we define the map r : N ~ {1,2 . . . . . k} by r (x) = i if and only if x ~ Ni and the map p" Inv(N, f ) --+ Sk by p ( x ) " - - { r ( f / (X))}i~=_~. The map p is continuous and the diagram (23) commutes. The hard part in completing the construction of the semiconjugacy is to show that p is surjective. This is guaranteed by the following theorem.
THEOREM 4.25. Let f : X ~ X be a homeomorphism on a locally compact metric space. Assume that N = No U Nl is an isolating neighborhood under f where No and N1 are disjoint compact sets and for k = O, 1
Con,l(Nk)=
(Q, id) 0
i f n ~ 1,
otherwise.
Then, Nkl :-- (Nk n f (Nk)) U (Nk n f (Nl)) U (N1 N f (Nl)) for k, 1 ~ {0, 1}, k 5/: l, are isolating neighborhoods, l f additionally X* (Nkl ) is not conjugate to the identity, then there
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K. Mischaikow and M. Mrozek
exists a d E 1~ and a continuous surjection p ' I n v ( N , f ) --+ .~v'2 such that the following diagram
Inv(N, f )
fd
> Inv(N, f ) (24)
~r,2
o"
> z~ 2
commutes.
A disadvantage in the above theorem is the fact that one needs four isolating neighborhoods and the corresponding Conley indices to verify the existence of a semiconjugacy onto the shift dynamics. Below we present a theorem which shows that two isolating neighborhoods are sufficient. Given a linear operator M let spec(M) denote the set of eigenvalues of M where the eigenvalues are repeated according to their multiplicity. Given two sets A and B, their amalgamation A H B is obtained by taking the union of A and B but treating elements common to both A and B as distinct elements in A H B. For example, {1, 2, 2, 3 } LI {0, 0, 2, 3 } = {0, 0, 1, 2, 2, 2, 3, 3}. A set Q c C is cyclic if Q = ~ - ~ P l Oi where Oi D Q j = ~ for i :~ j and Qi - {z E g lz ni - mi for some ni E Z +, mi E •}.
THEOREM 4.26. Let f " X --+ X be a homeomorphism on a locally compact metric space. Assume that N C X is an isolating neighborhood which is the disjoint union of compact sets No and N1, i.e., N= NoUNI. Let S "-- Inv(N, f ) and Si "-- Inv(Ni, f ) , i -- 0, 1. If f o r some positive integer n spec(xn(So)) H spec(xn(S,)) q2 spec(xn(S)) or
spec(xn(So))\(spec(xn(So)) H s p e c ( x n ( S , ) ) ) is not acyclic, then there exists a positive integer d and a continuous surjection p" S --+ r 2 such that diagram (24) commutes.
Recently Szymczak [79] presented a generalization of the Conley index theory which keeps track of the components of the isolating neighborhood. The generalization uses index pairs satisfying certain extra conditions and allows one to prove a criterion for a semiconjugacy onto the shift dynamics on k symbols restricted to a certain transition matrix which
Conley index
445
may be deduced from the generalized Conley index. The theorem requires the index computation for only one isolating neighborhood and additionally, guarantees the existence of periodic points corresponding to every periodic sequence of symbols. Unfortunately, the presentation of this generalization of the Conley index is beyond the scope of this paper.
4.5. Fast slow systems We finish this section with a discussion of a new direction of development in the Conley index theory. In Example 4.19 a Poincar6 section for an isolating neighborhood was described. Two points were asserted. The first was that the region described is an isolating neighborhood, and the second was that the index of the isolated invariant set is known. We shall now briefly mention the techniques used to obtain these results. The reader is referred to [8,58,32] for further details. The general framework is that of a family of differential equations on R " given by Ili
-
-
s0(
) +
+
i=1
where e ~> 0. For each s let ~0e:R x •" ~ IK" denote the flow generated by (1) with parameter value e. The reader can check that Examples 4.19 and 4.1.5 are of this form. In both cases for e - 0 there is no isolating neighborhood for the dynamics that is of interest for s > 0. These suggests the following definition. DEFINITION 4.27. A compact set N C R" is called a singular isolating neighborhood if N is not an isolating neighborhood for ~p0, but there is an ~ > 0 such that for all s 6 (0, ~], N is an isolating neighborhood for q)~. The first question that should be addressed is how can one verify that a compact set is a singular isolating neighborhood. The answer requires the following definitions. DEFINITION 4.28. The average of g on S, Ave(g, S), is the limit as T --+ cx~ of the set of numbers { T1 f J g(~oo(s, x)) ds I x ~ S}. g has strictly positive averages on S if Ave(g, S) C (o, ~ ) . DEFINITION 4.29. A point x E S is a C-slow exit point if there exists a compact set Kx C S invariant under (P0, a neighborhood U~ of TC(Kx), an ~ > 0 and a function g : cl(U~-) x [0, ~] --+ R such that the following conditions are satisfied.
(1) co(x, q)o) C Kx. (2) g is of the form g(z, e) = g0(z) + eel(z) + . . . + e mgm (z). (3) If L0 = {z I g0(z) = 0} then K~ n cl(U,-) = S N L0 A cl(U~) and furthermore
go[sncl(Ux) ~ O. (4) Let gj(z) = V=go(z) 9 f j ( z ) + V=gl(z) 9 f j - l (z) + " - + V=gj(z) 9 fo(z). Then for some m, gj -- 0 if j < m, and gm has strictly positive averages on T~(Kx).
K. Mischaikow and M. Mrozek
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A C-slow entrance point is a point which is a C-slow exit point in the backward flow. Given a compact set N C R n let S = Inv(N, ~00). Let S - (S +) denote the set of C-slow exit (entrance) points. Set S~ = S n 0 N and S f - S~ n S +. THEOREM 4.30. If S N ON C S +, then N is a singular isolating neighborhood. Following the standard assumption underlying the technique of singular perturbations, it is assumed that the dynamics of the "singular" system 2 = fo(x) is simpler to analyze than that of the perturbed system. Thus, the goal is to determine the Conley indices for isolated invariant sets for small e > 0 from the dynamics of ~P0. This motivates the following definition. DEFINITION 4.3 1. A pair of compact sets (N, L) with L C N is a singular index pair if cl(N \ L) is a singular isolating neighborhood, and there exists g > 0 such that for all e 9 (0, ~]
H.(N, L) ~" CH.(Inv(cl(N \ L), ~o~)). Again the crucial point is to obtain results which verify that a given pair is a singular index pair. The simplest approach is to mimic the three conditions in the definition of an index pair as closely as possible. With regard to the exit set requirement, it is clear that L will need to contain
N - "-- {x 9 ON I r
t), x) r N for all t > 0}.
Positive invariance leads to the following construction. Given Y C N its push forward set in N under the flow ~00 is defined to be
p(Y, N) := {x 9 N I 3z 9 Y, t ~> 0 such that ~00([0, t], z) C N, ~p0(t, z) -- x }. Observe that p(Y, N) consists of the set of points which can be reached from Y by a positive trajectory in N and that Y C p(Y, N). The unstable set of an invariant set Y C N under qg0 is
W~(Y) "- {x 9 N [ r
O),x) C N and or(x, r
C Y}.
Notice that Y C W~v(Y). It is easy to find examples of isolating neighborhoods N for which there does not exist a set L such that (N, L) is an index pair for Inv N. The following condition prevents this. A slow entrance point x is a strict slow entrance point if there exists a neighborhood Ox of x and an ~ > 0 such that if y 9 69x n N and e 9 (0, ~], then there exists ty(e) > 0 for which qg~([0, ty(e)], y) C N. Let S ++ denote the strict slow entrance points.
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447
THEOREM 4.32. Let N be a singular isolating neighborhood. Assume (1) S~ consists of C-slow exit points. (2) S a C S + + U S a. (3) (S ++ \ Sa-) A c l ( N - ) -- O. For each x ~ S~, let Kx denote a compact invariant set as in Definition 4.29. Define
L "--p(cl(N-), N, ~po)U W~( U ~(Kx)). x ~ S~
lf L is closed, then (N, L) is a singular index pair for the family of flows ~o~. Notice that all the computations used in the construction of L involve only the singular flow. The higher order terms in (1) are only used in determining which points are slow exit and slow entrance points.
4.6. References One of the pioneering applications of the Conley index is due to Conley and Smoller and involved the proof of the existence of shocks waves [11-14]. The proofs were based on arguments involving attractor-repeller pairs. For a more complete discussion on this topic see [76]. Since then there have been a large number of applications of these techniques to finding traveling waves [ 10,30,51,53,60] The idea of the connection matrix was due to Conley, but its existence was first established by Franzosa [25-28]. It has since been generalized to the setting of maps, first by Robbin and Salamon [71 ] and later by Richeson [69]. As was mentioned in the text transition matrices were first developed in the singular form [68,50]. The topological transition matrices were defined in [46]. A purely algebraic development of the transition matrix can be found in [29]. The results on the existence of periodic orbits and the long exact sequence relating the index of the Poincar6 map with that of the flow can be found in [48] The first characterization of symbolic dynamics in terms of the Conley index was given in [55]. Theorem 4.26 based on the spectrum of the Conley index was proved in [5]. The optimality of this result is shown in [6].
5. Multivalued dynamics 5.1. Multivalued maps The theorems in Section 4.4 characterizing chaos in terms of the Conley index cannot be applied directly to differential equations, because the automorphism part of the index is always the identity in this case. However, there is hope that they could be applied to a Poincar6 map if such a map exists. The problem is that in most concrete applications our knowledge of the Poincar6 map is based mostly on numerical experiments.
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The only way to obtain rigorous information from numerical computations is to perform rigorous error analysis by means of interval arithmetic. As an outcome one obtains a multivalued map: the exact values are not known but sets enclosing the exact values are computed rigorously. It turns out that the multivalued map may carry sufficient information to make rigorous claims about the dynamics of the differential equation. To keep things simple we will study only dynamics in compact subspaces of Euclidean space. Let X, Y C IRd be two such spaces. Formally, a multivalued (mv) map from X to Y is a map F : X ~ 79(Y), where 79(Y) stands for the family of all subsets of Y. To emphasize that we think of F as a generalization of a single valued map, we will write F : X Z Y to denote a m v map from X to Y. We will also consider mv maps of pairs. We say that F : (X, X0) ~ (Y, Y0) maps (X, X0) into (Y, Y0) if F : X ~ Y is a m v map such that F(X0) C Y0. We assume that all mv maps considered in this paper have non-empty compact values. For A C X we define the image of A under F by
F(A)'--U{F(x)Ix6A
t.
For B C Y we define the weak preimage of B under F by
F * - l ( B ) "-- {x ~ X I F(x) A B # 0} and the strong preimage by
F - l ( B ) "= {x ~ X I F ( x ) C B}. The inverse of a multivalued map F ' X ~ Y is a m v map F - I ' Y ~ X defined by F - l ( y ) "-- {x E X ly ~ F(x)}. We say that F is upper semicontinuous (usc) if F -1 (U) is open for any open U C Y. By a selector of F we mean a single-valued continuous map f " X --+ Y such that f (x) 6 F (x) for every x 6 X. Sometimes we will also consider mv selectors. A m v selector is a mv map G ' X ~ Y such that G(x) C F(x) for every x 6 X.
5.2. Representable multivalued maps The computer can handle only a finite amount of information. In particular all arithmetic operations in the computer are performed on a finite subset IR C R " - IR U {-cx~, co}. We will call R a representation of IR and the elements of ~ representable numbers. In concrete realizations of computer hardware the set R usually contains -cx~, 0, cx~ and a certain set of binary fractions. Since no non-trivial finite set is closed under arithmetic operations, rounding is used whenever the result of an operation is not in IR. To keep rigor in computations, instead of performing the usual rounding to the nearest representable number, the result may be stored as a pair of two representable numbers: a lower and an upper bound obtained via rounding down and rounding up, respectively. The process is known as interval computations or interval arithmetic and may be easily implemented
Conley index A
449 m
A
on most.., computers. We saAythat a representation R l C R is a refinement of I~ if for any x, y ~ R there exists a z e R i such that x < z < y. To work with multivalued maps on the cAomputer we need the class of representable sets and representable multivalued maps. An R-representable interval setAin R d is a Cartesian product of d intervals in R with endpoints in R. An elementary R-representable set is a representable interval set which cannot be decomposed as the union of two non-empty representable interval sets. An R-representable set is a finite union of elementary representable sets. A nice feature of the class of representable sets is that it is closed under the set-theoretic union, intersection and difference as well as under the operations of taking the interior or closure. This means that any finite combination of these operations on representable sets is algorithmizable. A m v map F ' X Z Y is R-representable if for every x e X the set F(x) is R-representable and F restricted to any elementary representable set is constant. The image and preimage of a representable set may be computed algorithmically as the following proposition shows. A
A
PROPOSITION 5.1. The image and preimage of an R-representable set in an Rrepresentable mv map is R-representable. A
In the sequel we will usually drop the prefix R when speaking about representable sets and numbers, although in applications it is usually necessary to work simultaneously with several re~esentations. When working with representable numbers usually it is sufficient to take as R the representation embedded in hardware, though sometimes it is necessary to implement via suitable software larger sets of representable numbers. On the other hand, when working with representable set and mv maps, one usually takes the set of representable numbers which is much smaller than representable numbers embedded in hardware. The reason is that representable sets and mv maps are stored as lists and then large IR results in very large lists.
5.3. Approximation and inheritance As we already mentioned, we would like to study the properties of ~ngle valued maps by enclosing them in multivalued maps. We say that F :X ~ Y is an R-representation of f " U --+ Y if U C X, F is R-representable and f is a selector of Flu. Assume that {IR, },,er~ is a given, monotonically incrAeasing sequence of representations of R such that R,,+l is a refinement of R,, and ~,,cr~ R,, is dense in R. A natural question is whether given a continuous map f : U --+ Y it is possible to find a sequence {F,, },,~r~ of R,,-representations of f such that F,, --+ f , i.e., for every x 6 X and for every e > 0 there exists an N 6 N such that n ~> N, y 6 F,, (x) ~
dist(y, f (x)) ~< e.
A positive answer to the above question gives the following simple theorem.
K. Mischaikow and M. Mrozek
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THEOREM 5.2. If f : U --+ Y is L-Lipschitz continuous, then there exists a sequence { Fn }ncl~ : X :z~ Y of usc convex-valued representations of f such that Fn --+ f . The above theorem shows that, given a sufficiently powerful computer, any Lipschitz continuous map admits an arbitrarily close multivalued representation. In other words Lipschitz maps may be approximated by representable mv maps. However, this approximation has a feature which distinguishes it from classical approximations. In classical approximations there is no way to extract properties of the approximated function from just one or any finite number of approximating functions. In case of mv approximations it is sometimes possible because of a feature we call inheritance. To explain this simple idea assume ~4 is a collection of mv maps and qg(F) is a property of such maps. We say that q9 is inheritable if for every single-valued selector f of F in r (p(F) ~
qg(f).
We say that an inheritable property q9 is strongly inheritable if for any single valued map f ~ ,A such that qg(f) and for any sequence {Fn} C At satisfying Fn --+ f we have qg(Fn) for n sufficiently large. If or(F) is a term then we say that ot is inheritable (strongly inheritable) if for any x the property or(F) = x is inheritable (strongly inheritable). As an example consider the family A = {F : ~ ~ R}. Then the property
or(F) ~
3x ~ R F(x) > 0
is strongly inheritable. Replacing the strong inequality in the above example by a weak inequality we obtain a property which is inheritable but is not strongly inheritable. A simple example of a non-inheritable property is
y(F) ~
3x E R F(x) ~ O.
Most properties of interest are non-inheritable. However there are cases when one can convert inheritable properties into non-inheritable properties via suitable theorems. For instance a theorem, which converts properties like ot in the above example to property y, is the Darboux Theorem. Note that in case of single-valued maps property y reduces to the existence of a zero of a function. Thus representable mv maps together with the Darboux Theorem provide a way of verifying algorithmically the existence of a zero of a continuous function. In the sequel we will show that the Conley index may be used in a similar way. To do this we need to extend the Conley index to mv maps.
5.4. Conley index for multivalued maps We begin with a definition of a discrete multivalued dynamical system. DEFINITION 5.3. An usc mv map F : X x Z ~ X with compact values is called a discrete multivalued dynamical system (dmds) if the following conditions are satisfied:
Conley index
451
(i) For all x 6 X, F ( x , 0) = {x}; (ii) For all n, m 6 Z with nm >~0 and all x ~ X, F ( F ( x , n), m) = F(x, n + m); (iii) For all x, y 6 X, y 6 F (x, - 1 ) ~ x 6 F (y, 1). It is convenient to use the notation F 'z (x) := F ( x , n). The map F n coincides with the superposition of n copies of F l : X ~ X or its inverse (F1) -1 . This implies that the dmds F is uniquely determined by F l , the generator of the dmds F. For simplicity we will denote the generator also by F. Let F : X x Z Z X be a dmds and let N C X be a compact subset. Let A C Z. We say that cr :A --+ N is a solution to F in N through x if 0 6 A, o-(0) = x and o-(n + 1) F (~r (n)) for n, n + 1 6 A. Define the sets Inv(N, F) := {x E N I 3a :Z --+ N a solution to F in N through x}, Inv+(N, F) := {x 6 N I 3o-:Z + --+ N a solution to F in N through x}, I n v - ( N , F) := {x E N I3a : Z - --+ N a solution to F in N through x}. The set N is an isolating neighborhood for F if
BdiamNF(Inv N)
C int N,
where diamN F is the maximal diameter of the values of F in N. A pair P = (PI, P0) of compact subsets P0 C P1 C N is called an index pair for F in N if the following conditions are satisfied:
F(Pi)ANCPi,
i = 0 , 1,
F(P1 \Po) C N, Inv N C int(P1 \Po). Note that unlike the single valued case we defined here the index pair relative to an isolating neighborhood. This is because of some fundamental problems we encounter when trying to define the index map. The problems come from the fact that the homotopy and homology of multivalued maps are not obvious to define. Probably the most extensively studied is the homology and cohomology theory for multivalued maps [35,34]. In general some assumptions, usually quite restrictive, must be imposed on the mv maps to guarantee that the map in homology makes sense. In most cases the assumptions are satisfied in applications but are lost when we want to form an analogue of the index map (5). Therefore in the multivalued case it easier to construct the index directly on the homology level. A simple class of mv maps which do induce maps in homology consists of star-shaped maps. We say that F : X ~ Y is star-shaped if there exists a selector f : X --+ Y such that for any other selector g : X --+ Y and any x 6 X the segment joining f ( x ) and g(x) is contained in F (x). Notice that convex-valued maps which admit at least one selector are in particular star-shaped. In the sequel we assume that the generator F is star shaped. For a star-shaped mv map we define the map induced in homology by F, := f , : H , ( X ) ~ H , ( Y )
452
K. Mischaikow and M. Mrozek
where f :X --+ Y is an arbitrary selector of F. The definition is correct, because F admits at least one selector and evidently any two selectors are homotopic. If F : ( X , Xo) ~ (Y, Yo) is a star-shaped multivalued map of pairs, i.e., a mv map F :X ~ Y such that F(Xo) C Yo, then in a similar way we define the map
F, : H,(X, Xo) ~ H,(Y, Yo). The process of collapsing the exit set to a point may expel us from the class of starshaped maps. Therefore we have to define the index map differently. First we define regular index pairs. An index pair P for an isolating neighborhood N C X is called regular if f ( U \ P0) C P1 for some U D P0 open in Pl and
cl(F(Po)\P1) A c l ( P l \ P o ) = 0 . One can show that every isolated invariant set admits regular index pairs. We put
T(P) "-- TN, F(P) "-- (P1U ( X \ intN), Po U (X\ intN)). The following proposition follows from the excision property of the homology theory. PROPOSITION 5.4. If P is a regular index pair for N then (i) F(P) C TN, F ( P ) , (ii) The inclusion i p, T(p) : P --+ T(P) induces an isomorphism in homology. Actually, the regular index pair in the above proposition may be replaced by an arbitrary index pair if one uses Alexander-Spanier cohomology instead of homology. Consider the mv map Fp,T(p):P ~ x ---+ F(x) C T(P). The endomorphism le := H,(ip,T(p)) -1 o H,(Fp,T(p)) of H,(P) is called the index map associated with the index pair P. The following theorem lets us extend the Conley index to multivalued maps. THEOREM 5.5. Let S be an isolated invariant set. Then the Leray reduction of Ip : H,(P) --+ H,(P) is independent of the choice of an isolating neighborhood N of S for F and a regular index pair P for N. Now Con, (S, F), the homological Conley index of S is defined similarly to the single valued case as the Leray reduction of Ie : H, (P) --+ H, (P) for any regular index pair P in any isolating neighborhood N of S. As we will see in the sequel, the importance of the generalization of the Conley index to mv maps lies in the fact that the index is inheritable. More precisely we have the following theorem. THEOREM 5.6. The isolating neighborhood, the index pair, and the Conley index are strongly inheritable terms.
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453
5.5. Continuation As was indicated in the introduction and stated explicitly in Theorem 3.10 the Conley index continues, i.e., as long as the invariant set S remains isolated its Conley index does not change. The same is true in the mv case. To be more precise consider a compact interval A C It~ and F : A x X x Z=~ X
be an usc star shaped map such that, for each X 6 A, F;~ : X x Z Z X given by F;~ (x, n) := F (~., x, n) is a dmds. The first observation is the following easy to prove theorem. THEOREM 5.7. Let Xo c A and let N be an isolating neighborhood for F~o. Then N is an isolating neighborhood for F~ for all X sufficiently close to Xo. Another way of stating Theorem 5.7 is that isolating neighborhoods are stable with respect to small perturbations of generators. To prove the stability of the Conley index itself it would be nice to have stability of index pairs. However, as pointed out in [70], this is not true in general. Never the less, in some settings stable index pairs do exist, as the following theorem shows. THEOREM 5.8. Let f : X --+ X be a homeomorphism, N an isolating neighborhood for f and W an open neighborhood of Inv N. Then there exists an index pair P f o r N with PI \Po C W and an ~ > 0 such that if G : X ~ T)(X) is an usc proper map with the property G(x) C B ~ ( f ( x ) )
for all x E X,
then P also is an index pair for G. In particular P is an index pair for any small perturbation of f .
Observe that an immediate corollary of the above Theorem is the Continuation Property of the Conley index in the setting of homeomorphisms and flows.
5.6. Computability of the index Of course, in order to be able to compute the Conley index of a multivalued representation of a map we need to know how to construct index pairs. Actually, all what we need is to show that a representable mv map in a representable isolating neighborhood admits representable index pairs and find a formula for such an index pair. This is relatively easy for special isolating neighborhoods, namely isolating blocks. A compact set N C R J is an isolating block for F if Bdiamx F ( F *-l (N) N N n F ( N ) ) C intN.
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K. Mischaikow and M. Mrozek
THEOREM 5.9. If F : X ::~ Y is usc representable and N is a representable isolating block f o r F, then N is an isolating neighborhood and (N,N\F-l(intN)) is an index pair in N consisting of representable sets. Hence, index pairs may be constructed algorithmically.
Unfortunately isolating blocks are much harder to find than isolating neighborhoods. To construct an index pair for an arbitrary isolating neighborhood N we need first the following theorem in which FN denotes the map FN : N g x --+ N A F(x) C N. THEOREM 5.10. If F and N are representable then there exist numbers p, q ~ N such that
Inv-(N, F) -- A F~ (N),
Inv + (N, F) -
i--O,p
A
FN-i(N).
i=O,q
In particular Inv- N, Inv + N and Inv N are representable.
For a given ~-representable set A C R d define the following two sets o~(A) := U {E I E is elementary representable a n d c l E A A # 0}, n~(A) "-- U {cl E I E is elementary representable andcl E N A :fi 0}.
One easily verifies that o~(A) C n~(A) are both neighborhoods of A. Moreover, one can check that they are both representable, therefore they are algorithmically computable. The following theorem shows how one can obtain an index pair from Inv-(N, F) and Inv + (N, F). THEOREM 5.1 1. Assume F : X ~ X is a representable usc multivalued map and N is a representable isolating neighborhood with respect to a representation ~ C IRd. Let ~1 be a refinement of II~. Put Pl " - n~, (Inv- N) N N, P0 "= P1 \o~, (Inv + N). Then P1, Po are A
representable with respect to IRl and P :-- (P1, P0) is an indexpairfor F in N.
Theorem 5.11 leads to the following algorithm constructing index pairs. ALGORITHM 5.12 function
begin A := N; B := N;
repeat A ' := A; B ' := B;
find_index_pair(N, F)
Conley index
455
,-1
A'---- FN(A); B :-- F N (B); until (A = A' and B -- B'); C := A • B ; r := diamN F;
if (B(C, r) C int N) then begin P1 := n ~ (A) fq N" P0 :-- Pl \o~j (B); return (Pt, P0); end else return "Failure"; end; THEOREM 5.13. If Algorithm 5.12 is called with N, a representable compact set and F, a representable multivalued dynamical system on input, then it always stops. If it does not output "Failure", then N is an isolating neighborhood for F and the algorithm outputs an index pair (PI, Po)for F in N. 5.7. References Representable multivalued maps as a tool for rigorous numerical computation of the Conley index appeared in [56,57]. The idea of inheritance comes from [63]. The Conley index for multivalued maps was constructed in [41 ] and the existence of stable index pairs was established in [42]. The algorithms for computing the Conley index were presented in [63] and [80].
6. Examples of computer assisted proofs The use of the computer to numerically approximate isolating neighborhoods and to compute the associated Conley indices represents yet another new direction in the Conley Index theory. The examples presented below are included to emphasize two points: first, it is feasible to employ the computer to obtain rigorous analysis of dynamical systems using the index theory; and two, this approach can be applied to a variety of different types of problems.
6.1. Lorenz equations Representable multivalued maps together with inheritable properties open the way to rigorous computations of the Conley index for concrete dynamical systems. Since there are various qualitative descriptions of dynamics in terms of the Conley index, many new interesting results may be obtained this way. As an example consider a result on chaotic dynamics in the Lorenz equations (see [59] and also [56,57]). The result is based on a generalization of the Conley index theory proposed by Szymczak [79].
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K. Mischaikow and M. Mrozek
For a k • k matrix A = (Aij)
over
Z2
put
Z ( A ) "-- {or-Z --+ {1, 2 . . . . . k} I'v'i ~ Z Ao~(i)ot(i+l) -- 1 } a:r(A)
9or--+ ( Z 9 i)--+ ot(i + 1) ~ {1,2 . . . . . k} ~ E'(A).
Let n(10,28,8/3) = 6,
A(10,28,8/3)
n(10,54,45) = 4,
n(10,60,8/3)
=
0 0 0
1 0 0
1 0 0
0 1 0
0 1 0
0 0
1
0
0
0
0
0 0
1 0
1 0
0 1
0 1
0 0 0
=
A(lO,6O,8/3) =
1
'
o0 o, 0 0 o 11
I 1
1
0
0
0
0
1
0
'
A(10,54,45)
=
01 I 1 0
1
1
0
0
1
"
We have the following theorem THEOREM 6.1. Consider the Lorenz equations k -- s(y - x), -- R x - y - x z ,
(25)
= xy-qz, and the plane P := {(x, y, z) I z = R - 1 }. For all parameter values in a sufficiently small neighborhood o f (a, R, b) there exists a Poincard section N C P such that the associated Poincard map f is Lipschitz and well defined. Furthermore, there is a continuous map p : Inv(N, f ) ~ Hn(~.R.b) such that po f = a op
and p(Inv(N, f ) D Z(A(a,R,b)). Moreover, f o r each periodic u E Z ( A ) there exists an x ~ Inv(N, f ) on a periodic trajectory o f the same minimal period such that p ( x ) = t~.
An outline of the algorithm which proves the above theorem is as follows: (1) Choose a candidate for a representable isolating neighborhood N for the Poincar6 (2) (3) (4) (5)
map f . Compute a m v representation F of f . Check if N is an isolating neighborhood for F. If no, find a better mv representation of f and go back to 3. If yes, compute the index pair and find the Conley index.
Conley index
457
(6) By inheritance, f has the same Conley index in N. (7) If the computed index satisfies the assumption of the chaos criterion, proclaim SUCCESS, otherwise, proclaim FAILURE.
6.2. The Hdnon map Recall that the H6non map is given by the formula f (x, y ) -
(1 - a x 2 + y, bx).
A. Szymczak gave a computer assisted proof of the following result. THEOREM 6.2. For a -- 1.4 and b principal periods except for 3 and 5.
0.3 the Hdnon map admits periodic orbits of all
The proof is based on rigorous numerical computation of the Conley index.
6.3. Kuramoto-Sivashinsky equations Consider the following system of ordinary differential equations with parameter )v. X'0
m
Xl,
X'I
--
X2,
(26)
-) x(~
x2 -- -1.Xl - T + 1. One easily verifies that the equation has exactly two stationary points: x - : = ( - v ~ , 0, 0), x + "-- (v/2, 0, 0). An interesting question is if the equation admits a solution x (t) such that lim x(t) - x - , t---+ - - O G
lim x(t) = x +.
(27)
t ---->(X),
In case of )~ = 1 Equation (26) is the ODE derived from the Kuramoto-Sivashinsky PDE and studied numerically by several authors. It is shown in [82] that for )v = 1 Equation (26) admits heteroclinic connections. In case of )v - 0 the equation becomes a variant of an equation studied by Conley [7], who shows that it also admits a heteroclinic connection. It turns out that combining the technique of Conley and some ideas of Troy with a computer assisted proof that certain sets are isolating neighborhoods one can show that (26) admits a heteroclinic connection for every )~ 6 [0, 1]. In other words the following generalization of the results of Troy and Conley may be proven. THEOREM 6.3. For every ;Z E [0, 1] there exists a solution xz(t) of Equation (26) such that (27) is satisfied.
K. Mischaikow and M. Mrozek
458
6.4.
References
The computer assisted proof of chaos in the Lorenz equations for various parameter values was presented in [56,57,59]. Rigorous numerical verification of Theorem 6.3 was presented in [65] and of Theorem 6.2 in [80].
References [1] A. Baker, Topological entropy and the first homological Conley index map, Preprint (1998). [2] V. Benci, A new approach to the Morse-Conley theory and some applications, Ann. Mat. Pura. Appl. 4 (1991), 231-305. [31 V. Benci and M. Degiovanni, Morse-Conley theory (in preparation). [4] R. Bott and L.W. Tu, Differential Forms in Algebraic Topology, Springer, Berlin (1982). [5] M. Carbinatto, J. Kwapisz and K. Mischaikow, Horseshoes and the Conley index spectrum, Ergodic Theory Dynamical Systems (to appear). [6] M. Carbinatto and K. Mischaikow, Horseshoes and the Conley index spectrum II: the theorem is sharp, Dis. Cont. Dynamical Systems 5 (1999), 599-616. [7] C. Conley, Isolated lnvariant Sets and the Morse Index, CBMS Lecture Notes, Vol. 38, Amer. Math. Soc. Providence, RI (1978). [81 C. Conley, A qualitative singular perturbation theorem, Global Theory of Dynamical Systems, Z. Nitecki and C. Robinson, eds, Lecture Notes in Math., Vol. 819, Springer, Berlin (1980), 65-89. [91 C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), 35-61. [10] C.C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction diffusion model, Indiana Univ. Math. J. 33 (1989), 319-343. [11] C. Conley and J. Smoller, Viscosity matrices for two dimensional nonlinear hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 867-884. [12] C. Conley and J. Smoller, Viscosity matrices for two dimensional nonlinear hyperbolic systems, H, Amer. J. Math. 94 (1972), 631-650. [131 C. Conley and J. Smoller, On the structure ofmagnetohydrodynamic shock waves, Comm. Pure Appl. Math. 27 (1974), 367-375. [141 C. Conley and J. Smoller, On the structure of magnetohydrodynamic shock waves, II, J. Math. Pures Appl. 54 (1975), 429-444. [15] C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V.L Arnold, Invent. Math. 73 (1983), 33-49. [16] C. Conley and E. Zehnder, Morse type index theory for flows and periodic solutions for Hamiltonian systems, Comm. Pure Appl. Math. 37 (1984), 207-253. [17] B. Fiedler and K. Mischaikow, Dynamics of bifurcations for variational problems with 0(3) equivariance: A Conley index approach, Arch. Rat. Mech. Anal. 119 (1992), 145-196. [18] A. Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergodic Theory Dynamical Systems 7 (1987), 93-103. [19] A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), 93-103. [20] A. Floer, A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41 (1988), 393-407. [21] A. Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), 775-813. [22] A. Floer, Wittens complex and infinite dimensional Morse theory, J. Differential Geom. 28 (1988), 93-103. [23] A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), 575-611. [24] J. Franks and D. Richeson, Shift equivalence and the Conley index, Preprint. [25] R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), 193-213.
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[26] R. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), 561-592. [27] R. Franzosa, The continuation theory for Morse decompositions and connection matrices, Trans. Amer. Math. Soc. 310 (1988), 781-803. [28] R. Franzosa and K. Mischaikow, The connection matrix theory, for semiflows on (not necessarily locally compact) metric spaces, J. Differential Equations 71 (1988), 270-287. [29] R. Franzosa and K. Mischaikow, Algebraic transition matrices in the Conley index theory, Trans. Amer. Math. Soc. 350 (1998), 889-912. [30] R. Gardner, Existence of travelling wave solution of predator-prey systems via the Conley index, SIAM J. Appl. Math. 44 (1984), 56-76. [31] T. Gedeon, Cyclic Feedback Systems, Mem. Amer. Math. Soc., No. 637 (1998). [32] T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka and J. Reineck, The Conlev index for fast-slow systems I." One-dimensional slow variable, J. Dynamics Differential Equations (to appear). [33] T. Gedeon and K. Mischaikow, Global dynamics of cvclic feedback systems, J. Dynamics Differential Equations 7 (1995), 141-190. [34] A. Granas and L. G6rniewicz, Some general theorems in coincidence theory, J. Math. Pure Appl. 60 (1981), 661-373. [35] L. G6rniewicz, Homological methods in fixed point theory of multi-valued maps, Dissertationes Math. Vol. 129, PWN, Warszawa (1976). [36] J.K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys, Vol. 25, Amer. Math. Soc., Providence, RI (1988). [37] J.K. Hale, L.T. Magalhhes and W.M. Oliva, An Introduction to Infinite Dimensional Dynamical SystemsGeometric Theory, Appl. Math. Sci., Vol. 47, Springer, Berlin (1984). [38] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkh~iuser, Boston, MA (1994). [39] M. Hurley, Chain recurrence and attraction in noncompact spaces, Ergodic Theory Dynamical Systems 11, 709-729. [40] M. Hurley, Chain recurrence and attraction in noncompact spaces, H, Proc. Amer. Math. Soc. 115, 11391148. [41] T. Kaczyfiski and M. Mrozek, Conlev index for discrete multivalued dynamical systems, Topology Appl. 65 (1995), 83-96. [42] T. Kaczyfiski and M. Mrozek, Stable index pairs for discrete dynamical systems, Canad. Math. Bull. 40 (1997), 448-455. [43] H. Kokubu, K. Mischaikow, Y. Nishiura, H. Oka and T. Takaishi, Connecting orbit structure of monotone solutions in the shadow system, J. Differential Equations 140 (1997), 309-364. [44] H. Kokubu, K. Mischaikow and H. Oka, Existence of infinitely many connecting orbits in a singularly perturbed ordinary differential equation, Nonlinearity 9 (1996), 1263-1280. [45] C. McCord, The connection map for attractor-repeller pairs, Trans. Amer. Math. Soc. 307 (1988), 195-203. [46] C. McCord and K. Mischaikow, Connected simple systems, transition matrices, and heteroclinic bifurcations, Trans. Amer. Math. Soc. 333 (1992), 379-422. [47] C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, J. Amer. Math. Soc. 9 (1996), 1095-1133. [48] C. McCord, K. Mischaikow and M. Mrozek, Zeta functions, periodic trajectories, and the Conlev index, J. Differential Equations 121 (1995), 258-292. [49] J. Milnor and J.D. Stasheff, Characteristic Classes, Princeton Univ. Press, Princeton, NJ (1974). [50] K. Mischaikow, Transition systems, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 155-175. [51] K. Mischaikow, Travelling waves for cooperative and competitive-cooperative systems, Viscous Profiles and Numerical Methods for Shock Waves, M. Shearer, ed., SIAM, Philadelphia (1991), 125-141. [52] K. Mischaikow, Global asymptotic dynamics of gradient-like bistable equations, SIAM Math. Anal. 26 (1995), 1199-1224. [53] K. Mischaikow and V. Hutson, Travelling waves for mutualist species, SIAM Math. Anal. 24 (1993), 9871008. [54] K. Mischaikow and Y. Morita, Dynamics on the global attractor of a gradient flow arising in the GinzburgLandau equation, Japan J. Indust. Appl. Math. 11 (1994), 185-202.
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[55] K. Mischaikow and M. Mrozek, Isolating neighborhoods and Chaos, Japan. J. Indust. Appl. Math. 12 (1995), 205-236. [56] K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc. (N.S.) 33 (1995), 66-72. [57] K. Mischaikow and M. Mrozek, Chaos in Lorenz equations: a computer assisted proof Part H: details, Math. Comp. 67 (1998), 1023-1046. [58] K. Mischaikow, M. Mrozek and J. Reineck, Singular index pairs, J. Dynamics Differential Equations 11 (1999), 399-426. [59] K. Mischaikow, M. Mrozek and A. Szymczak, Chaos in Lorenz equations: a computer assisted proof Part IlL" the classic case, J. Differential Equations (to appear). [60] K. Mischaikow and J. Reineck, Travelling waves for a predator-prey system, SIAM Math. Anal. 24 (1993), 1179-1214. [61 ] M. Mrozek, Index pairs and the fixed point index for semidynamical systems with discrete time, Fund. Math. 133 (1989), 177-192. [62] M. Mrozek, Leray functor and the cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 149-178. [63] M. Mrozek, Topological invariants, multivalued maps and computer assisted proofs in dynamics, Comput. Math. 32 (1996), 83-104. [64] M. Mrozek, An algorithmic approach to the Conley index theory, J. Dynamics Differential Equations 11 (1999), 711-734. [65] M. Mrozek, M. Zelawski, Heteroclinic connections in the Kuramoto-Sivashinsky equation, Reliable Computing 3 (1997), 277-285. [66] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Science Publ. (1994). [67] M. Po2niak, Lusternik-Schnirelmann category ofan isolated invariant set, Univ. Jag. Acta Math. 31 (1994), 129-139. [68] J. Reineck, Connecting orbits in one-parameter families offlows, Ergodic Theory Dynamical Systems 8* (1988), 359-374. [69] D. Richeson, Connection matrix pairs for the discrete Conley index, Preprint (1997). [70] J.W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8* (1988), 375-393. [71] J.W. Robbin and D. Salamon, Lyapunov maps, simplicial complexes and the Stone functor, Ergodic Theory Dynamics Systems 12 (1992), 153-183. [72] C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7, 425437. [73] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, FL (1995). [74] K.P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer, Berlin (1987). [75] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 1-41. [76] J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer, New York (1980). [77] E.H. Spanier, Algebraic Topology, McGraw-Hill, New York (1966), Springer, Berlin (1982). [78] A. Szymczak, The Conley index for discrete dynamical systems, Topology Appl. 66 (1995), 215-240. [79] A. Szymczak, The Conley index for decompositions of isolated invariant sets, Fund. Math. 148 (1995), 71-90. [80] A. Szymczak, A combinatorial procedure for finding isolating neighborhoods and index pairs, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997). [81] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York (1988). [82] W.C. Troy, The existence of steady state solutions of the Kuramoto-Shivashinsky equation, J. Differential Equations 82 (1989), 269-313.
CHAPTER
10
Functional Differential Equations Roger D. Nussbaum* Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The initial value problem for FDEs: existence and uniqueness theorems . . . . . . . . . . . . . . . . . . 3. Linear autonomous FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Locating zeros of characteristic equations: some examples . . . . . . . . . . . . . . . . . . . . . . . . . 5. The fixed point index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Periodic solutions of functional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
*Partially supported by N S F D M S 97-06891. H A N D B O O K OF D Y N A M I C A L SYSTEMS, VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 461
463 464 468 470 484 489 496
Functional differential equations
463
1. Introduction
Roughly speaking, a functional differential equation, or FDE, is a differential equation for which x'(t) depends not only on x(t) but also on the "past history" of the function s --+ x (s) for s ~< t. A precise formulation will be given later. Deceptively simple-looking examples are provided by
x ' ( t ) - - f ( x ( t ) , x ( t - 1)) and x'(t) - g ( x ( t ) , x ( t - 1), x(t - y)), where f and g are given functions and y > 0. Sometimes such equations are called differential-delay equations. Examples of functional differential equations can be traced back two hundred years. A 1911 article by Schmitt [70] lists a variety of early work on linear functional differential equations. However, both the systematic development of the theory of FDEs and the study of nonlinear FDEs are essentially twentieth century phenomena. The past forty years especially have seen an increasing flow of articles on the subject; and in recent years a number of books on FDEs have appeared. The interested reader should certainly consult Introduction to Functional Differential Equations by Hale and VerduynLunel [ 19] (an updating and extension of Hale's classic text [20]) and Delay Equations by Diekmann et al. [9]. The first of these books has a dense bibliography of twenty-five pages while the second has a bibliography of fifteen pages. An intriguing feature of the global study of nonlinear FDEs is that progress in understanding even the simplest-looking FDEs has been slow and has involved a combination of careful analysis of the equation and heavy machinery from functional analysis and algebraic topology. A partial list of tools which have been employed includes fixed point theory and the fixed point index (see [3,4,17,61-64]), global bifurcation theorems (see [40,53]), a global Hopf bifurcation theorem (see [7,55]), the Fuller index (see [7,12,13,15]), ideas related to the Conley index (see [ 14,39,50,69]), and equivariant degree theory. Nevertheless, even for the so-called Wright's equation,
x'(t) = - o t x ( t -
1)(1 + x(t)),
which has been an object of serious study for more than forty-five years, many questions remain open. It will be clear from the previous remarks that the title of this article is hopelessly immodest. A reasonably complete survey of the field of FDEs would require a large book. The goal here is much more limited. I shall first recall some of the basic theory of FDEs, particularly existence and uniqueness theorems for initial value problems and elementary linear theory. Then I shall describe the fixed point index, a tool from algebraic topology which has proved useful in studying nonlinear FDEs, and I shall sketch how the fixed point index can be applied to some equations to prove the existence of so-called "slowly oscillating periodic solutions". I shall also mention some open problems. My hope is that the reader will be encouraged by this introduction to explore further in the literature.
464
R.D. Nussbaum
2. The initial value problem for FDEs: existence and uniqueness theorems The proper abstract framework for the study of an FDE may well depend on the equation in question. However, the following framework is frequently useful. Given R > 0, let X = C ( [ - R , 0], R n) denote the Banach space of continuous maps x : [ - R , 0] --+ N n. As usual, i l l . I denotes a given norm on R n and x e X, we define the norm on X by Ilxll = sup{Ix(t)l: t e [ - R , 0]}. If M > 0 and y : [ - R , M ) --+ IRn is a continuous map, we shall follow standard notation (see [9,19]) and define for each t with 0 ~< t < M a function Yt E X by yt(s) = y ( t + s) for - R ~< s ~< 0. Now suppose that to < T are real numbers, that D is a closed subset of [to, T) x X and that f : D --+ IRn is a continuous map. Given 0 e X such that (to, 0) e D and T > to, one can ask whether there exists a continuous map x : [to - R, T] ~ R n such that (t, xt) e D for to ~< t ~< T and x ' ( t ) = f ( t , xt)
forto~
(1)
Equation (1) is an example of an FDE, and it will be general enough for our purposes. If g : R k ~ R is a continuous map and rl, r2 . . . . . rk are nonnegative reals, the differential delay equation x ' ( t ) -- g ( x ( t -- r l ) , X ( t -- r2) . . . . . x ( t -- rk))
is an example of Equation (1), with R = max{rj: 1 ~< j ~< k}, X = C ( [ - R , 0], R) and f : X --+ IR defined by f(~o) -- g(~o(--rl), ~o(-r2) . . . . . q)(--rk)). Equation (1), with the specified initial condition x(to + s) = O(s) for - R ~< s ~< 0, is an initial value problem for an FDE. If one uses the Schauder fixed point theorem in a manner similar to that often used in studying the initial value problem for ordinary differential equations, one obtains an existence theorem for the local initial value problem for FDEs. THEOREM 1. Suppose that R > O, X = C ( [ - R , 0], R n) and f : R x X ~ IRn is a continuous map. Suppose that to e N and 0 e X. Then there exists T > to and a continuous function x :[to - R, T] ~ R n such that x(to + s) = O(s) f o r - R <~ s <<.0 and x ' ( t ) = f (t, xt),
to <,t <, T.
I f there exist constants A and B such that Ilf(t, then one can choose T = oc.
qg)ll ~ Allgoll -+- B
f o r all (t, q)) e IR x X,
PROOF. Given 6 > 0 and M > 0, define Y(6) := Y : = C([to - R, to + 6], R n) and define V = V ( M , 6) by V := V ( M , 6) : = {x e C([to - R, to + 6], Rn): x(to + s) = O(s) for - R ~< s ~< 0 and Ix(t) - 0 ( 0 ) l <~ M for to ~< t ~< to + 6}.
Functional differential equations
465
Given eo > 0, the reader can verify that there exists 6o > 0 such that [Ixt - 0 [I < M + eo for all t with to <~ t ~< to + 6o and for all x ~ V (M, 6o). If we choose M sufficiently small, it follows from the continuity of f that there exists Ml such that [f(s, xs)] <~ Ml for all x 6 V (M, 6o) and all s with to ~< s ~< to + 6. Select 6o such that Ml 6 ~< M and 6 <~ 6o and define F : V(M, 6) --+ Y by
(Fx)(t) --0(0) +
f (s, xs) ds,
to <~t <~to+ 6,
and (Fx)(to + r) = O(r) for - R ~< r <~ 0. The selection of M, Ml and 6 insures that F ( V ( M , 6)) C V(M, 6). If x ~ V(M, 6) and y = F(x), then ly'(t)[ ~< M1 for to ~< t <~ to + 6, and y(to + r) = O(r) for - R ~< r ~< 0, so F ( V ( M , 6)) is a bounded, equicontinuous family of functions, and the Ascoli-Arzela theorem implies that F ( V (M, 6)) has compact closure. It follows that F : V(M, 6) ~ V(M, 6) is a compact map; and since V(M, 6) is a closed, bounded convex set in a Banach space, the Schauder fixed point theorem implies that F has a fixed point x in V (M, 6). The fixed point x gives the desired solution of the initial value problem. If If(t, q9)1 ~< Allqgll + B for all (t, qg) 6 R x X, one can obtain a uniform estimate for 6 above. If M > 0 and 6 > 0 and x ~ V (M, 6) (where V (M, 6) is as defined above), notice that IIx, II ~< II0 II + M for to ~< t ~< to + 6. It follows that
[f(t, xt)[ ~ A(II011 + M) + B for to ~< t ~< to + 6. We conclude from this inequality that for to ~< t <~ to + 6 and x V(M, 6)
I ( F x ) ( t ) - o(o)1 ~
f
t
If(s, xs)lds <~A(II011 + M)~ + B~. )
Select 6 (independent of 0, to and M) so that 6A ~< 1. Then, if M is chosen with M ~> 2(A [10116 + B6), it follows that
I(Fx)(t) -0(0)1 ~ A(IIOII + M)6 + B6 ~ M for to ~< t ~< to + 6, and F ( V (M, 6) C V (M, 6). shows that F has a fixed point x in V (M, 6), and [to, to + 6]. It remains to show that a solution of the initial R, oe]. Suppose that k is a positive integer, xk :[to for - R ~< s ~< 0, and
x'k(t) = f ( t , (xk)t)
Now the same argument given before xto = 0 and x satisfies Equation (1) on value problem can be defined on [to -
- R, to + k6] --+ R n, xk(to + s) = O(s)
for to ~< t ~< to + k6.
466
R.D. Nussbaum
Define tk = to + k6 and define Ok 6 X by Ok(s) = xk(tk + s) for - R ~< s ~< 0. By the previous work, there exists a continuous map x : [tk - R, tk + 6] ~ IR" such that x(tk + s) = Ok(s) = xk(tk + s) for - R ~< s ~< 0 and x ' ( t ) = f (t, xt)
fortk~
If one defines Xk+l (t) = x ( t ) for tk ~< t ~< tk + 6 and Xk+l (t) = xk(t) for to - R ~< t ~< tk, one sees that xk+l(t0 + s) = 0 ( s ) for - R ~< s ~< 0, Xk+l [[t0 -- R, tk] = xkl[t0 -- R, tk] and x~+l(t) -- f ( t , (xk+l)t)
for to ~< t ~< to + (k + 1)6.
By mathematical induction, the functions xk can be defined for all k ~> 1. If x ( t ) is defined by x ( t ) = x k ( t ) for to - R ~< t <~ to + k6, x : [to - R, oo) ~ R is the desired solution of the initial value problem. D One also wants theorems which insure that the initial value problem associated with Equation (1) has a unique solution. If X := C ( [ - R , 0], IRn), J : = [to - R , oo] and f : J x X ~ ]~n is a map, recall that f is called locally Lipschitzian in the X-variable if, for every t E J and 0 ~ X, there exists 6o,t = 6 > 0 and Co,t = C > 0 such that for all ~ ~ X and ~o ~ X with I[lP - 011 ~< 6 and 11q9- 011 ~< 6 and all real s with Is - t[ ~< 6 one has [f ( s , r
- f ( s , qg)[ ~< CIl~ -goll.
If f : J x X --+ ]~n is continuous and locally Lipschitzian in the X-variable and if F and V (M, 6) are defined as in the proof of Theorem 1, one can prove that M and 6 can be chosen so that F ( V ( M , 6)) C V ( M , 6) and F I V ( M , 6) is a Lipschitz map with Lipschitz constant k < 1 (a contraction mapping). It follows from the contraction mapping principle that F has a unique fixed point in V (M, 6). By exploiting this fact one can obtain the following uniqueness result, whose detailed proof is omitted. THEOREM 2. Let J and X be as above and suppose that f : J • X ~ R n is a continuous map which is locally Lipschitzian in the X-variable. Suppose that T > to and that x : [to - R, T] --+ •n and y:[t0 - R, T] --+ R n are continuous functions such that x(to + s) = y(to + s) f o r - R <~ s <<.0 and x ' ( t ) -- f (t, xt) and y ' ( t ) -- f (t, Yt) f o r to <<.t <~ T. Then it f o l l o w s that x ( t ) = y ( t ) f o r to - R <<.t <<.T. For simple differential-delay equations, Theorems 1 and 2 can be supplanted by use of the corresponding existence and uniqueness theorems for the initial value problem for ordinary differential equations. For example, suppose that g" 1--Ikj=0 ]l~n ~ ]l~n is a continuous map and that rj, 1 <<.j ~ k, are given positive numbers. Let R = max{rj [ 1 ~< j ~ k} and r = min{rj [ 1 ~< j ~< k}. If to 6 IR and O : [ - R , 0] --+ R" is a given continuous function, one can try to solve, for to ~< t ~< to + r, the special case of Equation (1) given by x ' ( t ) -- g ( x ( t ) , x ( t -- r l ) , x ( t -- r2) . . . . . x ( t -- rk)),
Xto = 0 .
(2)
Functional differential equations
467
However, for to <~ t <~ to + r, Equation (2) is equivalent to an ordinary differential equation (or ODE)" x ' (t) = 9=
g ( x ( t ) , O(t -- rl -- to), O(t -- r2 -- to) . . . . . O(t -- rk -- to)) h(t, x(t)),
(3)
Xto - - O.
It follows from the existence theory for the initial value problem for ordinary differential equations that Equation (3) (and hence Equation (2)) has a local solution defined on some interval [to, to + 3], 0 < 6 ~< r. If 6 = r, one can iterate this procedure, but with to replaced by tl = to + r and 0 replaced by 01(s) = x ( t l + s), - R <<, s <~ O. In this way one can replace the initial value problem for the FDEs given by Equation (2) by a succession of initial value problems for ODEs. Furthermore, one actually obtains a sharper uniqueness result from this kind of argument than from Theorem 2: If g is continuous and also locally Lipschitzian in the x ( t ) variable, then the solution of Equation (2) is unique. However, there are many examples for which one needs the full strength of Theorems 1 and 2 or for which Theorems 1 and 2 are inadequate. In studying certain classes of FDEs (for example, "differential-delay equations with state dependent time lags" as in [44,45, 47]), the function f in Equation (1) may be naturally defined on a proper subset D of X = C ( [ - R , 0], R"), R may have to be determined, and the domain D may need to be further restricted to insure uniqueness for solutions of the initial value problem. A simplelooking example which illustrates all these problems and which plays a central role in the analysis in [45] is given by /
e x (t) = - x ( t )
- k x ( t - r),
r "-- 1 + c x ( t ) ,
(4)
where e > 0, k > 1 and c > 0. To make sense of (4) as an FDE, one needs to insure that 1 + c x ( t ) >~ 0 for all t. Let R := k / c and let D = {0 6 C ( [ - R , 0], IK): - 1 / c <~ O(s) <<,k / c for - R ~< s ~< 0}. Given to 6 ]K and 0 6 D, one can prove that there is a continuous map x : I t 0 - R , cx~) --+ [ - 1 / c , k / c ] such t h a t x ( t o + s ) = 0 ( s ) f o r - R ~< s <~ 0 and x(t) satisfies Equation (4) for all t >~ to. If 0 is Lipschitzian, x is unique. The reader is referred to [45,47] for a discussion of Equation (4) and of more general examples. Equation (1) does not include some interesting cases. One may want to include equations for which x'(t) depends on both the past history of x on [t - R, t] and the past history of x' on [t - R, t]. Such equations are often called neutral FDEs. For neutral FDEs, the analogue of the map F in the proof of Theorem 1 is no longer compact, and fixed point theorems more sophisticated than Schauder's are needed. Further details and references to the literature can be found in [19]. The previous discussion may have overemphasized the similarity between the initial value problem for ODEs and FDEs. For ODEs, solving the initial value problem forward in time t is equivalent to solving backward in time. For FDEs this is false: the problem of backward continuation is nontrivial and is analogous to the same problem for parabolic PDEs. Indeed, given a nonlinear FDE x' (t) = f ( x t ) , even one as simple as Wright's equation, it is a nontrivial problem to find a nonconstant function x : ~ --~ ~" which solves the equation for all t. Roughly, speaking (see [59] for details), if f is analytic and x : ~ --+ ]Kn is bounded, then x is necessarily real analytic.
468
R.D. Nussbaum
3. L i n e a r a u t o n o m o u s F D E s
Suppose that L : C ( [ - R , 0], R n) ~ R n is a bounded linear map; L extends uniquely in the obvious way to a bounded, complex linear map, which we shall also denote by L, from C ( [ - R , 0], C n) to C n. More generally, we assume throughout this section that L : C ( [ - R, 0], C n) :-- Y --+ C n is a bounded, complex linear map. If 0 E Y and we identify C n with ~2n, the results of the previous section prove that there is a unique, continuous function x : [ - R , cx~) --+ C n with x f(t) = L(xt)
(5)
for t >~ 0 and x0 = 0.
For each 0 E Y, let x(.; 0) :-- x denote the corresponding solution of Equation (5) and for t ~> 0 define T ( t ) : Y ~ Y by T(t)(O) = xt. One can prove easily that for each t ~> 0, T (t) : Y ~ Y is a bounded, complex linear map and satisfies the following properties: (1) T ( 0 ) = I; (2) T (t + s) = T (t) T (s) for all t, s ~> 0; (3) limt~s+ T ( t ) ( O ) = T(s)(O) for s ~> 0 and all 0 ~ Y. In general, a family of bounded linear maps T (t), t ~> 0, of a Banach space Z into itself is called a C0-semigroup if it satisfies properties (1), (2), and (3). In our case, an application of the Ascoli-Arzela theorem shows that T(t) is compact for all t ~> R. In general, a C0semigroup (T(t): t ~> 0) is called "eventually compact" if there exists to > 0 such that T ( t ) is compact for all t ~> to. There is an extensive literature concerning linear semigroups. We refer to [5,16,67,72, 80] for expositions of the general theory and to [9,19] for aspects of the theory for linear, autonomous FDEs. Here we recall only a few facts and refer the reader to [9,19] and the general references for proofs and further details. In general, if (T(t): t ~> 0) is a C0-semigroup on a Banach space Z, one defines an operator A, called the infinitesimal generator of (T(t): t ~> 0), by
A(qg) = 7z ~
lim t-+0 +
T (t)q9 - q9
=0.
(6)
By definition, D(A), the domain of A, is the set of q9 E Z for which the limit in Equation (6) exists. One proves that A is a closed, densely defined linear operator. In our case, D ( A ) = {rp 6 Y I 99 is continuously differentiable on [ - R , 0] and qg'(0) = L(qg)} and A(rp) = q9I. In general, for a closed, densely defined linear operator A : D ( A ) C Z --+ Z (Z a complex Banach space), p ( A ) , the resolvent set of A, is the set of z E C such that z l - A is a one-to-one map of D ( A ) onto Z. The spectrum of A, o-(A), is the complement of p ( A ) in C; and crp (A), the point spectrum of A, is the set of z E C such that z I - A is not one-toone. In our case, o-(A) = ~re(a), cr(T(t)) \ {0} = cre(T(t)) \ {0} = exp(tcr(a)), and one can give an explicit description of cr (A). For each z E C and v 6 C n, define e(z, v) E Y by e(z, v)(t) = exp(zt)v,
- R <~ t <<,O.
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469
For z 6 C, define A(z) : C" ~ C" by
a(z)(v) = zv-
C(e(z, v)),
(7)
and note that A(z) is a complex linear map and A(z)(v) = 0 if and only if e(z, v) ~ D(A). One can prove that z ~ a ( A ) if and only if there exists v ~ C n, v -~ O, with A(z)(v) = O; and since A (z) is linear, the latter condition is equivalent to det(A (z)) - - 0 ,
(8)
the so-called characteristic equation. If z E C and v 6 C n satisfy A(z)(v) - - 0 , one easily checks that y(t) := eZtv satisfies y'(t) = L(yt) for all t 6 R. For each real number #, it is relatively easy to prove that the set { z l d e t ( A ( z ) ) = 0 and Re(z) > / z } is finite. Thus, for each X 6 or(A), there is a number rx > 0 and ball Bx = {w 6 C [[w - X[ ~< rz} such that Bz A Bx, = 0 whenever X, X' ~ or(A) and X 5/= X'. Let Fx denote the boundary of Bx, oriented counterclockwise and, using the functional calculus for linear operators (see [30,80]), define
Pz--
~
(~ l -
A ) - l d~ 9
It follows from the functional calculus (see [30,80]) that Px :Y --+ Y is a bounded linear projection (so p2 = pz) and that Pz Px' - 0 whenever X, X' 6 a ( A ) and X #- X'. If Mx "-R(Px) := the range of Px, Mx is a closed linear subspace of Y, Mx is contained in the domain of A and A Px = Px A. Furthermore, X is a pole of the resolvent operator ~" --+ (ff - A ) - I ; and if the order of this pole is k, then
Mx = {y E Y I(XI - A) k ( y ) - O } . Indeed, all of the above facts have analogues for a closed, densely defined linear operator A on a general Banach space and a isolated point X of or(A): see [30,80]. If A is a finite subset of ~r(A), one can define PA = Y~X~A Pz and easily check that PA is a bounded, linear projection, MA := R (PA) is a closed linear subspace of Y, M A is the direct sum of Mx for X 6 A, and A PA = PA A. In our case A is the infinitesimal generator of an eventually compact C0-semigroup (T (t): t ~> s), and one can derive from this that each Mx is finite dimensional and T (t) Px = Pz T (t) for all t ~> 0. A deeper fact is that the dimension of Mx is the multiplicity of X as a zero of the equation det(A(z)) = 0: see [9,19,26,36]. Of course it is desirable to have explicit formulas for Px (y), and the reader is referred to [9,19] for such formulas. Just the knowledge that )~ 6 cr (A) is an isolated point of ~r (A) implies that Mx is contained in D(A), the domain of A, and APz = PxA; so Ax :-- A IMx is a closed operator defined on all of Mz, and the closed graph theorem implies that A x : M x ~ Mx is a bounded linear operator. It follows that Tx(t) := T(t) [ Mx is given by Tz(t) = exp(tAx) for t >~ 0 and the semigroup (Tx (t): t ~> 0) extends to a C0-group given by exp(tAz) for t6R.
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470
If co 6 IR and A = {~. 6 or(A) I Re(1.) ~> co} and R ( I - PA) denotes the range of I - PA, one can prove that T(t) I R ( I - P a ) is a C0-semigroup with infinitesimal generator A I R ( I - PA). Furthermore, there is a constant M such that I I T ( t ) ( I - PAll <. M exp(cot) for all t ~> 0. Using PA, one can write
T ( t ) y = T ( t ) ( l - PA)Y + T ( t ) P A y . One has control of the growth of T ( t ) ( I - PA) as t --+ oe, and T ( t ) P A can be explicitly computed. The above remarks provide a sketchy introduction to the theory of linear autonomous FDEs. There remain intriguing questions which we have not mentioned. For example, do there exist nonzero solutions of Equation (5) which converge to zero at a "superexponential rate" as t --+ cx~? The reader is referred to [ 19] for further details and references to the literature.
4. Locating zeros of characteristic equations- some examples The results of the previous section suggest the importance of locating the zeros of the characteristic equation d e t ( A ( z ) ) - 0 for a linear, autonomous FDE y'(t) = L(yt). Recall that this is equivalent to finding those z 6 C for which there exists v 6 C n, v :/: 0, such that y(t) := eZtv satisfies the equation y'(t) - L(yt) for all t 6 R. Even for relatively simple-looking, linear, autonomous FDEs, locating the zeros of the characteristic equation precisely enough to establish various results may be highly nontrivial. In most cases of which we are aware, the tools used for analyzing characteristic equations and their close relatives have been essentially elementary: a judicious use of Rouch6's theorem combined with a close analysis (basically calculus) exploiting the particular form of the equation. This comment applies, for example, to classical work of Pontryagin [68]. Pontryagin, studies (among other things) trigonometric polynomials of the form
m ~_~ f (z) "-- Z aJkzJ ekz'
amn 7/=O,
j = 0 k=0
and asks when all solutions z of f (z) = 0 satisfy Re(z) < 0. He succeeds in generalizing earlier work of Hurwitz (for polynomials) and of N.G. (~ebatarev (n - 1) and gives useful necessary and sufficient conditions that every solution of f ( z ) = 0 satisfy Re(z) < 0: see [68] and the appendix of [19]. Here we shall be content to illustrate the sorts of methods typically used by discussing three important examples:
(9)
z = -/~ e - : , z =-ct-/~e-:
and
z = -a - b exp(-rz) - c exp(-sz).
(10) (11)
Functional differential equations
471
The constants c~,/3, a, b, c, r and s will always be assumed real. These equations arise, respectively, as the characteristic equations of the linear differential-delay equations 1),
x'(t) =-~x(t-
x ' ( t ) = - o t x ( t ) - ~ x ( t - 1)
and
x ' ( t ) = - a x ( t ) - b x ( t - r) - c x ( t - s),
where x (t) 6 C for all t. One can easily show that for any real number d, there are only finitely many solutions z of Equations (9)-(11) which satisfy Re(z) > d. The reader should keep in mind that there are important examples which may not have this property. Typically, the characteristic equation of a linear neutral functional differential equation does not have this property, as one can see by considering the neutral FDE x'(t)-
1)--~x(t-
kx'(t-
1)
and its associated characteristic equation z-kze
-~ - - ~ e - : .
Another intriguing class of examples arises in work in progress by J. Mallet-Paret and S. Verduyn-Lunel. Mallet-Paret and Verduyn-Lunel consider x' (t) = a x ( t ) + b x ( t - 1) + c x ( t + 1),
and the associated characteristic equation z-a+be--+ce:.
In view of the central role played by Rouch6's theorem in the analysis of characteristic equations, we begin by stating Rouch6's theorem in a general form. Typically, the theorem is stated less generally in complex variables textbooks, but the usual proofs yield the version given below. THEOREM 4.1. Let G C C be a bounded, open set whose boundary OG comprises a finite number o f simple, closed, rectifiable Jordan curves which are given the standard positive orientation. I f H is an open neighborhood o f G and f " H --+ C is an analytic function such that f (z) =/=0 f o r all z E OG, then f has a finite number n o f zeros in G (counting multiplicity) and
n =
~i
a
f(z)
dz " - deg(f, G, 0).
I f F" H x [0, 1] ~ C is a continuous map such that Ft(z) "-- F ( z , t) =/=Of o r all (z, t) E OG x [0, 1] and z --+ Ft(z) is analytic on H f o r 0 <~ t <, 1, then if nt, t - O, 1, denotes the number o f zeros o f Ft in G (counting multiplicity), it follows that no - n l.
R.D. Nussbaum
472
If one uses the Brouwer degree, one can prove that the second part of Theorem 4.1 remains true without the assumption that Ft is analytic on H for 0 ~< t ~< 1 and without any assumptions on 0 G. It is often useful to give an alternate, more geometrical integral for deg(f, G, 0). Suppose that OG is a simple, closed rectifiable Jordan curve and that OG is given parametrically (with proper orientation) by a continuous, piecewise C 1 curve z(t) = x(t) + iy(t), a ~< t ~< b. Assume that f is as in Theorem 4.1 and that f ( z ) = u(x, y) + iv(x, y), where z -- x + iy and u and v are real-valued. Assuming that f ( z ) ~ 0 for z ~ OG, it is well known that there exists a continuous, piecewise C l function O(t), a <~t <<,b, such that f(z(t))-
lf(z('))lexp(iO(t)), a ~
~< b;
and 0 (t) is uniquely determined to within integer multiples of 2re. As usual, one can write
O(t) ~ arg(f (z(t))), a <~t <<,b. It is easy to show that 1 f
f'(z)
--dz
2zri J~c f (z)
-
l fab -dtd o(t) dt
l fab f ' (z(t)) f (z(t)) z' (t) dt - 27c
2:rri
l fb[u(x(t),y(t))d-~ v(x(t), y (t))
2re
-
d (x
v(x(t), y(t)) ~7 u
Thus, deg(f, G, 0) is simply (1/2rc)[O(b) - 0(a)]. Notice that in general
f:(z(t)) d f (z(t)) z' (t) r -dt o(t), so, depending on f , one integral formula may be more useful than another. The basic approach which will be used in analyzing Equations (9)-(11) will be to homotope them to simpler equations and apply Theorem 4.1. Occasionally the implicit function theorem will also be employed. To analyze Equation (9), define for each real number/3, f/~ (z) - z + 13e x p ( - z ) .
(12)
For each integer n define
Gn-
{z ~ C l ( 2 n - 1)re < Im(z) < (2n § 1)re }.
(13)
As usual, for a complex number z - x + iy, ~ - x - iy will denote the complex conjugate of z. We shall assume that 13 > 0; analogous arguments apply to the case/3 < 0. Our first lemma is a simple calculus exercise. LEMMA 4.1. l f fl > O, the equation f~(x) - 0 has a real root x if and only ifO < 13 < 1/e. If 0 < fl < 1/e, f~ (x) = 0 has precisely two real roots, z-(fl) and z+ (fl), with z-(fl) <
Functional differential equations
473
- 1 < z+(fl) < 0; a n d z - ( f l ) a n d z+(fl) are roots o f multiplicity one. I f fl -- l / e , x = - 1 is a root o f multiplicity two o f the equation f ~ (x ) = O. PROOF. Notice that ffi (x) = 1 - / 3 e x p ( - x ) for x 6 IR, so f~ is strictly decreasing on ( - c o , ln(fl)], strictly increasing on [ln(fl), oc) and has its m i n i m u m on IR at ln(fi), with m i n i m u m value ln(fl) + 1. The assertions of the l e m m a follow from these remarks. D If fl < 0, the reader can easily verify that the equation f/~ (z) - 0 has precisely one real root, and that root is positive and of multiplicity one. LEMMA 4 . 2 . / f f l E IR, z E (2, f/~(Z) - - 0 and f~(z) --0, then i t f o l l o w s that z = - 1 and / 3 - 1/e. PROOF. By adding the equations f/~ (z) -- z + f le - = -- 0 and f~ (z) -- 1 - f l e - = -- 0, one obtains z = - 1 , which then implies that fl - 1/e. [] L e m m a 4.2 implies that, with the exception fi -- 1/e and z = - 1 , any root of f~ (z) - - 0 has multiplicity one. For m o r e general equations like Equation (11), nothing so simple is true, and analysis of the equations is c o r r e s p o n d i n g l y m o r e complicated. LEMMA 4.3. I f fi E IR, n E Z, a n d z E O(G,,), then f ~ ( z ) 7k O. For each nonnegative integer n, define fi,, = 2njr + Jr/2. For n >~ 1, z -- fi,, i is the only solution z E G,, o f f~,, (z) - O. For n --O, z -- + f l 0 i a n d z - - f l 0 i are the only solutions z E Go o f f~o (z) - O . PROOF. If Z E O(G,,), then z = # + (2n - 1)jr i or z = # + (2n + 1)jr i for some lz E R. It follows in either case that I m ( f ~ (z)) = (2n 4- 1)jr i 7~ 0. Suppose next that n ~> 1, z E Gn and f/~,, (z) = 0. One easily shows that f/~,, (i/~,,) = 0. It remains to prove that z = ifl,,. Writing z = # + iv, with (2n - 1)jr < v < (2n + 1)Jr and /z E IR, the equation f~,, (z) = 0 gives # -- -fi,, e - u cos(v)
and
(14)
v --/3,, e - u sin(v).
(15)
Since sin(v) ~< 0 for (2n - 1)jr < v ~< 2njr and v > 0, Equation (15) implies that 2njr < v < ( 2 n + 1)jr. If (2njr + j r / 2 ) < v < ( 2 n + 1)jr, cos(v) < 0 and Equation (14) implies that # > 0. It follows that fin e - u < fi,,, so Equation (15) implies that v < fin = (2njr + Jr/2), a contradiction. Thus one must have that 2njr < v ~< fi,,. If v = fl,,, Equation (14) implies that # = 0. Thus one can assume that 2njr < v < fi,,, cos(v) > 0 and (from Equation (14)) /z < 0. B e c a u s e Iz[ 2 - - fl,21 e x p ( - 2 z ) [ , one has "9 fl,2 e x p ( - 2 / ~ ) - # 2 _ V-.
One can easily check t h a t / z --+/3,2 e x p ( - 2 # ) /3,2, e x p ( - 2 / z ) -
- #2 is strictly decreasing on ( - o o , 0], so
# 2 > fl21,
and since v 2 < fl,2, this is a contradiction, which proves the l e m m a w h e n n 7> 1.
R.D. Nussbaum
474
If n = 0, it suffices (recalling that f~0 (z) = f~0 ( 7 ) for all z and that ft~0 has no real roots) to prove that i/30 is the only solution z 9 Go of f~0 (z) = 0 with Im(z) > 0. However, if z = # + iv with/z 9 R and 0 < v < rr, the same argument used when n ~> 1 proves that z = i/30. [] THEOREM 4.2. For each fl > 0 and nonzero integer n, the equation 0 - - f ~ ( z ) := z - / 3 e x p ( - z ) has precisely one zero (counting multiplicity) z = Zn(fl) 9 Gn :-- {z 9 C I (2n - 1)7r < Im(z) < (2n + 1)7r}. The map fl --+ Zn(fl) is infinitely differentiable. For each fl > O, the equation f ~ ( z ) = 0 has precisely two solutions (counting multiplicity) in Go. For fl # 1/e and fl > O, these solutions are distinct and of multiplicity one; and f o r fl = 1 / e , - 1 is a zero of multiplicity two. For fl > 1/e, the solutions in Go are complex conjugates, zo(fl) and zo(fl), with Im(z0(fl)) > 0; and the map fl --+ zo(fl) is infinitely differentiable. For n ~ O, and fl > O, there exists z 9 Gn with f ~ (z) = 0 and Re(z) > 0 if and only if ~ > ~,, := 2nrr + rr/2. PROOF. For/3 > 0 and a fixed integer n, consider the homotopy on Gn given by
Ft,n(Z) "-- z - [(1 - t)~ -4- t ~ n ] e x p ( - z ) ,
0 ~ t ~ 1.
The idea is to use Rouch6's theorem, but care is needed because G,, is unbounded. Thus define Gn,R = {z 9 Gn I IRe(z)l < e}. For a fixed n, one can prove that there exists R = Rn > 0 such that Ft,n(Z) ~: 0 for all z 9 Gn with IRe(z)l /> R. L e m m a 4.3 implies that Ft,n(Z) # 0 for z 9 O(Gn,R) and 0 ~< t ~< 1. Because Gn,R is bounded, T h e o r e m 4.1 is applicable and implies that f~ (z) and f~,, (z) have the same number of zeros in Gn,R. By L e m m a s 4.3 and 4.2, f~,, has exactly one zero in G,, and that zero has multiplicity one. Therefore, T h e o r e m 4.1 implies that f~ (z) has precisely one zero (counting multiplicity) in G,~. The same argument shows that f~ (z) = 0 has precisely two (counting multiplicity) solutions in Go. Using L e m m a s 4.1 and 4.2, one sees that these solutions are distinct for /~ ~ 1/e and that z = - 1 is a double root for/~ = 1/e. For/~ > l / e , the solutions are not real and hence are complex conjugates z0(/~) and z0(/~), with Im(z0(/~)) > 0. For each n ~> l, let zn(/~) denote the unique solution z 9 Gn of f ~ ( z ) = 0. The corresponding solution z-n(/~) 9 G - n is given by z-,,(/~) = z,,(/~). Using L e m m a 4.2, f ~ ( z ) - - 0 implies that f ~ ( z ) # 0, u n l e s s / ~ - 1/e and z - - l , so the implicit function theorem implies that for n ~> 1 and/~ > 0, the map/~ ~ zn (/~) is C ~ and for n = 0 and /~ > 1/e, the map/~ --+ z0(/~) is C ~ . Furthermore, writing z(/~) = Zn (/~), the implicit function theorem gives
z'(r
=
z(~) + Iz(r /311 + z(/3)l 2 "
It follows from this equation that if z(fl) = iv is pure imaginary, then Re(z'(fl)) > 0 and there exist 0 < fl, < fl < fl* with Re(z(fl,)) < 0 and Re(z(fl*)) > 0. If f~ (iv) = 0 for v 9 ]I{ and fl > 0, one easily checks that cos(v) = 0 and v = fl sin(v). If iv 9 Gn, where n is a positive integer, one concludes that v = 2nrr + re/2 = fin and
Functional differential equations
475
fl = fin. Similarly, if iv E Go, one concludes that v = -+-zr/2 and fl = re/2 = 130. Our previous remarks imply that there exists e = en > 0 such that Re(zn (/3,, -Jr-en)) > 0 and Re(zn(fin - en)) < 0 for n >~ 0. If n >~ 0 and there exists 13 > fin with Re(zn(fl)) ~ O, then Re(zn (fl)) < 0. By continuity of the map 9 / ~ Re(z,7(9/)) for 9/ E [/3, + e,,, fl], there must exist 9/ E [/3,, + en, fl] with Re(zn (9/)) = 0, which implies that 9 / = fi,7 and gives a contradiction. A similar argument shows that for n ~> 1, Re(zn(fi)) < 0 for 0 < 13 < fin. If n = 0, L e m m a 4.1 implies that f ~ ( z ) has two (counting multiplicity) negative real roots in Go for 0 < 13 ~< 1/e. Our previous argument shows that f ~ ( z ) = 0 has two roots in Go for /~ > 0, so f~ has only real roots in Go for 0 0 and n a positive integer, Equation (9) has no solution z with (2n - 1)Jr ~< Im(z) <~ 2nrr. For 13 < 0, Equation (9) has no solution z with 2nTr ~< Im(z) ~< (2n + 1)7r. Thus, for n >~ 1, the solution z,(/3) in Theorem 4.2 satisfies 2nrr < Im(zn (/3)) < (2n + 1)rr. REMARK 4.2. The proof of T h e o r e m 4.2 depends on L e m m a 4.3, and L e m m a 4.3 is almost a direct proof of T h e o r e m 4.2 in the case 13 :--/~,, := 2nTr + 7r/2, n >~ 0. One can ask whether a different argument, more in the spirit of Rouch6's theorem, can be given; and in fact, there is such an argument. Let n be a nonnegative integer. By using T h e o r e m 4.1 and the homotopy z + [(1 - t)/~0 + t/~l] e - : = 0, one can prove that (assuming/~0/31 > 0) z +/~0 e - : = 0 and z +/~l e - : have the same number of solutions in G,z. As usual, one must work on G,,.R = {z E Gn ]IRe(z)[ < R} for R sufficiently large. Simple estimates comparing the size of Izl and I/~ e - : l for z E G , and Re(z) ~< 0 show that there is a number y, > 0 such that z + 13 e - : r 0 for all z E Gn with Re(z) ~< 0 and all/3 > y,,. It is also true that z +/~ e -~ --/-0 for all z with Ira(z) = 2nTr and 13 > ?',7. Define F, = {z E Gn I Re(z) > 0 and 2nTr < Im(z) < (2n + 1)Tr }. The above remarks show that in order to prove z + 13 e -~ = 0 has precisely one zero in Gn for n ~> 1 and 13 > 0 and precisely two zeros in Go for 13 > 0, it suffices to prove that for n ~> 0 and all sufficiently large/~ > 9/,, z + / ~ e -~ = 0 has precisely one solution in F,,. To prove the latter statement, define 0,7 = 2nzr + (37r/4), and for 13 > 9/,, and 0 <~ t ~< 1 consider the homotopy
Ft(z) = z -
tO,, - itO,, + (1 - t ) ~ e - : .
One can prove that there exists fin ~> Y,, such that Ft (z) =/=0 for z E 0 F,,, 0 <~ t ~< 1 and fl > fl,7. The delicate part of the argument is to show that Ft(iv) ~ 0 for 2nzr ~< v ~< (2n + 1)7r. Using the homotopy and T h e o r e m 4.1, one concludes that for /~ > fl,, z + 13 e -~ = 0 has precisely one solution in F,,. The case fl < 0 can be handled by similar arguments and yields the following theorem. Details are left to the reader. THEOREM 4.3. For each ~ < 0 and each integer n, the equation 0 = f ~ ( z ) := z + / ~ e x p ( - z ) has (counting multiplicity) exactly one solution z : z,,(~) E G,, = {z E C, I
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(2n - 1)7r < Im(z) < (2n + 1)Tr}. For n = O, zo(~) is a positive real number. The map ~ Zn(~), ~ "< O, is C ~ for all n 9 Z. For n ~ 1 and ~ < O, there exists z 9 Gn with f~(z) = 0 a n d R e ( z ) > 0 if andonly if ~ < -(2nTr - re/2).
Notice that Theorems 4.2 and 4.3 immediately imply that Re(z) < 0 for every solution z of Equation (9) if and only if 0 < 13 < rr/2. One hardly need say that the results of Theorems 4.2 and 4.3 are not new. Equation (9) was already exhaustively analyzed by L6meray [35] in 1897; see, also, Wright's classic article [74]. There may, however, be some novelty in the approach given here. The great advantage of Rouch6's theorem is that homotopy arguments can be used to obtain information about complicated equations from simple equations. The next theorem illustrates this point by deriving information about Equation (10) from Equation (9). Of course one can also observe that z is a solution of Equation (10) if and only if w := z + ot is a solution of w + (/3 e ~) e -w = 0, so results from Theorems 4.2 and 4.3 (with/3 replaced by/3 e ~) give corresponding results for Theorem 4.4 below. THEOREM 4.4. Assume that ot 9 R, ~ > 0 and z 9 C and define g(z; ~, ~) := z + ot + /3 e x p ( - z ) . The equation g(z; or, ~) -- 0 has a real root if and only ifO
of Equation (10), and this solution is of multiplicity one. The map (or, ~) --+ z,,(ot, ~) is C ~. There exist precisely two (counting multiplicity) solutions z 9 Go of Equation (10). If/3 > exp(-ot - 1), these solutions can be written zo(ot, ~) 9 Go and zo(ot, ~), with Im(z0(ot,/3)) > 0. The map (or, ~) --+ zo(ot, 13) is C ~ f o r / 3 > exp(-ot - 1). If ot ~ ~, Equation (10) has no solution z with Re(z) ~> 0. If or <<,- ~ , Equation (10) has precisely two (counting multiplicity) real solutions in Go, one solution nonnegative and one solution nonpositive. If ot <~ - ~ and n ~ O, one has Re(zn (c~,/3)) < 0. PROOF. By arguing as in Lemma 4.1, one can prove the assertions about real roots of
g(z; or,/3) = 0. Details of the calculus argument are left to the reader. For R > 0, define G,,R as in the proof of Theorem 4.2 and on Gn,R consider the homotopy
Ft(z) -- g(z; ( 1 - t)ot + ~),
0~
If Im(z) = (2n -+- 1)re, a calculation shows that Im(Ft(z)) -- (2n 4- 1)re, so Ft(z) ~ 0 for z 9 OGn. Simple estimates show that there exists R = Rn such that Ft (z) ~ 0 for 0 <~ t ~< 1 and for z 9 Gn with IRe(z)l ~> Rn. Theorem 4.1 now implies that g(z; ct, 13) = 0 has the same number of solutions in Gn as g(z; 0,/~) = f~ (z) = 0. Thus Theorem 4.2 implies that Equation (10) has (counting multiplicity) precisely one solution z := Zn(t~, ~) 9 G,, for n -~ 0 and precisely two solutions z 9 Go. If/3 > exp(-c~ - 1), the two solutions in Go are not real and must be complex conjugates, z0(ot,/~) and z0(ct,/3), with Im(z0(ot,/3)) > 0. Og A simple calculation shows that if g(z; or, ~ ) - 0 and ~ ( z , ot,/~) = 0, then z -c~ - 1 and /3 = exp(-c~ - 1). It follows that if z 9 Gn, n ~ O, and g(z; ~, ~) = O,
Functional differential equations Og
then ~ ( z , ot,/3) r 0; and if z E Go and g(z; or,/3) = 0 and 13 > e x p ( - o t -
477 1), then
0g (z ' ot ' /3) =/: 0. Thus the implicit function t h e o r e m implies that (or,/3) --+ z,7 (or,/3) is C ~ Oz for n -r 0 and (c~,/3) --+ zo(a, fl) is C ~ for/3 > e x p ( - o t - 1). If ot >~ /3 and g(z'c~,/3) = 0 for z = # + i v , with Re(z) = # >~ 0, one obtains Re(g(z" or, 13)) -- # + ot + / 3 e - u cos v. This gives lz + o r = l# + ~ l -
I/3e - " c o s v l - / 3 e - ~ l
cosvl ~ / 3 .
If ot > / 3 or # > O, we obtain a contradiction, so the only possibility is ot - / 3 , / z cos v - - 1 and sin v - O. Using sin v - 0 one finds that
- 0 and
I m ( g ( z ; ot, fl)) -- v - - O , so z = 0, which contradicts g(0; or,/3) = ot + I3 > 0. If/3 6 IR, one can check that/3 ~< exp(/3 - 1), with equality if and only if/3 = 1. If ot ~< - / 3 , it follows that/3 <~ exp(-c~ - 1), with equality if and only if ot = - / 3 = - 1 , so g(z; c~, fl) has precisely two (counting multiplicity) real solutions in Go, and these solutions are distinct unless ot = - / 3 = - 1 . Because g(0, or, r = ot + [3 ~< 0, these solutions are, respectively, nonpositive and nonnegative. Define H, + = {z E G,, I R e ( z ) > 0} and H,~- = {z ~ G,, IRe(z) < 0}"' and for R > 0 define H n+, R -- {z E H, + I IRe(z)l ~< R}, with a similar formula for H,~.R. For n ~ 0 and z E H +,,,R and ot ~< - / 3 , consider the h o m o t o p y
~ t ( z ) = g(z" ot, ( 1 - t)fl). The reader can verify that q~t(z) ~ 0 for all z ~ 0(14,+). F u r t h e r m o r e there exists R -Rn > 0 such that ~ t ( z ) =/=0 f o r 0 ~< t ~< 1 and for all t c H, + with Re(z) ~> R. T h e o r e m 4.1 implies that g(z; or,/3) = 0 has the same n u m b e r of solutions i n / 4 , + as z + ot - 0. Thus g(z; or, fl) -- 0 has no solution in the closure o f / 4 , + for n :/= 0. D One is also interested in determining w h e n the solution z,, (or, fl) 6 G,,, n ~ 0, satisfies Re(z,,(c~,/3)) > 0. With the aid of T h e o r e m 4.4, this question can be answered with the same kind of reasoning used in T h e o r e m 4.2. THEOREM 4.5. A s s u m e that ot c ~, fl > 0 and z ~ C and define g(z; or, fl) = z + ot + /3 e x p ( - z ) . For each nonzero integer n and real n u m b e r ot there is a unique positive real = ~,7(ot) such that Equation (10) has a pure imaginary root z -- iv ~ G,,. l f n >~ 1 and v -- v,~ (o~) is the unique real n u m b e r with 2nTr < v < (2n + 1)Tr such that v cos(v)
sin(v)
-- -c~,
(16)
then fl,,(ot) = fl-,z(~) -- v,,(ot)/sin(v,l(ot)). For fl > 0 and n ~ O, the equation g(z; c~, fl) = 0 has a unique solution zn(ot, fl) ~ G,,, and Re(z,,(ot, 13)) > 0 if and only if fl > fl,,(ot). I f - 1 < or, there is a unique real n u m b e r v -- vo(ot) ~ (0, re) which satisfies Equation (16). I f - 1 < ~ and fl >/30(or) : = vo(ot)/sin(vo(ot)), Equation (10) has two
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(counting multiplicity) roots zo(u, fl) E Go and zo(u, fl) E Go, and Re(z0(u, fl)) > 0 and Im(z0(u,/3)) > 0. IfO <<.u and 0 < fl < flo(u), Equation (10) has no solution z ~ Go with Re(z) ~> 0. I f - 1 < u < 0 and - u < ,6 < flo(u), Equation (10) has no solution z E Go with Re(z) ~> 0. I f - 1 < u < 0 and 0 < fl < - u , Equation (10) has precisely one solution z E Go with Re(z) > 0, and this solution is real. I f u <~ - 1 , and fl > - u , Equation (10) has (counting multiplicity) two solutions z E Go with Re(z) > 0; and if u <<.- 1 and 0 < ~ < - u , all solutions z E Go o f Equation (10) are real and one solution is positive and one negative.
PROOF. The argument that (assuming fl > 0 and n -76 0) Re(zn (U, /3)) > 0 if and only if /3 > fin(u) is essentially the same as in Theorem 4.2 and is left to the reader. The point is to show that iv 6 Gn is a solution of Equation (10) if and only if v = v,7 (u) and/3 = fin(u). We restrict attention to n = 0 and let H~-, H o , H +0,R and H0~R be as in the proof of Theorem 4.4. By using Theorem 4.4 for any fixed u 6 R and/3 > 0 we see that there are precisely two root of Equation (10), z+ (u,/3) and z_(u,/3), in Go. These roots are real for 0 < fl < e x p ( - u - 1) and z - ( u , / 3 ) < z+(u,/3). At/3 = e x p ( - u - 1) these roots coalesce into a double real root; and for/3 > e x p ( - u - 1), the roots become a complex conjugate pair with z_(u, fl) = z+(u,/3) and Im(z+ (u, /3)) > 0. By using Theorems 4.1 and 4.4 one can prove that the map (u, fl) -+ z+ (u,/3), u 6 R,/3 > 0, is continuous. The problem is to translate this simple picture into the statement of the theorem. First, assume that u > - 1 and /31 >/30(u). The reader can verify that there exists /32 >/31 such that 1[32e x p ( - z ) l > Izl + I~1
for all z ~ closure ( n o ) .
It follows that for all fl ~> fie, Equation (10) has no solution in H o . Select R > 0 so that Equation (10) has no solution z ~ Go for IRe(z)l ~> R and for/31 ~
'
H 0+, R ' 0)
----deg(g2, H+O,R,0 ) + deg(g2, H~, R, O) = deg(g2, G0,R, 0) -- 2. It follows that if u > - 1 and/3 :=/31 >/30(u), Equation (10) has (counting multiplicities) precisely two solutions in H +. If u ~> 0, these solutions are clearly not real. In general, if - 1 < u, the reader can verify a calculus lemma, namely, v sin v
> exp
(vcosv) sin v
1
Jr f o r 0 < v < -2'
and conclude that/3 > fl0(u) > e x p ( - u - 1). It follows that for/3 > fl0(u), Equation (10) has no real roots. The function v --~ v cos v/sin v is strictly decreasing on [0, Jr) and achieves its maximum value of 1 at v = 0. Thus, if u ~< - 1 , the only possible pure imaginary solution of
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479
Equation (1 0) in Go is z = 0, and this is a solution if and only if/3 = -or. The same homotopy argument used above shows that if oe ~< - 1 and/3 > -oe, Equation (10) has, counting multiplicity, precisely two solutions in H~-. If ot >7 0, one can argue with the aid of Theorem 4.1 that the number of solutions z 9 H o of Equation (10) (counting multiplicity) is constant for 0 < fl < fl0(ot). For fl sufficiently small and/3 > 0, Equation (10) has two, distinct, negative real solutions. Theorem 4.4 implies that there are exactly two solutions of Equation (10) in Go, so Equation (1 0) has no solutions in H~- for 0 < fl < fl0(ot) and ot >~ 0. The remainder of the proof is left to the reader. [] As an easy consequence of Theorem 4.5, one can determine exactly when all solutions of Equation (10) have negative real part. We state the following corollary for the case/~ > 0, but the same result is true in general. COROLLARY 4.1. Assume that c~ 9 R and ~ > 0 in Equation (10). Then all solutions z o f Equation (10) satisfy Re(z) < 0 if and only if ~ > - 1 , ot + fl > O, and ~ < flo(~), where ~o(oe) "= vo(ot)/sin(vo(ot)) and vo(o~) 9 (0, Jr) is the unique solution v 9 (0, Jr) o f vcos(v)
toO/.
sin(v) PROOF. The case ot - - 0 is covered by Theorems 4.2 and 4.3, so we assume ot 7~ 0. If o t - - / 3 , z - 0 solves Equation (10). If ot ~< - 1 and either/3 > -or or/3 < -or, Theorem 4.5 implies that Equation (10) has a root with positive real part. Thus, if ol ~< - 1 , Equation (10) has a root z with Re(z) >~ 0. If ot +/3 ~< 0, it is clear that Equation (10) has real root z ~> 0. If/3 >/~0(oe), Theorem 4.5 implies that Equation (10) has a root z 9 Go with Re(z) > 0; and if/3 -/30(or), Equation (10) has a pure imaginary root in Go. Conversely, assume that o~ > - 1 , ot +/~ > 0 and 0 0 if and only if/3/>/3, (or). By using the facts that w --+ w / s i n ( w ) is strictly increasing on [0, Jr) and w ~ 1/cos(w) is strictly increasing on [0, 7r/2) and on (7r/2, Jr], one can prove that/~, (or) >/~m(Ot) >/30(or) for 0 < m < n. Details are left to the reader. It follows, since 0 0, then z 9 Go. If such a z exists, Theorem 4.5 implies that - 1 < ot < 0. I f - 1 < ot < 0, Theorem 4.5 implies that Equation (10) has no solution with Re(z) >~ 0 i f - o r 0. [-1 The results of Theorems 4.4 and 4.5 and Corollary 4.1 are not new, although there may be some novelty in the proof. The reader is referred to Hayes's article [23] for further details. See also the appendix of [1,19]. In passing from Equation (10) to Equation (11), one enters terra incognita. Equation (11) has been studied by several authors: see, for example, [18,38], Chapter III of [57], and references in these papers. In [18,38], one seeks a "stability analysis" of Equation (11), i.e., one wishes to determine (for J := [0, ec)) S := { ( a , b , c , r , s ) e R 3 x J • J IRe(z) < 0 for all z 9 C which satisfy Equation (11) }.
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If ( a , b , c , r , s ) ~ S and x ( t ) is a solution of an initial value problem for x ' ( t ) = a + b x ( t - r) + c x ( t - s), the results of Section 3 imply that limt--,oc x(t) = 0. One can assume for stability analysis that a + b + c > 0, since otherwise Equation (11) clearly has a nonnegative real root. However, numerical and theoretical studies (see [ 18,38]) show that any complete stability analysis for Equation (11) must necessarily be extremely complicated. Hale and Huang [ 18] fix (a, b, c) ~ R 3 with a + b + c > 0 and consider the "stable region" Da,b,c := D for Equation (l l). If Va.b,c := V := {(r, s) 6 J x J: ( a , b , c , r , s ) ~ S}, D is the maximal connected component of V which contains (0, 0). It is known (see [ 18]) that the boundary of Da,b,c may be complicated, with many "kinks". To make the problems more manageable, Hale and Huang ask asymptotic questions. Given r > 0 and s > 0, when is ()~r,)~s) ~ Da,b,c for all large ~.? When is ()~r,)~s) ~ Da,b,c for all large X? Even these questions are not easy. Aside from stability analysis, one may also need to prove that Equation (11) has a solution which lies in a specified region of C. If I is a given interval of reals, one may need to prove existence of a solution z of Equation (11) with Re(z) > 0 and Im(z) 6 I. Even for simple, special cases of Equation (11), precise answers to such questions may be very complicated: see, for example, the discussion of z + oee -z + / 3 e -3~ = 0 in Theorem 3.3 of [57]. If s / r = n / m , where n and m are positive integers, then writing rz = m w (so sz = nw), Equation (11) is equivalent to w +
(r) --
a +
(
r
) ( r ) b e -mw +
c e -nw - - 0 .
m
The latter equation is of the form studied by Pontryagin, so one can try to apply results from [68], but typically these results only provide a starting point for detailed analysis. Here, we restrict ourselves to some general comments about Equation (11). Define sets U, M a n d M ~ as follows: U = {(a, b, c, r, s) EIR 5 [ a 4 - b + c ~ > 0 , r~>0ands~>0}, M - - { ( a , b , c , r , s , v) ~IR6 l ( a , b , c , r , s ) ~ U a n d i v + a + b e - i r v + c e - i s v - - O } andM ~ = { ( a , b , c , r , s , v) ~ M [a + b + c > 0, r > 0 a n d s > 0}. If (a, b, c) 6 N 3 and a + b + c > 0, define sets Na,b,c and N a,b, c o as follows: Na.b,c -- {(r,s, v) E N 3 1 ( a , b , c , r , s ,
v) e M }
and
N aO, b , c -- {(r,s ' v) e Na.b.c I r > 0 a n d s > 0} The above notation will be used for the remainder of this section. LEMMA 4.4. M ~ is a f o u r dimensional C ~ manifold; and if a + b + c > 0 and N~
7/: 0,
then Na,b. 0 c is a one dimensional C ~ manifold. I f p" R 6 ~ R 5 and q" R 3 ~ R 2 are defined by p ( a , b, c, r, s, v) - (a, b, c, r, s) and q(r, s, v) - (r, s), then p ( M ) is a closed subset o f U, q(Na,b,c) is a closed subset o f R 2 and p ( M ) D {(a, b, c, r, s) 6 R 5 [ a + b + c - 0, r >>.0 and s >~ 0}. PROOF. Take fixed real numbers a, b, c with a + b + c > 0 and define gl (r, s, v) - a + b c o s ( r v ) + c c o s ( s v ) and g2(r, s, v) - v - b s i n ( r v ) - c s i n ( s v ) . If we write x - (r, s, v),
Functional differential equations
481
then Na,t),c o 0 c is a -- {x E ]t~3 I r > 0, s > 0, gl (x) - - 0 and g2(x) ~- 0}. It follows that Na,b, C a , one dimensional manifold if Vgl (x) and V g2(x) are linearly independent for every 0 c. The condition a + b + c > 0 implies that gl (r, s, 0) > 0; so if (r, s, v) E NOa,b,c, x ~ Na,b, v :/: 0. The equation g2 (r, s, v) = 0 then implies (since v :/: 0) that (b, c) r (0, 0). If x = 0 c, a calculation gives Vgl (x) = ( - b y sin(rv), - c v sin(sv), - b r sin(rv) (r, s, v) ~ Na,b, cs sin(sv)) and Vg2(x) = ( - b v c o s ( r v ) , - c v c o s ( s v ) , 1 - br cos(rv) - cs cos(sv)). The above remarks show that if x ~ N~ , then XTgl (x) ~- 0 and Vg2 (x) ~ 0. If Vgl (x) and Vg2 (x) are linearly dependent, there must exist X :/: 0 with Vgl (x) = XVg2 (x). The latter equation yields b sin(rv) - X b c o s ( r v ) and c sin(sv) = )~ccos(sv). Substituting these equalities in the equation X(1 - br cos(rv) - cs cos(sv) = - b r sin(rv) - cs sin(sv), one finds that X = 0, which is a contradiction. Note that it may easily happen that NOa,b,c -- 0. If a > Ibl + Icl, then la + ivl /> a > Ibl + Icl/> [be -i'v + ce-isvl, and Equation (11) has no pure imaginary solution. The proof that M ~ is a (nonempty), four dimensional, C a manifold is similar but easier, and is left to the reader. If v = 0, r ~> 0, s ~> 0 and a + b + c = 0, then (a, b, c, r, s, 0) E M, which implies the final statement of L e m m a 4.4. If (a, b, c, r, s, v) 6 M, then I v l - livl -- la + b e x p ( - i r v )
+ cexp(-isv)]
<~ lal + Ibl + Icl.
If (ak, bk, ck, rk, sk, vk) := (Yk, Vk) is a sequence of points in M and Yk --+ Y, it follows that y 6 U and Irk [, k >~ 1, is a bounded sequence. By taking a subsequence, we can assume that vk --+ v; and since M is closed, (y, v) 6 M and p ( y , v) = y E p ( M ) . This proves that p ( M ) is closed. A similar argument proves that q(N~.b.~) is closed. D DEFINITION 4.1. If x, y E U - p ( M ) , we shall say that "x and y are pathwise connected in U - p ( M ) " if there exists a continuous map ~" : [0, 1] --+ R 5 with ~'(0) = x, ~'(1) = y and ( ( t ) ~ U - p ( M ) for 0 ~< t ~< 1. If Q -- {(r, s) E ]R 2 I r ~> 0 and s ~> 0} and a + b + c > 0, two points x ~ Q - q(Na.b.c) and y E Q - q(Na.b.c) are pathwise connected in Q - q(Na.b.c) if there is a continuous map 0:[0, 1] --+ Q - q(Na.b.c) with 7/(0) = x and ~(1) = y. The stability region Da.b.c defined earlier is the set of points x = (r, s) ~ Q - q ( N a . b , c ) which are pathwise connected to (0, 0) in Q - q(Na.b,~). THEOREM 4.6. A s s u m e that (ao, bo, co, ro, so) ~ U - p ( M ) a n d (al, bl, Cl, rl, S1) E U - p ( M ) are p a t h w i s e c o n n e c t e d in U - p ( M ) . Then the equations z + aj q- bj e rjz qcj e - s j : o , j = O, 1, have the same n u m b e r o f solutions z (counting multiplicity) with Re(z) > 0. PROOF. Let ~" : [0, 1] --+ U - p ( M ) be the continuous map in Definition 4.1 and write (t) = (at, bt, ct, rt, st) for 0 <~ t ~< 1. For 0 ~< t ~< 1, consider the homotopy gt (Z) := Z + at -+- bt e x p ( - r t z) -Jr-ct e x p ( - s t z).
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By definition, gt(iv) r 0 for v 6 IR and 0 ~< t ~< 1. Also, if R, = max{latl + Ibtl + Ictl: 0 t <~ 1 }, gt (z) 7~ 0 for Re(z) ~> 0 and Izl > e , . If H = {z 6 C IRe(z) > 0} and H e = {z H I lzl < R l, it follows that for R > R , , gt(z) 5~ 0 for Z 6 0 H e and 0 ~< t ~< 1. Using Theorem 4.1, one concludes that go and g l have the same number of zeros in H. D As a trivial corollary of Theorem 4.6, we have the following well-known result. COROLLARY 4.2. l f a > Ibl + Icl, E q u a t i o n (11) has no solutions z with Re(z) ~> 0. PROOF. We apply Theorem 4.6 to the homotopy Ft(z) := z + a nt- t b e -rz -q- tc e -sz for 0 ~< t ~< 1. By Theorem 4.6, it suffices to prove that Ft (iv) 7~ 0 for 0 ~< t <~ 1 and v E R. However, the latter fact is obvious, because liv + al ~ a > Ibl + Icl ~ I t b e -irv -+- t c e - i s v [ .
D
For purposes of locating solutions of Equation (11), we can always assume that bc 7~ 0, rs > 0 and r 7~ s. Otherwise, by a transformation of the form #z -- w, # > 0, we can transform Equation (11) to an equivalent equation of the form of Equation (10), and Equation (10) has already been analyzed. If the numbers r and s in Equation (11) are commensurable, we have already noted that Equation (11) falls in the class of equations treated in [68]. The next theorem shows that Equation (11) is more tractable in this case, and the general analysis of the equation reduces to analysis on certain horizontal strips in C. THEOREM 4.7. Let a, b, c, r a n d s be real n u m b e r s as in Equation (1 l) a n d a s s u m e that c ~ O, 0 < r < s a n d s / r -- n / m , where n a n d m are positive integers a n d n > m. For each integer k define Gk -- {z ~ C I m / r ( 2 k 1)~ < Im(z) < m / r ( 2 k + 1)7r}. I f k 7~ O, there exist n (counting multiplicity) n u m b e r s z E Gk which solve E q u a t i o n (11). I f c > 0 a n d n is even or if c < 0 a n d n is odd, there exist n (counting multiplicity) n u m b e r s z ~ Go which solve E q u a t i o n (11). I f c > 0 a n d n is o d d or if c < 0 a n d n is even, there exist (n + 1) (counting multiplicity) n u m b e r s z E Go which solve Equation (11). PROOF. For z E C, define w ~ C by rz = m w , so sz = n w . We see that z + a + b e -rz -kc e -sz - 0 and z ~ Gk if and only if w E Gk and w + al + bl e -mw -t- Cl e -nw - O, where al = ( r / m ) a , bl = ( r / m ) b and cl = ( r / m ) c . Thus we shall work with the second equation and count solutions in G k. On Gk, consider the homotopy Ft(w) = w + (1-
t)al + ( 1 -
t)bl e -mw + cl e -nu',
O <~ t <. l.
Because I m ( F t ( w ) ) = Im(w) for w ~ OGk, one sees that F t ( w ) :/= 0 for w ~ OGk and 0 ~< t ~< 1. Also, because cl -r 0 and n > m ~> 1, there exists Rk > 0 such that Re(w) ~> - R k if w 6 Gk and F t ( w ) = 0 for some t with 0 ~< t ~< 1. If F t ( w ) = 0 and Re(w) ~> 0, easy estimates imply that Iw[ ~< l a l [ + I b l l + ICl I. Thus Theorem 4.1 is applicable and implies that Fo(w) = 0 has the same number of solutions in Gk as FI (w) = w + cl e -nw = 0.
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If we write n w = ~', FI (w) = 0 has the same number of solutions in Gk as f(~') := + ( n c l ) e -~ = 0 has in the strip {~" I n ( 2 k 1)7r < Im(~') < n ( 2 k + 1)Tr}. By using Theorems 4.2 and 4.3 and Remark 4.1, one sees that for each j ~> 1, the strip {~" I jTr < Im(~') < (j + 2)7r} contains precisely one solution of the equation f(~') = 0 . By taking complex conjugates, one obtains an analogous statement for j ~< - 1. Also, if j ~> 1, Remark 4.1 implies that f(~') ~ 0 if (2j - 1)7r ~< Im(~') ~< 2jzr and c > 0 and f(~') :/: 0 if 2jrr <~ Im(~') ~< (2j + 1)7r a n d c < 0. Finally, we know that f(~') = 0 has precisely two solutions in Go if c > 0 and precisely one solution in Go if c < 0. Combining these facts, one can count the number of solutions ~" of f(~') = 0 with n ( 2 k - 1)Tr < Im(~') < n ( 2 k + 1)Tr and obtain the conclusions of the theorem. [] It is sometimes useful to study a parametrized version of Equation (11): z + )~a + ~.be - ' : + )~c e -s: = 0,
(17)
~. > 0.
Indeed, this is the approach in [ 18] and (after a change of variables) in [57]. If a + b + c > 0, one can easily prove with the aid of the implicit function theorem that there exists ~ > 0 such that for 0 < )~ ~< 6, every solution z of Equation (17) satisfies Re(z) < 0. Generally, given (a, b, c) :/: (0, 0, 0) and 0 < r < s, one can ask for what values of)~ > 0 Equation (17) has a solution z with Re(z) > 0. Obviously, if the hypotheses of T h e o r e m 4.7 are satisfied, this question can be posed much more precisely on each of the strips Gk, k >~ 0. If one poses these questions asymptotically for large )~, the analysis becomes more manageable. ,-,,.,
THEOREM 4.8. A s s u m e that a, b a n d c are real n u m b e r s with (a, b, c) =/= (0, O, O) a n d r and s are nonnegative reals with 0 < r < s. Suppose that z , ~ C is a solution o f multiplicity v o f the equation a + b e -rz -Jr-c e -sz := g(z) = O. There exists 6, > 0 such that g ( z ) ~ 0 f o r 0 < Iz - z,I <~ 6,; a n d if 0 < 6 <~ ~*, there exists )~, = )~,(z,, 6) such that f o r every )~ >~ ~., Equation (17) has precisely v solutions z with Iz - z , I <. 6. PROOF. Under the given hypotheses, g ( z ) := a + b e x p ( - r z ) + c e x p ( - s z ) is a nonconstant function, so z, is an isolated zero of g. If we write e = )~-l for )~ large, Equation (17) is equivalent to ez + g(z) = 0,
and the conclusion of T h e o r e m 4.8 follows by a straightforward application of Rouch6's theorem (Theorem 4.1). l--1 It follows from T h e o r e m 4.8 that if the equation a + b e - r : + c e -s: - 0 has a solution with positive real part, then for all sufficiently large )~, Equation (17) has a solution with positive real part. One can prove a variety of other results of this type. For example, if a = 0 and the hypotheses of T h e o r e m 4.7 hold, then necessarily, for all ,k sufficiently large, there exists a solution z0~) of Equation (17) with Re(z(~.)) --+ + e c as )~ ~ oc. We leave the detailed statement of such a theorem to the reader.
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5. The fixed point index There has been a great deal of work over the past forty years concerning global analysis of various nonlinear FDEs. It would be impossible in a paper of this length to summarize even a small fraction of this research or of the topological tools which play a role in the analysis. We mention in passing the Fuller index (see [7,12,13,15]) and ideas related to the Conley index [14,39,50,69]. Here we shall briefly describe one topological tool which has proved very useful, namely, the fixed point index. In the next section we shall describe how the fixed point index can be combined with a close analysis of the equations in question to prove existence and establish properties of certain kinds of periodic solutions. We shall also mention some open questions. If X is a topological space, U is an open subset of X and f : U --+ X is a continuous map such that S := {x 6 U I f ( x ) - x} is compact (possibly empty), one would like to assign to the triple (f, U, X) an integer ix (f, U), which should be an "algebraic count" of the number of fixed points of f in U and will be called the fixed point index of f : U ~ X. In order to define such an integer ix (f, U) and insure that the assignment (f, U, X) --+ ix (f, U) has desirable properties, it is necessary to restrict attention to "appropriate" classes of spaces X and maps f . The question of what is meant by an "appropriate" space X or "appropriate" map f in this context remains an object of research: see [63,64]. Here we shall restrict attention to the more-or-less classical case (see [3,17]) that the spaces X are metrizable absolute neighborhood retracts (or ANRs) and the maps f are locally compact. Recall that a metrizable topological space X is called a (metric) ANR, or (metric) absolute neighborhood retract, if whenever X is homeomorphic to a closed subset X l of a metric space MI, there exists an open neighborhood U1 of X I in Ml and a continuous retraction rl of Ul onto Xl (so rl(Ul) C XI and rl(y) = y for all y ~ Xl). A metrizable topological space X is called a (metric) AR, or (metric) absolute retract, if whenever X is homeomorphic to a closed subset X l of a metric space Ml, there exists a continuous retraction rl of Ml onto Xl. O. Hanner (see [21,22]) has proved that if a metric space X is an ANR, A is a closed subset of a metric space M and f :A -+ X is a continuous map, then there exists an open neighborhood U of A in M and a continuous map F : U --+ X with F [ A = f . If A is a closed subset of a metric space M, X is an A R, and f :A --~ X is continuous, then there exists a continuous map F : M -+ X with F I A = f . Thus one can give alternate definitions of an ANR or an AR: If X is a metric space, X is an ANR (respectively, AR) if, whenever A is a closed subset of a metric space M and f :A -+ X is a continuous map, there exists an open neighborhood U of A and a continuous map F : U --+ X (respectively, there exists a continuous map F : M --+ X) with F I A = f . Using this second definition, it is easy to see that an open subset of a metric ANR is a metric ANR. Conversely, suppose that X is a metric space and that for each x 6 X there exists an open neighborhood Ux of x such that Ux is an ANR. O. Hanner has proved (see [21,22]) that under these assumptions X is an ANR. In particular, any metrizable Banach manifold is an ANR; and in general the property of being an ANR is a local property. If X is a metric space and X = X I t2 X2, where X l, X2 are closed subsets of X and X l, X2 and X l N X2 are ANRs, Borsuk [2] has proved that X is an ANR. If C is a convex subset of a normed linear space, Dugundji [ 10] has proved that C is an AR. Using the Borsuk and Dugundji results, an easy induction argument shows that if X is a subset of a normed linear
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space Y and X -- uin_=l C i , where Ci is a closed, convex subset of Y for 1 ~< i ~< n < co, then X is an ANR. In fact, with the aid of Hanner's theorem one can show that if X is a subset of normed linear space Y and there exists a locally finite covering {Ca" ot 6 A } of X by closed convex sets Cc~ C X, oe E A, then X is an ANR. If X l and X2 are Hausdorff topological spaces and f ' X l --+ X2 is a continuous map, we shall say that f is "compact" if the closure of f ( X l ) in X2 is compact. We shall say that f is "locally compact" if for each x E X l there exists an open neighborhood N,- of x in X l such that the closure of f (N,-) in X2 is compact. If S is a compact subset of X I and f ' X l --+ X2 is locally compact, then there exists an open neighborhood V of S such that f] V" V --+ X2 is a compact map. Now suppose that X is a metric ANR, U is an open subset of X and f ' U --+ X is a continuous map. Assume that S -- {x 6 U ] f ( x ) -- x} is compact (possibly empty) and that there exists an open neighborhood V of S such that f l V is a compact map. Note that the latter condition will be satisfied if f " U --~ X is locally compact. Under these assumptions one can assign an integer ix (f, U) to the map f ' U --+ X. The integer i x ( f , U) is called the fixed point index of f ' U --+ X and is, roughly speaking, an algebraic count of the number of fixed points of f in U. We allow the possibility that U is empty, in which case ix (f, U) "--O. If X -- R '~ and I denotes the identity map on R '~, then the fixed point i x ( f , U) is related to the more familiar Brouwer degree of the map g(x) "- (I - f ) ( x ) "x - f (x) by ix (f, U) -- deg(I - f, V, 0), where V is a bounded open neighborhood of S and V C U. More generally, if X is a Banach space and V is a bounded open neighborhood of S such that V C U and f ( V ) is compact, then ix (f, U) -- deg(l - f, V, 0), where the right-hand side now denotes the Leray-Schauder degree. The so-called additivity property of degree theory implies that d e g ( l - f, V, 0) is independent of the particular open neighborhood V D S which is chosen. The fixed point index satisfies four properties, the additivity property, the homotopy property, the commutativity property and the normalization property; and these properties axiomatically determine the fixed point index. 5.1. The additivity property. Assume that X is a metric ANR, U is an open subset of X and f : U -~ X is a continuous map such that S - {x ~ U I f ( x ) - x } is compact (possibly empty) and f ( V ) is compact for some open neighborhood V of S with V C U. If i x ( f , U) ~ 0, then f has a fixed point in U. If U1 and U2 are open subsets of U, S C U1 U U2 and S A Ul A U2 is empty, then i x ( f , U) - i x ( f ,
UI) + i x ( f , U2).
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5.2. The homotopyproperty. Let X be a metric ANR, U an open subset of X and F : U x [0, 1] --+ X a continuous map. Define Ft (x) = F (x, t) for 0 ~< t ~< 1, x 6 U. Assume that 2; = {(x, t) 6 U x [0, 1] : F ( x , t) = x } is compact (possibly empty) and that there exists an open neighborhood G of 27 in U x [0, 1] such that F ( G ) is compact. Then i x ( F t , U) is defined and constant for 0 ~< t ~< 1. In particular we have i x ( F o , U) = i x ( F 1 , U).
As is the case for the Leray-Schauder degree, the homotopy property provides a powerful method for computing the fixed point index of a given map by homotoping it to a map for which the fixed point index is known. We illustrate this point by discussing the fixed point index of certain "positive" linear operators. If Y is a Banach space, we shall call a closed, convex set C C Y such that (1) tC C C for all t ~> 0 and (2) C A ( - C ) = {0} a closed cone (with vertex at 0). A closed cone C is called "total" if Y is the closed linear span of C. If L : Y ~ Y is a bounded linear map and L ( C ) C C, L is sometimes called "positive" (with respect to C). Of course L is "compact" if L maps bounded sets into sets whose closure is compact. The Krein-Rutman theorem asserts that if C is a closed, total cone in a Banach space Y and L : Y ~ Y is a bounded, compact linear operator with L ( C ) C C and r ( L ) :-- r, the spectral radius of L, strictly positive, then there exists x 6 C - {0} with L x = rx. THEOREM 5.1. Let C be a closed, total cone in a Banach space Y and L : Y --+ Y a compact, bounded linear map with L ( C ) C C. Let U = {x E C ]]]x 11< 1} and f : U --+ C be defined by f (x) = L ( x ) . A s s u m e that r ( L ) > 1 and that L ( y ) =/=y f o r all y E C - {0}. Then it f o l l o w s that i c ( f , U) = O. PROOF. By using the compactness of L, there exists 6 > 0 such that IIx - L(x)ll ~ ~ for all x 6 C with IIx II - 1. The Krein-Rutman theorem implies that there exists y 6 C - {0} with L ( y ) = ry, r := r(L) > 1. Multiplying y by a positive constant, we can assume that IIY II < ~. For 0 ~< t ~< 1 consider the homotopy ft (x) = f (x) + ty, x ~ U. The reader can verify that this homotopy satisfies the conditions of the homotopy property, so i c ( f , U ) = i c ( f o , U) = i c ( f l , U).
Suppose, by way of contradiction, that ic ( f , U) ~ O. The additivity property implies that there exists x E U with fl (x) = x, so Lx + y=x.
Applying fl to both sides of the equation, we find that L 2 (x) nt- y + L y -- x,
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and in general, a simple induction yields that
L ''+t (x) + ~
L k (y) : x
k=0
for every n ~> 0. Since L k (y) = rky and r > 1, we conclude that
x-X,+y 6C,
~,n "-- ~
rk ,
k=0
for all n >~ 0. Dividing by )~,, and letting n approach infinity, we conclude that - y ~ C, which contradicts the assumption that y ~ C - {0} and C is a cone. D Here we have used the Krein-Rutman theorem to derive a statement about the fixed point index. In [61 ] (see, also, Section 2 of [62]), the apparatus of the fixed point index is used to prove a generalization of the Krein-Rutman theorem. 5.3. The commutativity property. Suppose that X j, j = 1, 2, is a metric ANR, that Uj is an open subset of X j and that fl :U1 ~ X2 and fz:U2 --+ X l are continuous maps. Let VI = f 1 - 1 ( U 2 ) , V2-- f 2 l(Ul), Sl = {x E Vl" f 2 ( f l ( x ) ) = x } and S 2 - - { y E V21 fl (f2(Y)) = Y}. Assume that S1 is compact and there exists an open neighborhood W1 of Sl such that fl (W1) is compact. Then $2 is homeomorphic to S1 and
ixl (f2 f l , Vl) = ix2 ( f l f2, V2). In the case that X1 and X2 are Banach spaces, the commutativity property corresponds to a basic fact about the Leray-Schauder degree. Oddly enough, this fact is rarely explicitly stated in treatments of degree theory. Starting from the Leray-Schauder degree and using the commutativity property for the Leray-Schauder degree, one can exploit an observation which goes back to J. Leray and develop the fixed point index from the Leray-Schauder degree. See [ 17,3 ]. The normalization property of the fixed point index connects the fixed point index and the Lefschetz number of a map f : X --+ X. Suppose that X is a compact Hausdorff space and that Hi (X) denotes the singular homology of X with rational coefficients. Assume that Hi (X) is a finite dimensional vector space over the rationals for all i ~> 0 and that Hi(X) - - 0 for all sufficiently large i. All these conditions will be satisfied if X happens to be a compact, metric ANR. A continuous map f : X -+ X induces a vector space endomorphism f,i : Hi (X) ~ Hi (X), and one can compute tr(f,i), the trace of the map f,i. By definition L ( f ) , the Lefschetz number of f , is given by
L ( f ) -- Z ( - 1 ) i t r ( f , i ) . i >~o
R.D. Nussbaum
488 5.4. The normalization property. continuous map. Then one has
Let X be a compact, metric ANR and f : X
---> X a
ix (f, X) = L ( f ) . The normalization property implies the Lefschetz fixed point theorem: If X is a compact, metric ANR and f :X ---> X is a continuous map with L ( f ) :/: O, then f has a fixed point in X. It is interesting to note that the Lefschetz fixed point theorem is false without some assumption on X. There is an example of a compact, Hausdorff space X with Ho(X) = Q and Hi (X) = 0 for i > 0 and a continuous map f : X --+ X with L ( f ) = 1 such that f has no fixed point in X: see [31]. If U is an open subset of a Banach space Y and f : U ---> U is a continuous map such that f ( U ) C X, where X C U is a compact, metric ANR, then one can prove with the aid of the commutativity property and the normalization property that deg(l - f, U, 0) = i y ( f , U) = L ( f : X --+ X), where L ( f : X ~ X) denotes the Lefschetz number of the map g : X --+ X defined by g(x) = f ( x ) for x 6 X. For many applications of the fixed point index, it suffices to consider the situation that the ANR X is a closed, convex subset of a Banach space Y. In this case one can give a more straightforward definition of the fixed point index. First, recall a slight generalization of the Leray-Schauder degree. Suppose that G is an open subset of a Banach space Y and f : G --+ Y is a continuous map such that S := {x E G [x = f ( x ) } is compact (possibly empty). Assume that there exists a bounded open neighborhood V of S, V C G, such that f (V) is compact. Then one can define deg(l - f, G, 0 ) = d e g ( l -
f, V, 0),
where the right-hand side denotes the usual Leray-Schauder degree. This definition is independent of the particular open neighborhood V of S and agrees with the ordinary L e r a y Schauder degree. It has the advantage that it sometimes allows the degree to be defined when G is unbounded or even when x - f (x) = 0 for some x 6 0 G. Now suppose that X is a closed, convex subset of a Banach space Y and that U is a relatively open subset of X, so U = G M X, where G is an open subset of Y. Notice that we are not assuming that X has a nonempty interior in Y. Assume that f : U ---> X is a continuous map, that S : - {x 6 U ] f ( x ) = x} is compact and that there exists an open neighborhood V of S in X, S C V C U, such that f ( V ) is compact. A theorem of Dugundji [ 10] implies that there exists a continuous retraction r : Y ~ X of Y onto X. We can define i x ( f , U) in this situation by
i x ( f , U) -- d e g ( l - f r, r -1 (U), 0).
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A little thought shows that S = {y e r - l (U) I Y = f ( r ( y ) ) } , so the right-hand side is defined. If W is any open neighborhood of S in Y and W C r -1 (U), one can easily see that deg(I - f r , r -I (U), O) - deg(l - f r , W, 0). It may, of course, happen that there are two, different, continuous retractions r l and r2 which map Y onto X, and one wants to prove that ix (f, U) is independent of the particular retraction chosen. To see this, let W be an open neighborhood of S in Y such that W C rj | (U) for j - 1,2. Then it suffices to prove that deg(l-
f rl , W, 0 ) - - d e g ( l - f r2, W, 0).
Using the convexity of X, consider the homotopy (1 - t ) f ( r l (y)) + tf(r2(y)) := F(y, t) for y ~< W, 0 ~< t ~< 1. One can easily check that if F ( y , t ) = y for some y E W and some t, 0 ~< t <~ 1, then x E S, so the homotopy property for degree theory implies that the definition of ix (f, U) is independent of the retraction r : Y --+ X. Further details about this kind of approach are given in [62, pp. 34-36] and [51 ].
6. Periodic solutions of functional differential equations Early computer numerical studies by Jones [25] of Wright's equation
x ' ( t ~ - - Z x ( t - l~(1 + x(t~)
(18)
suggested that the equation has a nonconstant, periodic solution for every )~ > rr/2 and that the periodic solution has strong stability properties. In [25] Jones gave a proof of existence of a nonconstant, periodic solution of Equation (18) for every )~ > zr/2. Here we shall first describe a generalization of Jones's existence result for some equations of the form
x'(t)--Xh(x(t),x(t-
1)).
(19)
The theorems we shall describe are very special cases of existence theorems for periodic solutions in [47], and we refer the reader to [47] for generalizations and further details. Roughly speaking, we seek periodic solutions of Equation (19) which oscillate about a constant solution xc(t) = c of Equation (19). We shall assume that such a constant solution has already been found, define g(x, y) := h(x + c, y + c) (so g(0, 0) = 0), and talk about solutions of
x' (t) -- - X g ( x ( t ) , x(t - 1))
(20)
which oscillate about 0. In practice, verifying hypotheses on g may involve nontrivial calculus problems: see [43].
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It sometimes happens that Equation (19) can be simplified by a change of variables and some technical difficulties can be avoided. For example, if a continuous function x : [ - 1 , c~) --+ R solves Equation (6.1) for t ~> 0 and x(t) > - 1 for - 1 ~< t ~< 0, then one can easily prove that x(t) > - 1 for all t ~> 0. If one now writes x(t) = exp(y(t)) - 1, one finds that
y'(t)--~[exp(y(t-
1))-
1],
(21)
and Equation (21) is frequently more convenient to work with than Equation (18). More generally, consider an equation of the form
x'(t)--~g(x(t-
1))N(x(t)).
(22)
We assume that g : R ~ ~ and N : R ~ R are continuous maps and that N ( x ) > 0 for - b < x < a, where 0 < a ~< cx~ and 0 < b ~< cx~. If a < cx~ (respectively, b < e~) we assume that N ( a ) = 0 and N is Lipschitzian on some neighborhood of a (respectively, N ( - b ) = 0 and N is Lipschitzian on some neighborhood of - b ) . If 9/ > 0 and x : [ - 1 , y] --+ R is a continuous function which solves Equation (22) for 0 ~< t ~< y and satisfies - b < x (0) < a, one can prove that - b < x(t) < a for 0 ~< t ~< y. Consider now the initial value problem
f ' ( y ) - N ( f (y)),
f(0)--0.
(23)
Elementary existence theory for ordinary differential equations implies that Equation (23) has a solution f (y) defined on a maximal interval ( - d , c), and that limy__._d+ f (y) = - b and l i m y ~ c - f ( y ) = a. If a < cx~ or if a = cx~ and there exist constants A1 and A2 with N ( x ) <<,AlX -k- A2 for 0 ~< x < cx~, then c = ~ . Similarly, d = ~ if b < cx~ or if b = and there exist constants Bl and B2 with N ( x ) <~ B11xl + B2 for -cx~ < x ~< 0. Of course, it may happen that c < cx~ or d < c~, e.g., if N (x) = x 2 + x + 1. If y > 0 and x : [ - 1 , y] --~ R is a continuous function which solves Equation (22) for 0 ~< t ~< y and satisfies - b < x(t) < a f o r - 1 ~< t ~< 0, then our previous remarks show that we can write x(t) = f ( y ( t ) ) , where f is a maximal solution of Equation (23) as above. Making this substitution gives
y'(t)--)~g(f(y(t-
1))).
(24)
In the case that c = d = cx~, e.g., when N ( x ) = (a - x)(b + x) or N ( x ) = 1 + x, this substitution has proven useful: see [65]. W h e n c < c~ or d < cx~, e.g., when N (x) = x 2 + x + 1, less work has been done. We now return to Equation (63). A periodic solution x(t) of Equation (20) will be called a "slowly oscillating periodic solution" or SOP solution if there exist real numbers z l, z2, Z3 with Z2 - - Zl > 1, Z3 - - Z2 > 1, x(t) > 0 for Zl < t < Z2, x(t) < 0 for Z2 < t < Z3 and x(t + z3 - zi) = x(t) for all t. The word "slowly" refers here to the fact that the zeros of x(t) are separated by distances greater than one, which is the time lag in Equation (20). The "SOP" terminology is not ideal but is widely used. We seek conditions on ~ and g in Equation (20) which insure existence of an SOP solution.
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It will be crucial for our work that -)~g := h satisfy a "negative feedback condition" on an interval [ - B, A]. DEFINITION 6.1. Suppose that A and B are positive reals and that h : [ - B , A ] x [ - B , A] ~ R is a map. We shall say that h satisfies a negative feedback condition on [ - B , A ] if (1) h(x, y) < 0 whenever 0 < x ~< A and 0 < y <~ A, (2) h(x, y) > 0 whenever - B ~< x < 0 a n d - B ~< y < 0 and (3) yh(O, y) < 0 f o r - B ~< y <~ A and y - ~ 0 . If h : R • R --+ R is a map, we shall say that h is "locally Lipschitzian in the first variable" if, for each (x0, y0) E ~ • R, there exist 3 = 3(x0, y0) > 0 and C = C(xo, yo) such that
Ih(xl, y) - h(x2, Y)I <~ Clxl - x2l for all x l, x2, and y with [x l - x01 <~ 3 and Ix2 - x01 ~< 3 and lY - Y0I ~< 3. If g is locally Lipschitzian in the first variable and 0 : [ - 1 , 0] --+ R is a continuous function, then there is a unique continuous function x : [ - 1 , 9/) -+ R, defined on a maximal interval [ - 1 , y), y > 0, such that x [ [ - 1 , 0] = 0 and x satisfies Equation (20) for 1 ~< t < y. If sup{ Ix (t) l : 0 ~< t < y} < cxz, then we have y = ~ . If A > 0, it is convenient to define a closed, convex set KA C C ( [ - 1, 0]) by
KA'={OcC([-1,O])"
O<,O(t)<,aforalltc[-1,O]andO(O)--O}.
(25)
THEOREM 6.1. Suppose that k > 0 and g : ]R x R --+ R and make the following assump-
tions: (1) g is locally Lipschitzian in the first variable. (2) There exist positive reals A and B such that - k g satisfies a negative feedback condition on [ - B, A ]. Og Og (3) g is Frdchet differentiable at (0, O) with ~ ( 0 , O) --oto >~ 0 and ~ ( 0 , O) - ~o > O. (4) The characteristic equation z = -koto - ,kilo, e -z has a root z ~ C with Re(z) > 0. (See Theorem 4.5.) (5) For each 0 E C ( [ - 1 , 0]), denote by x(t) := x(t; O) the function such that x l [ - 1 , 0] = 0 and x'(t) = - ) ~ g ( x ( t ) , x ( t - 1)) for 0 <~ t < Y, where y is maximal. Assume that f o r each 0 E KA (see Equation (22)), - B <~x(t; O) <~ A for all t ~ O. Then there exists an SOP solution x(t) of Equation (20) such that - B <~ x(t) <~ A f o r
all t. We cannot present a detailed proof of T h e o r e m 6.1 here, but an outline of the proof may be helpful. A key step is to define a map 05:KA ~ K A whose nonzero fixed points correspond to SOP solutions of Equation (6.3). If 0 = 0 E KA, define 05 (0) = 0. If 0 6 KA and 0 ~ 0, define z0 = sup{t Ix (s; 0) = 0 for all s with 0 ~< s ~< t }. One proves that z0 = zo(O) < 1. If there exists t > z0 with x(t; 0) = 0, define
zl " - zl(O) - inf{t > zo l x(t" 0 ) k - 0 } . One proves that zl - z0 > 1. If there does not exist t > z0 with x(t; 0) = 0, define 05(0) = 0. If there exists t > zl with x(t; 0) = 0, define z2 = z2(0) by 2 2 = inf{t > zl I x(t; 0) = 0}. In this case one proves that z2 - zl > 1 and one defines 05(0) = 7t 6 KA by
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~p(t) = x(z2 + t; 0), --1 ~< t ~< 0. If there does not exist t > zl with x(t; 0) = 0, one defines 4 (0) = 0. One proves that 4 : KA ~ K A is a continuous, compact map. If 4 (0) = 0 and 0 r 0, then x(t; O) extends to an SOP solution of period z2 -- z2(0). Note that both the fixed point 0 and the period z2 (0) are typically not explicitly known. The definition of 4 is elementary and does not involve information about the characteristic equation. However, it is precisely this part of the proof which fails for more complicated equations like x'(t) = - ) ~ g ( x ( t ) , x ( t - 1), x ( t -
y)).
Once 4 is defined, the problem reduces to proving that 4 has a nonzero fixed point in K A. Roughly speaking, we expect condition (4) of Theorem 6.1 to imply that 0 is an unstable fixed point of 4 , and we hope that this instability will imply existence of a nonzero fixed point. If we can find a relatively open subset U of KA such that 0 r U and iKa (4, U) is defined and nonzero, then 4 will have a fixed point in U, and we will be done. The normalization property of the fixed point index implies that i Xa (4, KA) ~- 1. For 3 > 0, define Bs = {0 E KA I I]0]] < 3} and Us = {0 ~ KA: ]]0]l > 3}. If we can prove that there exists 6, 0 < 6 < A, such that 4 ( 0 ) 7~ 0 for 0 6 g z with II011 - ~ and iKz (4, B3) = m ~ 1, then it will follow that iKz ( 4 , U6) = 1 - m r 0, and we will be done. Thus it suffices to find 6, 0 < 6 < A, with iKz (t~, B~) = 0. For notational convenience, define go(x, y) = g(x, y) and gl (x, y) = otox + floy and gs(x, y) = (1 - s)go(x, y) + sgl (, y). Consider the initial value problem
x'(t)---)~gs(x(t),x(t-
1))
f o r t ~>0;
xl[-1,0]--0
E
KA.
(26)
Define 40 = 4 : K A ~ KA. There exists 6, > 0 such that if 0 6 KA and II011 ~ 6,, one can define 4s (0) = 7r E KA corresponding to Equation (26)just as we described above for the original equation (20). One can prove that there exists 6 > 0 such that 4s (0) 7~ 0 for all 0 E K A with 0 < II0 II ~< ~ and all s, 0 ~< s ~< 1. By using the homotopy property, one concludes that
iKA (40, B~)
:=
iKA ( 4 , B~)
=
iKA ( 4 1 , Bs).
Thus it suffices to prove that iKA (41' B~) = 0. Select/~ > 0 with )~/3 > zr/2 and define g2(x, y) = ~y and gs(x, y) = (2 - s)gl (x, y) + (s - 1)gz(x, y). Consider the initial value problem
x'(t)---)~gs(x(t),x(t-
1))
fort ~0;
x][-1,0]--0
E KA.
(27)
Again, there exists 31 > 0 such that if 0 6 KA and II011 ~ ~ and 1 ~< s ~< 2, then one can define 4 s ( 0 ) = 7r E KA just as in the case s = 0. One easily proves that 4 s ( 0 ) ~ 0 for 0 < II0 II ~< ~ and 1 ~ s ~ 2. It follows from the homotopy property that
iKA ( 4 1 ,
B~) - -
iKA ( 4 2 ,
B,S)
for 0 < 6 ~< 31, and it suffices to prove that iKA (42' Bs) = 0.
Functional differential equations
493
N o w define C = {0 6 C ( [ - 1 , 0 ] ) I 0 ( - 1 ) = 0 and 0 is increasing}. Note that by "increasing" we mean that O(sl) <, 0(s2) whenever - 1 ~< sl ~< s2 ~< 0. Similarly define K = {0 E C ( [ - 1 , 0 ] ) : 0(0) = 0 and O(t) >~ 0 for - 1 <~ t ~< 0}. ~ 2 c a n be considered as a map of K to K; and by using the commutativity property, one proves that
iKA(~P2,
Bs) = i x ( ~ 2 , Bs).
If 0 E C, 0 # 0, and if we write x(t; O) for the solution of
x'(t) = - ) ~ f l x ( t -
1)
for t ) 0;
x ] [ - 1 , 0] -- 0,
one can show (1 (0) -- (1 = inf{t > 0 I x(t; O) -- 0} and (2(0) - (2 "-- inf{t > (l I x(t" O) = 0} are well-defined and finite. For 0 E C - {0}, we define
(O (O))(t) -- x((2 -+- 1 q- t; 0), and O (0) = 0. We observe that O actually defines a continuous map of C into C, and O takes b o u n d e d sets to precompact sets. If B = {0 E K III011 < 1} and V = {0 E C [ II0 II < 1 }, one can prove with the aid of the additivity and commutativity properties of the fixed point index that
iK(~2, B~) -- iK(~2, B) -- ic(O, V). Although, O arises from a linear differential-delay equation, O is not linear. However, one can see that O is h o m o g e n e o u s o f d e g r e e one, i.e., O(sO) = s O (0) for 0 E C a n d s ~> 0. The constant/~ in the definition of O can be chosen as large as desired, and for sufficiently large 15 simple estimates show that I1~ (0)II > #110 II for all 0 E C, w h e r e / z > 1 and # is independent of 0. Furthermore, there exists rl > 0 such that II45 (0)II/> rl > 0 for all 0 E C with II0 II -- 1. If 27 = {0 E C I II0 II -- 1 }, there exists a continuous retraction p of C onto I7. (To construct p, let j : C --~ [0, ec) be a continuous function such that j (0) = 0 for II0 II/> 1 and j(O) = 1 for II011 <~ (1/2). Define 00 E C by 00(t) = t + 1, define k(0) = (0 + j(O)Oo) and define p(O) = k(O)llk(O)ll-l.) For o- ~> 0 and 0 6 V we consider the h o m o t o p y
q,~(o)- ~,(p(o)) + q,(o). If O~ (0) -- 0 for some 0 E V with !]0 ]] - 1, we have that p(O) -- 0 and
O(0)--
1
1 +~r
)0,
which contradicts the fact that }]O(0)[[ > ]10ll. It follows that ic(O~, V) is constant for cr >~ 0. However, if II~ (0)II ~< M for all 0 6 V, then
IlO~ (O) [[ >~a rl - M,
494
R.D. Nussbaum
so tp~ has no fixed point in V for o- > ( M + 1)7/-1 and ic(tP~, V ) = 0 for o" > ( M + 1)0 -1. It follows that ic (q/~, V) -- 0 for all cr ~> 0, which gives the desired result. Other proofs of results like Theorem 6.1 have used somewhat more sophisticated arguments involving "asymptotic fixed point theory" (see [4,51,53,63]). Whichever argument one uses, actually showing that i XA (~, Ba) -- 0 proves very useful in establishing further results about Equation (20), e.g., "global bifurcation theorems" as in [53,62]. COROLLARY 6.1. Let X and g be as in Theorem 6.1, except replace assumption (5) by (5)' g ( A , y) ~ Of o r - B <<.y <~ A and g ( - B , y) <~ Of o r - B <<.y <<.A. Then Equation (20) has an S O P solution x ( t ) such that - B <<.x ( t ) <~ A f o r all t. PROOF. The reader can verify that if 0 9 C ( [ - 1 , 0]) and - B then - B <<.x(t; O) <~ A for all t ~> 0.
<~ O(t) <<.A for - 1 ~< t ~< 0, D
COROLLARY 6.2. Let f " R --+ R be a continuous map such that f (O) -- O, x f (x) > Of o r all x ~ O, and f is either b o u n d e d above or b o u n d e d below. A s s u m e that f is differentiable at 0 and f ' ( O ) - 1. lf)~ > re~2, the equation x ' ( t ) - - X f ( x ( t - 1)) has a slowly oscillating periodic solution. In recent years it has been increasingly recognized that it is important for both theoretical and practical reasons to study functional differential equations with "state dependent delays", i.e., functional differential equations in which time lags appear which depend on the unknown function x. A simple-looking class of examples is provided by e x ' ( t ) -- f ( x ( t ) , x ( t
-- r l ) , X ( t -- r2) . . . . . x ( t -- rm)),
(28)
where ri in the preceding equation denotes r i ( x ( t ) ) , 1 <<.i <<.m, and ri, 1 <~ i <<.m, and f are given functions, and e > 0. An important special case is e x ' ( t ) -- f ( x ( t ) , x ( t
- r)),
(29)
where r - r ( x ( t ) ) and r and f are given functions and e > 0. The quasilinear example ex' (t) -- - - x ( t ) -- k x ( t - 1 - c x ( t ) ) ,
(30)
for e > 0, c > 0, k > 1, already illustrates many difficulties. However, one should also note that it is important to allow more complicated examples in which the time lags depend in a more complicated way on the history of the unknown function. The fixed point index ideas we have described can be used (under appropriate assumptions) to prove the existence of SOP solutions of equations like (6.11). We refer particularly to [47] for a discussion of existence of SOP solutions of Equation (28). Further discussion, particularly with regard to behaviour of solutions of (29) as e ~ 0 +, can be found in [44, 45] and the references there. The reader should also see work of Kuang and Smith [34] and Smith [71] and references in these papers.
Functional differential equations
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In general one wants to understand the full dynamics of nonlinear functional differential equations; but since these equations are infinite dimensional, nonlinear dynamical systems, a complete understanding is difficult. There exist some very useful general tools, e.g., discrete Lyapunov functions and Poincar6-Bendixson methods: see [39,48,49] and see Section 2 of [44] for an application of these ideas. The application of various global bifurcation theorems has also proved helpful: see [40,53]. However, even if one restricts attention to equations like x'(t) = - ) ~ f (x(t), x ( t - 1)) or x'(t) = - ~ . f (x(t - 1)), our detailed understanding is incomplete. We mention some open questions. If the function f satisfies the hypotheses of Corollaries 6.1 or 6.2, we know that, for appropriate Z, x'(t) = - Z f ( x ( t ) , x ( t - 1)) has an SOP solution x ( t ) , and we can normalize this SOP solution so that x(0) = 0 and x ( t ) > 0 for 0 < t ~ 1. Under what assumptions on f is this normalized SOP solution unique? Less generally, one can ask for what Z and f the equation
x' (t) = - ) ~ f ( x ( t - 1))
(31)
has a unique, normalized SOP solution. If f is odd, increasing and satisfies an appropriate concavity assumption on [0, co), then it is known that Equation (31) has at most one normalized SOP solution and it is known for exactly what range of Z > 0 an SOP solution exists. The first such result was proved in [58], and refinements can be found in [6,8,24]. However, the arguments in [8,24,58] depend strongly on the oddness of f and the consequent existence of special SOP solutions of period 4 (see [27,56]). Cao [6] has weakened the oddness assumption but still must impose very restrictive conditions on f . For more general functions f such that x f (x) > 0 for all x =/= 0 and such that f (x) has nonzero limits as x --+ i e c , Xie [76-79] has proved theorems which insure that, for Z sufficiently large, Equation (31) has a unique SOP solution. It is interesting to note what these theorems give for the well-studied class of equations =
-
-
+
(32)
where )~ > 0, a > 0 and b > 0, and for Wright's equation,
x'(t) = - Z x ( t -
1)(1 + x(t)).
(33)
As previously noted, both equations can be transformed to the form y' (t) = - Z f ( y ( t 1)), and f will be odd if a = b in Equation (32). If a = b, theorems in [58] imply that Equation (32) has a unique, normalized SOP solution for Z > 7r/2ab and possesses no SOP solution for 0 < Z < 7r/2ab. If Z is sufficiently large, Xie's results imply that Equations (32) and (33) have unique, normalized SOP solutions; and these solutions are "stable". In particular (see [79]), Wright's equation has a unique, normalized SOP solution for Z >/30. For Z < 30 and Z not near to rr/2, say 2 ~< )~ < 30, nothing is known about uniqueness. One should note that even if x f ( x ) > 0 for all x =/= 0 and f is odd, Equation (31) may have multiple SOP solutions: see [58,60]. Indeed, the dynamical system generated by Equation (31) may display "chaotic" behaviour. The reader should see, for instance, papers by H.-O. Walther cited in [9,19].
R.D. Nussbaum
496
If f is as in Corollary 6.2 and limx_~+~ f (x) = A and limx~-e~ f (x) = - B , where A > 0 and B > 0 (A -- ~ or B -- c~ is allowed), then it is proved in [54] that the period Pz of any SOP solution xz (t) of Equation (31) satisfies lim p z - - 2 + ~
A B
B
+--.
A
Furthermore (see [53]), for any p with 4 < p < 2 + A / B + B / A , there exists ~. > 0 such that Equation (32) has an SOP solution of minimal period p. One can prove that as )~ zr/2, there are SOP solutions of Equation (32) whose period approaches 4. Furthermore, under the hypotheses of Corollary 6.2, if f is odd, Equation (28) has an SOP solution of period 4 for every )~ > 7r/2: see [27,56]. This implies that Equation (32) has an SOP solution of period 4 if a = b and )~ab > zr/2. However, it has been conjectured that if a ~: b, Equation (32) has no SOP solution of period 4 and, in fact, no nonconstant periodic solution of any type of period 4. Can one prove this? More generally, it often appears the case that Equation (31) has no nonconstant periodic solution of period 4. Can one give useful conditions on f which imply this fact and imply the conjecture for Equation (32)? It has been proved in [65] that Wright's equation has no nonconstant periodic solution of period 4, but the proof has resisted generalization. What can be said about the dynamics of
ex'(t) -- f ( x ( t ) , x ( t - r l ) , x ( t - r2)) for e > 0 small and for rl and r2 positive constants? As already noted, even a complete analysis of the associated characteristic equation for general e > 0, r l > 0 and r2 > 0 is a daunting task. For the equation
e x ' ( t ) - - o t g ( x ( t - r,)) - f l g ( x ( t - r2)), where there is no x(t) dependence, the analysis in [57] shows that some interesting theorems can be obtained for e > 0 small. If there is an explicit x (t) dependence, as in equations of the form
ex'(,)- -yx(t)-ug(x(t-
rl)) - f i g ( x ( , - r2)),
for or,/3, y, e, rl, and r2 positive reals, it is unlikely that the methods of [57] will apply, but one can still hope for some sort of asymptotic analysis as e --+ 0 + as in [40].
References [1] E G. Boese, Some stability charts and stability conditions for a class of difference-differential equations, Z. Angew. Math. Mech. 67 (1987), 56-59. [2] K. Borsuk, Theory of Retracts, Polish Sci. Publ., Warsaw (1967). [3] EE. Browder, Local and global properties of nonlinear mappings in Banach spaces, Institute Naz. di Alta Math. Symposia Math. 2 (1968), 13-35.
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[4] F.E. Browder, Asymptotic fixed point theorems, Math. Ann. 185 (1970), 38-60. [5] EL. Butzer and H. Berens, Semigroups of Operators and Approximation, Springer, New York (1967). [6] Y. Cao, Uniqueness of periodic solutions for differential delay equations, J. Differential Equations 128 (1996), 46-57. [7] S.-N. Chow and J. Mallet-Paret, The Fuller index and global Hopfbifurcation, J. Differential Equations 29 (1978), 66-85. [8] S.-N. Chow and H.-O. Walther, Characteristic multipliers and stabili~ of periodic solutions of x f(t) g(x(t - 1)), Trans. Amer. Math. Soc. 307 (1988), 127-142. [9] O. Diekmann, S.A. van Gils, S.M. Verduyn-Lunel and H.-O. Walther, Delay Equations, Appl. Math. Sci., Vol. 100, New York (1995). [10] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367. [1 l] K. Engelborghs and D. Roose, Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations, Adv. Comput. Math. 10 (1999), 271-289. [ 12] C. Fenske, An index for periodic orbits of local semidynamical systems, Trans. Amer. Math. Soc. 350 (1998), 4973-4991. [13] C. Fenske, An index for periodic orbits of functional differential equations, Math. Ann. 285 (19895, 381392. [14] B. Fiedler and J. Mallet-Paret, Connections between Morse sets for delay differential equations, J. Reine Angew. Math. 397 (1989), 23--41. [15] EB. Fuller, An index offixed-point type for periodic orbits, Amer. J. Math. 89 (1967), 133-148. [16] J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, Oxford (1985). [17] A. Granas, The Leray-Schauder index and the fixed point theo~ for arbitrary ANRs, Bull. Soc. Math. France 100 (1972), 209-229. [ 18] J. Hale and W. Huang, Global geometry, of the stable regions for two delay differential equations, J. Math. Anal. Appl. 178 (19935, 344-362. [ 19] J. Hale and S. Verduyn-Lunel, Introduction to Functional Differential Equations, Appl. Math. Sci., Vol. 99, Springer, New York (1993). [20] J. Hale, Functional Differential Equations, Appl. Math. Sci., Vol. 3, Springer, New York (1971). [21] O. Hanner, Some theorems on absolute neighborhood retracts, Ark. Mat. 1 (19515, 389-408. [22] O. Hanner, Retraction and extension of mappings of metric and non-metric spaces, Ark. Mat. 2 (19521954), 315-360. [23] N.D. Hayes, Roots of the transcendental equation associated with certain differential difference equations, J. London Math. Soc. 25 (19505, 226-232. [24] A. Herz, Solutions of x r(t) = - g ( x ( t - 1)) approach the Kaplan-Yorke orbits for odd sigmoid g, J. Differential Equations 118 (1995), 36-53. [25] G.S. Jones, The existence ofperiodic solutions of f ' ( x ) -- - o t f ( x - 1)[1 + f(x)], J. Math. Anal. Appl. 5 (1962), 435-450. [26] M.A. Kaashock and S.M. Verduyn-Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc. 334 (1992), 479-517. [27] J. Kaplan and J. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Anal. Appl. 48 (1974), 317-325. [28] J. Kaplan and J. Yorke, On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal. 6 (1975), 268-282. [29] J. Kaplan and J. Yorke, On the nonlinear differential delay equation x'(t) = - f (x(t),x(t - 1)), J. Differential Equations 23 (1977), 293-314. [30] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1966). [31] S. Kinoshita, On some contractible continua without the fixed point property, Fund. Math. 40 (1953), 96-98. [32] T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, Preprint (1998). [33] T. Krisztin, H.-O. Walther and J. Wu, The structure of an attracting set defined by delayed and monotone positive feedback, Preprint (1998). [34] Y. Kuang and H.L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear Analysis 19 (19925, 855-872. [35] E.M. L6meray, La quatribme algorithme naturel, Proc. Edinburgh Math. Soc. 16 (18975, 13-35.
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[36] B.W. Levinger, A folk theorem in functional differential equations, J. Differential Equations 4 (1968), 612619. [37] T. Luzyanina, K. Engelborghs, K. Lust and D. Roose, Computation, continuation and bifurcation analysis ofperiodic solutions of delay differential equations, Internat. J. Bifur. Chaos 7 (1997), 2547-2560. [38] J. Mahaffy, A geometric analysis of stability regions for a linear differential equation with two delays, Internat. J. Bifur. Chaos 5 (1995), 779-796. [39] J. Mallet-Paret, Morse decomposition for differential delay equations, J. Differential Equations 72 (1988), 270-315. [40] J. Mallet-Paret and R.D. Nussbaum, Global continuation and asymptotic behaviour of periodic solutions of a differential delay equation, Ann. Mat. Pura Appl. 45 (1986), 33-128. [41 ] J. Mallet-Paret and R.D. Nussbaum, A bifurcation gap for a singularly perturbed delay equation, Chaotic Dynamics and Fractals, M. Barnsley and S. Demko, eds, Academic Press, New York (1986), 263-287. [42] J. Mallet-Paret and R.D. Nussbaum, Global continuation and complicated trajectories of periodic solutions of a differential delay equation, Proc. Sympos. Pure Math., Vol. 45, Part 2, Amer. Math. Soc., Providence, RI (1986), 155-167. [43] J. Mallet-Paret and R.D. Nussbaum, A differential-delay equation arising in optics and physiology, SIAM J. Math. Anal. 20 (1989), 249-292. [44] J. Mallet-Paret and R.D. Nussbaum, Boundary layer phenomena for differential delay equations with state dependent time lags I, Arch. Rat. Mech. Anal. 120 (1992), 99-146. [45] J. Mallet-Paret and R.D. Nussbaum, Boundary layer phenomena for differential delay equations with state dependent time lags II, J. Reine Angew. Math. 447 (1996), 129-197. [46] J. Mallet-Paret and R.D. Nussbaum, Multiple transition layers in a singularly perturbed differential-delay equation, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 1119-1134. [47] J. Mallet-Paret, R.D. Nussbaum and E Paraskevopoulos, Periodic solutions for functional differential equations with state-dependent time lags, Topological Meth. Nonlinear Anal. 3 (1994), 101-162. [48] J. Mallet-Paret and G. Sell, Systems of differential delay equations I, J. Differential Equations 125 (1996), 385-440. [49] J. Mallet-Paret and G. Sell, The Poincar~-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations 125 (1996), 441-489. [50] C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, J. Amer. Math. Soc. 9 (1996), 1095-1133. [51] R.D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations II, J. Differential Equations 14 (1973), 368-394. [52] R.D. Nussbaum, A correction of periodic solutions of some nonlinear autonomous functional differential equations//, J. Differential Equations 16 (1974), 548-549. [53] R.D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal. 19 (1975), 319-339. [54] R.D. Nussbaum, The range of periods of periodic solutions of x I (t) = -o~f (x(t - 1)), J. Math. Anal. Appl. 58 (1977), 280-292. [55] R.D. Nussbaum, A Hopf global bifurcation theorem for retarded functional differential equations, Transl. Amer. Math. Soc. 238 (1978), 139-163. [56] R.D. Nussbaum, Periodic solutions of special differential delay equations: an example in nonlinear functional analysis, Proc. Roy. Soc. Edinburgh 81 (1978), 131-151. [57] R.D. Nussbaum, Differential-delay equations with two time lags, Mem. Amer. Math. Soc. 205 (1978). [58] R.D. Nussbaum, Uniqueness and Nonuniqueness for periodic solutions of x ~(t) = - g ( x ( t - 1)), J. Differential Equations 34 (1979), 25-54. [59] R.D. Nussbaum, Periodic solutions of analytic functional differential equations are analytic, Michigan Math. J. 20 (1973), 249-255. [60] R.D. Nussbaum, Asymptotic analysis of functional differential equations and solutions of long period, Arch. Rational Mech. Anal. 81 (1983), 373-397. [61 ] R.D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Fixed Point Theory, Lecture Notes in Math., Vol. 886, Springer, Berlin, 309-330. [62] R.D. Nussbaum, The Fixed Point Index and Some Applications, Lecture Notes, Seminaire de Math. Superieures at the Univ. of Montr6al (1983), Les Presses de l'Univ, de Montr6a194 (1985), 1-145.
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[63] R.D. Nussbaum, The fixed point index and fixed point theorems, Topological Methods for Ordinary Differential Equations, Lecture Notes in Math., Vol. 1537, Springer, Berlin (1993), 143-205. [64] R.D. Nussbaum, Generalizing the fixed point index, Math. Ann. 228 (1977), 259-278. [65] R.D. Nussbaum, Wright's equation has no solutions of period four, Proc. Roy. Soc. Edinburgh 113 (1989), 281-288. [66] P. Paraskevopoulos, Delay differential equations with state-dependent time lags, Ph.D. dissertation, Brown University (1993). [67] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983). [68] L.S. Pontryagin, On the zeros of some elementary transcendental functions, Amer. Math. Soc. Transl. (2) 1 (1955), 95-110. [69] K. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, New York (1987). [70] E. Schmitt, Ober eine Klasse linearer funktionaler Differentialgleichungen, Math. Ann. 70 (1911), 499524. [71] H.L. Smith, Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study, Math. Biosci. 113 (1993). [72] J. van Neerven, The Adjoint of a Semigroup of Linear Operators, Lecture Notes in Math., Vol. 1529, Springer, Berlin. [73] H.-O. Walther, An invariant manifold of slowly oscillating solutions for x ' ( t ) - - - # x ( t ) + f ( x ( t - 1)), J. Reine Angew. Math. 414 (1991), 67-112. [74] E.M. Wright, A nonlinear differential-difference equation, J. Reine Angew. Math. 194 (1955), 66-87. [75] J. Wu, Theo9 and Applications of Partial Functional Differential Equations, Springer, New York (1996). [76] X. Xie, Uniqueness and stabili~ of slowly oscillating periodic solutions of delay equations with bounded nonlineari~,, J. Dynamics Differential Equations 3 (1992), 515-540. [77] X. Xie, The multiplier equation and its application to S-solutions of differential delay equations, J. Differential Equations 95 (1992), 259-281. [78] X. Xie, Uniqueness and stability of slowly oscillating periodic solutions of delay equations with unbounded nonlinearity, J. Differential Equations 103 (1993), 350-375. [79] X. Xie, Ph.D. dissertation, Dept. of Mathematics, Rutgers University (1991). [80] K. Yosida, Functional Analysis, 6th edn., Springer, New York (1980).
CHAPTER
11
Navier-Stokes Equations and Dynamical Systems Claude Bardos Universi~ Denis Diderot and LAN, Universi~ Pierre et Marie Curie, Paris, France Email : bardos @math.jussieu.fr
Basile Nicolaenko Arizona State Universi~, Tempe, AZ 85287-1804, USA E-mail: byn @math. la.asu, edu
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Euler and Navier-Stokes equations: scaling parameters, regularity and stability results, theorems and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Compressible and incompressible equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Entropy and the stability of the compressible Euler equations . . . . . . . . . . . . . . . . . . . . . 2.4. Stability and instability of the incompressible Euler equation . . . . . . . . . . . . . . . . . . . . . 2.5. Existence and regularity results for the 3d Navier-Stokes equation. The weak solution of J. L e r a y . 3. Hierarchy of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The Boltzmann-Grad limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The fluid dynamics limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Turbulence and turbulence modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Introduction and the example of the k-e model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Wigner transform and defect measures for the Reynolds tensor . . . . . . . . . . . . . . . . . . . . 4.3. The Kolmogorov-Kraichnan theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Invariant measures, attractors, and evaluation of the number of degree of freedom of the flow . . . . . . 5.1. Introduction and formal derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Kolmogorov and Kraichnan inertial range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Kolmogorov-Kraichnan waves numbers and asymptotic degrees of freedom . . . . . . . . . . . . . 5.4. Mathematical tools for rigorous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Exponential attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Coherent structures in two space variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Stability of stationary solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Criteria for attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H A N D B O O K OF D Y N A M I C A L SYSTEMS, VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 503
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6.3. S o m e heuristic justification for the construction of the attractors Acknowledgements References
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1. Introduction There are many deep reasons why the Mathematical Analysis of the Navier-Stokes equations fits in the theory of Dynamical Systems. At the level of first principles the Navier-Stokes equations can be deduced from the Boltzmann equation which is obtained from a Hamiltonian system describing the evolution of molecules of gas. To do so, one takes in account the magnitude of N the Avogadro number of the order of 10 26 and considers the Boltzmann-Grad limit N --+ ec. On the other hand, fluids described by the incompressible Navier-Stokes may exhibit very complicated chaotic or self organized structures when the Reynolds number turns out to be very large. Commonly these situations are called "turbulent" and the present challenge is the construction of equations that will be used to compute the evolution of some type of averaged quantities. Therefore the Navier-Stokes equations appear to be one of the main pieces of a sequence of equations: Hamiltonian system of particles Boltzmann equation Navier-Stokes equations Models of turbulence each of them being deduced from the previous one by some averaging process where the notion of irreversibility is embedded. Irreversibility at the level of the Boltzmann equation and its relation with the irreversibility at the level of the compressible Euler equation, the compressible or incompressible Navier-Stokes equations, are by now well understood and will be recalled in Section 3.3; the solutions are related to the notion of semiflow and global attractors which can be extended from finite to infinite dimensional systems. It is more difficult to understand how the Boltzmann equation can be obtained as the limit of the genuinely reversible system, which describes the flow at the level of molecular dynamics. As will be shown, this can be done by some averaging process where the self interaction of the molecules and therefore the non linearity of the problem plays a crucial role so that the limit is in agreement with the appearance of irreversibility. This will be explained in Section 3.2. Much more difficult and unsolved questions arise when the macroscopic fluid becomes turbulent and when some type of averaging is necessary for quantitative or qualitative results. In spite of being the very end of the hierarchy, this step shares in common some points with the previous one. It is an averaging process and the "turbulent model" starts to be efficient when the original Navier-Stokes are outside the reach of direct numerical simulations. In this averaging appears a problem of moments and of closure and the search for something that would play the role of the thermodynamical equilibrium.
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However this is not easy for the following reason. There is up to now no well defined notion of equilibrium and relaxation to this equilibrium with something that would play the role of the entropy. The parameters that would lead to turbulent phenomena are not so clearly identified as in the previous steps of the hierarchy. In some sense they are less universal and more local. In this process the dynamical point of view is also essential: The introduction of randomness requires the construction of a "canonical measure" on the set of solutions. This leads to the adaptation of the Birkhoff ergodic theorem to the Navier-Stokes equations. The structure of the turbulent spectra which would play the role of the thermodynamical equilibrium has been the object of phenomenological studies initiated by Kolmogorov and Kraichnan (for the two-dimensional flow) and it is only to the best of our knowledge, in the frame of statistical semiflows defined on a periodic box that some "spin-offs" of this theory can be proven in full rigor. The structure of the turbulent spectra also leads to the notion of degrees of freedom and exponential decay after the Kolmogorov or Kraichan cut-off wave number. Here also some counterpart of these notions can be proven in full rigor provided the global attractors or exponential attractors of the semiflow are introduced. Eventually (this is the last chapter of this presentation) at very large scales one observes coherent structures (the classical examples are the Jupiter red spot or the anticyclone of the A~ores). These structures are generated through turbulent processes but play the role of metastable thermodynamical equilibriums. Up to now there has been no dynamical derivation for their appearance and stability; however, some notions of entropy and "negative temperature" inherited from statistical mechanics are used and motivated by comparison with the evolution of point vortices which are a "canonical Hamiltonian" system. As a consequence this contribution is organized as follow. In Section 2 the basic mathematical properties of the Navier-Stokes equations are presented. Relations between compressible and incompressible equations are given and the emphasis is put on the finite time stability (which very often in its mathematical formulation concerns the regularity and uniqueness of solutions). One should keep in mind that local results for smooth solution go back to Lichtenstein (1927). The fact that these results cannot always be global in time is well understood on the example of the appearance of singularities (in particular shock waves) for the compressible Euler equation. On the other hand: (i) The existence of a global in time weak solution for the 3-dimensional Navier-Stokes equation has been proven by Leray. His notion of weak solution (1934) preceded both the introduction of the Sobolev spaces (Sobolev 1936) and the distributions (Schwartz 1944). However, in spite of several interesting improvements, the question of the existence of a global in time smooth solution remains essentially open. An interesting instability result of Lions and DiPerna described in Section 2.4 may give some clues to the reason why the regularity of the solution of the Navier-Stokes equation is "hard" to prove. (ii) For the compressible Euler equation the only convenient global solution is the weak solution. Here also one should keep in mind that the only available result goes back to J. Glimm (1965) and that it has never been improved.
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The existing and non existing results for these macroscopic equations which are at the center of the hierarchy give some indication on what could be proven above and below. Section 3 is concerned with the hierarchy from the Hamiltonian system of particles to the macroscopic equation with an essential intermediate step of the introduction of the Boltzmann equation. First it is shown how to derive the Boltzmann equation from a Hamiltonian system of particles using the BBGKY hierarchy. It is important to observe in this section how the fact that the initial problem is non linear is in agreement with the appearance of an irreversible process with a nontrivial entropy from a reversible process. Then, following Hilbert, Chapman, Enskog and a series of more recent contributions, the relations between kinetic and macroscopic equations are explained. It is important to notice that most of the rigorous results are the counterpart of the classical results of the previous section for the Navier-Stokes equation. Section 4 is a short introduction to turbulence. To introduce the Reynolds stress tensor a classical model of turbulence (the k, e model) is presented. Through the study of the Reynold stress tensor, with the use of the Wigner transform, a local notion of turbulent spectra appears. The necessity of introducing some randomness is compared with the use of defect measures. In Section 5 connection is made with dynamical systems to prove some of the basic properties of turbulent spectra. For sake of simplicity (many results of this section have been adapted to other configurations) one considers a flow in a periodic box with some time independent low frequency external force. First the classical phenomenological derivation of the Kolmogorov and Kraichnan inertial range and dissipative wave numbers is given. Then the global attractor and some rigorous properties for the invariant measure are given and counterparts (in this setting) of the phenomenological results are p r o v e n - some in full rigor, others with natural hypotheses. The comparison of the evolution of the flow with the solution of a finite dimensional dynamical system has led many authors to the introduction of the notion of "inertial manifold". This notion, which works well for a series of equations (for instance KuramotoSivashinsky) as discussed in Section 5.5, does not seem effective for the Navier-Stokes equation due to intermittency in turbulence. This section is concluded with a description of a more robust and flexible object: the exponential attractor, which with some generalization of the Man6 projection theorem yields equivalent finite-dimensional "inertial" dynamical systems. The last chapter concludes the description of the hierarchy by the introduction of objects which in some sense are even more coarser: the coherent structures. A short state of the art in conjunction with the dynamical system of interacting particles is given. As a large part of the material of this contribution is classical it is useful to conclude the introduction with some references: The up to date but classical theory on Navier-Stokes equation can be found in the books of E Constantin and C. Foias, "Navier-Stokes Equations" and the book of EL. Lions, "Mathematics Topics in Fluid Mechanics, Volume 1, incompressible models". The presentation of the E-k model of Launder and Spalding follows the book of Mohammadi and Pironneau Analysis of the K-Epsilon Turbulence Model. Of course this is not the only (or the always more relevant) model. Besides the ideas given here many other approaches have been tried including the use of renormalization group (cf. Orzag and Yakhot
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Jean Leray By courtesy of the French Academy of Sciences, "Service des Archives".
Andrei Nicolaevich Kolmogorov By courtesy of Professor Albert N. Shiryaev, Moscow State University.
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[ 144]). However the k-e model seemed well adapted to the introduction of the problematic of turbulence. The authors found in the technical report of Besnard, Harlow and Rauenzahn, Spectral Transport Model for Turbulence [31] the use of the Wigner transform for the analysis of the local turbulent spectra. In spite of the fact that this is a very natural approach it does not seem to have appeared anywhere else. Section 5 borrows many ideas and most of the presentation to the review article of C. Foias What do the Navier-Stokes equations tells us about turbulence [74]. Eventually basic ideas and a systematic presentation on the notion of attractors and of inertial manifolds are an essential part of the books by A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, and by Temam on Infinite Dimensional Dynamical System in Mechanics and Physics [ 17]. The notion of the exponential attractor itself appears in the more recent book by Eden, Foias et al., Exponential attractor for dissipative evolution equations [67]. Up to now very few mathematical books have considered the question of coherent structure; however, most of the material of Section 6 can be found in Chorin, Vorticity and Turbulence [47] or in Marchioro and Pulvirenti, Mathematical theory of incompressible nonviscous fluids [ 136]. In 1987 and 1999 years passed away two scientists whose contributions, as we try to explain, were corner stones for the present theory: Jean Leray and Andrei Nikolaevich Kolmogorov. We think that their pictures should be present in this review article on NavierStokes equations.
2. Euler and Navier-Stokes equations" scaling parameters, regularity and stability results, theorems and counterexamples 2.1. Introduction The macroscopic description of the fluid is the cornerstone of the analysis of the hierarchy. It is at this level that the Reynolds number and Mach number are the most easily defined. Relations between compressible and incompressible equations are a clue to the understanding of the different types of limits of the kinetic equation as described in the next chapter. The present chapter is organized as follows. In the second section as an introduction, the relation between compressible and incompressible equations is derived at a formal level. Rigorous proof of convergence can be found in Klainerman and Majda [ 106] or in Benabdallah-Lagha [28]. The next section is devoted to the entropy which is used to show that the compressible Euler equation is an hyperbolic system, to prove some results of uniqueness and finite speed of propagation. These properties will be used in the next section where physical sufficient conditions for loss of regularity are given. The Section 2.4 is devoted to the incompressible Euler equation in 2 and 3 space variables. In 2 space variables the conservation of the vorticity along the trajectories of the flow is a precious information which gives in particular global (in time) regularity results. Nevertheless the question of the large time stability (in higher norms) remains open and this is in full agreement with the ideas exposed in the Section 6.
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In 3 space variables, the local in time existence of a smooth solution can be obtained by an adaptation to pseudodifferential operators of the Cauchy-Kowalevski theorem and by now it is a very classical result. On the other hand the existence of a solution in the large and the possible loss of regularity remains a completely open problem. The difficulty of this problem can be illustrated by a very explicit example of instability due to EL. Lions and R. DiPerna which is given. This example is also used to illustrate the difficulty of analyzing the regularity of the weak solution of the 3-dimensional incompressible Navier-Stokes equations which is considered in the Section 2.5, where some conditional results are given. For the authors, at present, the most striking one is the contribution of Constantin and Fefferman [51 ]. They have shown that loss of regularity (or stability) is induced by strong oscillations in the direction of the vorticity. This result should be combined with the a complementary point of view contained in a series of papers by Babin, Mahalov and Nicolaenko, motivated by the rotating Euler and Navier-Stokes equations in the atmosphere. These authors have studied the effect of the presence of a large Coriolis term (or large rotation frequency). They have shown that high-frequency oscillations induced by this term do stabilize the three-dimensional Euler or Navier-Stokes equation [6-8]. Of course in this situation the Coriolis force is an external force. However it may appear that large vorticity could play the same role and in the end lead to regularity results for the classical Navier-Stokes equations; for some results along this line, see [13] for a class of 3D initial data.
2.2. Compressible and incompressible equations At the macroscopic level, the most universal equations of fluid dynamics are the compressible Navier-Stokes equations. They involve, as unknowns, p, u, 0 and p the density the velocity, the temperature and the pressure. In this contribution emphasis is put on the notion of hierarchy of equations therefore the state law which gives the pressure in term of temperature and density will be the Mariotte law for the perfect gases: p
RO ~
m
o
#
R is the gas constant and # its molecular weight. As it will be shown in Section 3.3 the evolution equation for a perfect gas are derived from the Boltzmann equation when the Knudsen number goes to zero. With a convenient scaling this limit produces the following equations: Otps + Vx 9 (psus) = 0 , ps(Ot + u s . Vx)us + Vxps - - s V x .
(2.2.1) [wr(us)],
ps --psOs,
(2.2.2)
3 1 -~p~(at + u~ . V~)O~ + p~O~Vx . u~ -- s-~vcr(u~) " ~(u~) + SVx 9 [xV~0~].
(2.2.3)
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The numbers s v and s x are the viscosity and thermal diffusivity they are proportional to s the Knudsen number which is the ratio between the mean free path (the mean distance travelled by a molecule of fluid between two successive collisions and the characteristic size of the domain where the interaction takes place). Observe that the ratio between the viscosity and the thermal diffusivity is an s independent number. In fact it is one of the characteristic number of the fluid and it is called the Prandtl number; o-(u) denotes the strain-rate tensor given by 9
2
(2.2.4)
O'ij (U) = (Uirj -~ U{i ) -- -~ Vx " u r~ij .
In a compressible fluid one also introduces the Mach number M a which is the ratio of the bulk velocity to the sound speed and the Reynolds number Re which is a dimensionless reciprocal viscosity s v of the fluid. These numbers in consistency with the derivation of the above equations from the Boltzmann equations (cf. Section 3.3 and [116]) are related by the formula s --
ma Re
.
(2.2.5)
When the Reynolds number goes to infinity Equations (2.2.1)-(2.2.3)reduce to the compressible Euler equations O, p + Vx 9 ( p u ) --O,
(2.2.6)
p(Ot + u 9 Vx)u + XTx(p0) = 0,
(2.2.7)
3
p(at + u. Vx)O + pOVx 9u = O.
(2.2.8)
On the other hand the incompressible Navier-Stokes equation can also be deduced from the above equations when the Mach number goes to zero. More precisely, consider in three space variables, for time of the order of (s) -I the solutions of Equations (2.2.1)-(2.2.3). Assume that the velocity, the fluctuation of density and temperature are also of the order of s, introduce the change of functions: u , = erie(st, x),
pe = po + erie(st, x ) ,
O~ - Oo + sOe(st, x ) ,
(2.2.9)
and observe that if these functions converge (for s ~ 0) in a convenient topology their limit satisfy the following system of equations at f i + f i . V x f i + V x p - - v A f t ,
Vx (p0p + 000) - 0,
5- (a,0 + a. v 0) 2
Vx.fi--0,
(2.2.10) (2.2.11)
,AO,
(2.2.12)
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where (2.2.10) is a standard version of the incompressible Navier-Stokes equation, (2.2.11) is the Boussinesq relation between the fluctuations of density and temperature and (2.2.12) is the equation for the temperature. For v = 0 the system becomes the incompressible Euler equation. Most of the above equations are non linear and this is one of the main reasons why analytical solutions almost never exist. Therefore the analysis relies on estimates usually called a priori estimates and the connection between the different type of equations also explain if these estimates are difficult to obtain for some equation (E) it will also be difficult to obtain, for any other family (Ee) which in some sense converge to (E), e-independent estimates of the same type.
2.3. Entropy and the stability o f the compressible Euler equations Observe any smooth solution of (2.2.6)-(2.2.8) satisfies the entropy relation OtpS + Vx . (upS) = 0
with
p2/3 p S = p log - - .
0
(2.3.1)
Since S is a convex function of the principal variables
p, pu, p
lu12 3 0 ) -~+2
one can show that the corresponding linearized system is hyperbolic. As a consequence one obtains the existence and stability of smooth solutions of the system (2.2.6)-(2.2.8) for smooth initial data. One can also in the same situation prove the finite speed of propagation of localized perturbations of the constant state, and (cf. Sideris [152]) the appearance, after a finite time, of singularities. This correspond in particular to the generation of shock waves. In the presence of such singularities the relation (2.3.1) is no more valid and both on physical and mathematical ground it has to be replaced by the relation: Otp S + V~ 9 (upS) <~0
(2.3.2)
which describes the decay of entropy (observe that the mathematical and physical entropy are of opposed sign, this is due to the fact that mathematicians do prefer to consider convex functions). The entropy decay can also be used to prove a stability result between regular solutions and weak solutions which satisfy (2.3.2) (cf. Dafermos [60]). However from the mathematical point of view the situation is far from being satisfactory. The existence for all time of weak solutions has only been proved in one space variable by Glimm [90] in 1965 and in spite of tremendous efforts involving the best mathematicians of our generation the problem remains widely open. No progress has been made concerning the existence of global in time weak solution of the genuine compressible Euler equation since the work of Glimm. Furthermore one of the basic tools of this approach, in one space dimension, is the introduction of the space of functions with bounded variation. This approach seems quite natural to deal with shocks. Unfortunately it has been proven by Rauch [ 145], that no estimate of this type would be valid in higher dimension.
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2.4. Stability a n d instabilio, o f the incompressible E u l e r equation In the previous section it has been shown that the incompressible Navier-Stokes or Euler (v -- 0) equations (Equation (2.2.10) above) are the incompressible limit of the corresponding compressible equations. The necessity to have at our disposal viscosity independent results lead to the consideration of the incompressible Euler equation which in 2 and 3 space variables is Otu + u . V x u -- - V x p,
(2.4.1)
V~ " u - O,
with, if a boundary is present, an "impermeability boundary condition" u .~-0
on the boundary of the domain (~ is the outward normal to the boundary). However for sake of simplicity some of the present analysis is done for domains with no boundary (all space or space periodic solutions). The relation Vx 9u = 0 can be viewed as a constraint and p the pressure is in this point of view the Lagrange multiplier of this constraint. With the introduction of the vorticity (or V x u) co = V x u the above equation can also be written Orco + u 9 Vxco - co" Vxu -- O, Vx 9u -- O, curl u -- co,
u.~--0
on0s
(2.4.2)
The second line of (2.4.2) defines a - 1 order pseudo-differential operator co ~ K (co) = u. This observation has important consequences both at the level of abstract geometry and analysis: "(i)" The expression {u, co} - u 9 Vxco - co. Vxu has the formal properties of a Poisson bracket and the Euler equation can be written in the form:
r S - -{u, o9}. Therefore it may have in some sense an Hamiltonian structure. In fact it is natural, and this will be used in others subsections, to introduce the trajectories of the particles of the flow (or the Lagrangian coordinates) defined by the relation" 2(t,X)
=u(t,x(t,X)),
The mapping Gt " X ~
x(t, X)
x ( O ) - x.
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is a volume preserving (use the relation Vx 9u = 0 and u 9~ = 0 on the boundary) isomorphism of the domain of definition I2 of the fluid. On the other hand one can define in terms of energy the Riemannian distance between the identity and any volume preserving GT isomorphism of S2 according to the formula:
EG -- min
f fo TIG(t, x)l 2 d S dt
(2.4.3)
with G(t, .) ranging over the C 1 volume preserving transformation of $2 with the initial and final conditions:
G(O,X)=X,
G(T,X)=GT(X).
A standard variational computation shows that if (X, t) w-~ G(t, X) is an extremal for the action given by (2.4.3) then
u(t, x) -- (3(t, G-l (t, x)) is a solution of the Euler equation. Systematic extensions of this point of view can be found in Abraham and Marsden [ 1], and Arnold and Khesin [3] and are used to characterize the stability of some stationary solutions. "(ii)" It is easy to see that, in d-dimension, for initial data in a enough regular Sobolev space (for instance Hs(IR d) with s > d/2 + 1) the problem (2.4.1) has a unique solution continuously defined in the same space for a finite time interval (0, T) with
T~C
Ilu(., 0) [IHS(R,t)
Results of this type were already obtained by Lichtenstein in 1925 [ 123] (in a less elaborate language). However the problem of the existence of a solution in the large is still widely open and there is no, at variance with the compressible case, a proof of the appearance of singularity or a good physical reason for such an event. When the space dimension is equal to 2 the term co. Vxu disappears from Equation (2.4.1). In fact co = curl u remains perpendicular to the plane where the fluid evolves. Equation (2.4.1) becomes: Otco(X, t) + U" Vxco(X , t) = 0
(2.4.4)
and co is conserved along the trajectories of the particles of the fluid:
and remains bounded in L ~ for all time. This is enough to prove the existence of a weak solution (cf. Yudovich [ 169]). However the proof of the uniqueness in a convenient class
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515
is slightly more elaborated (cf. also [ 169]) and to prove the persistence of the regularity of the solution with smooth initial data one has to face the following problem: The estimate co = V x u ~ Lee
(2.4.5)
which comes from (2.4.4) is simply not enough to imply that Vxu 6 L ee
(2.4.6)
and (2.4.6) seems compulsory for any boot-strap argument for the proof of the regularity. However a more precise use of (2.4.5) gives according to Wolibner [ 168] a regularity result as follows: THEOREM 2.1. Consider the solution of the 2d incompressible Euler equation in a
bounded domain I2 of diameter L with an "impermeability boundary condition". Assume that the initial vorticity is in the HOlder space C ~ (I2). Then one has the following uniform (in time) estimate: [IV x u(.,t)llco.~,,, <. CIIv x u(.,
0>11o
(2.4.7)
with c~(t) = otexp{-CtllV x u(., 0)IIL~(~)}. PROOF. Observe that the Green function of the Laplace operator with Dirichlet boundary condition is of the following form
G(x, z)
= -~
2zr
log Ix - zl + g (x, z)
(2.4.8)
with 9/(x, y) being a smooth function and that (2.4.9)
u(x, t) = fs? V x G(x, z)V x u(z, t) dz. From (2.4.8) and (2.4.9) one deduces the estimate: IVr- u (x) .
Vr
Clgxulg~(x~)lx-yllog(
I x -) yDl
(2.4.10)
with D denoting the diameter of S2. Since the vorticity is constant along the trajectories of the flow one can use in the relation (2.4.10) the estimate
IV x u(.,t)lt ~ - - I V x u(.,0)lL ~ and for the H61der norm of the curl, the estimate:
IV x u(x(t), t) - V x u(y(t), t)l Ix(t)-y(t)l ~lt)
(2.4.11)
516
C. Bardos and B. Nicolaenko IV • u(x(O),t)- V • u(y(O),t)l ( Ix(O)- y(O)l~ ) ~?()) - y(0)i g
where
x(t)
and
p(t)-
y(t)
(2.4.12)
I x ( t ) - y(t)l a(t)
denotes Lagrangian coordinates as introduced above. With
Ix(t) - y(t)[,
uses the estimate (2.4.10) and (2.4.11) and by comparison with the solution of the differential equation
pit) D
+cP(t) log(p(t) ) D
--D--
(2.4.13)
obtains:
-CtlV •
( IX(0) -- y(0) l ) D
L OC
e
CtlV • (r
L OC ( ~2 )
Ix(t) - y(t)l /> (IX(0) -- Y(0)') e ~> D D (2.4.14)
Since D is the diameter of the domain s where the particles live one has Ix(0) - y(0)l <1 D and therefore the first term of (2.4.14) goes very rapidly to 1 and the last one very rapidly to zero when t -+ oc. Eventually one deduces form (2.4.14) that: I x ( 0 ) - y(0)l ~ Ix(t) - y(t)l ~(t)
<"
D~_~(t) (2.4.15)
which gives (2.4.6) and by classical H61der estimates for elliptic operators:
eCtlV xulL~(S2) IV~ulL~(~ ~< C
IV x u(., t)lc(~e-C,lv•
~~
IV x u(., 0)lc~. (2.4.16)
Now this relation can be used to prove that for any finite time the solution remains in the same regularity class as the initial data. F] REMARK 2.2. The estimates (2.4.16) do not prevent the measure of the regularity to deteriorate very rapidly for t --> oc even according to the rate lu(t)lc~+= -exp{exp{Ct }}. Such a behavior is not incompatible with the following facts: (i) An example due to Bahouri and Chemin [18] of a flow with an initial vorticity in L ~ (but not in an HOlder class) shows that (2.4.15) is optimal.
Navier-Stokes equations and dynamical systems
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(ii) The singular behavior (for t --~ cx~) of the H61der norm of the curl implies that due to the corresponding loss of compactness the omega limit set of the family V • u (x, t) which exist for the weak*L ~ topology may not be approached in the strong L p norm. Such an observation would be consistent with the appearance of some coherent structures as described in the Section 6. As said above in three space variable the problem is locally in time well posed in H s (N3) for s > 3 (or in Cl'~). In fact it seems much more "physical" to believe that eventually the loss of regularity would be governed by the "sup norm" of the vorticity. It turns out that such a result is true and has been proven by Beale, Kato and Majda [26] with an extension of the method of the proof of the Theorem 2.1 For sake of simplicity the proof is done for periodic solutions defined in the "flat torus" ]~3\Z3, extension to bounded domain or to the whole space are just technical. THEOREM 2.3 (Beale, Kato, Majda). Let u e C~ T[; H3(R3\Z3)) be a solution o f the three dimensional incompressible Euler equation. Suppose that there exists a time T, such that the solution cannot be continued up to T = T, and assume that T, is the first such time. Then one has f o r co(x, t) = V x u(x, t),
f0T*II
oo,
(2.4.17)
lim sup [Ico(t) ]l L ~ -- o c . tl"T,
(2.4.18)
at
-
and in particular
PROOF. Start from the standard estimates in H 3(~3\Z3) ld
---Ilull 2H3 ~ 2dt
CllVxullL~ Ilull~3.
(2.4.19)
Then as in the proof of the Theorem 2.1 introduce the Green function G ( x , y) of the Laplacian, defined on the function with mean value zero and observe that one has: f u(x, t) -- I V x G ( x , y) co(y, t) dy. JR 3\g~
(2.4.20)
Furthermore for x ~ y G(x, y) is very smooth (analytic) and for x near y one has: G(x, y) =
1
1
4re Ix - Yl
~- y ( x , y)
(2.4.21)
with V (x, y) smooth. As a consequence one can prove the following estimate:
IIV,-ulIL~ ~
Clog(1 +
IlUllH~)llcollL~.
(2.4.22)
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C. Bardos and B. Nicolaenko
Therefore one deduces from (2.4.19) and (2.4.22) and for I1u1123 > 1 an estimate of the following type: 0tllull 2,,~
~< c
IIo~r
Ilu II2H3 log Ilu II2MS
(2.4.23)
which gives by integration:
IIu <,)11 .~ ~ (1[ u <,0) 11.3) e~ ~o ~.~. ,~ ~s
(2.4.24)
and proves the theorem.
D
Eventually the fact that existence of a regular solution is a difficult problem is illustrated by the following THEOREM 2.4 (E-L. Lions and R. DiPerna). (i) For each 1 < p < oe , t > 0 and each e > O, 6 > 0 there exists a smooth periodic solution of the 3d periodic incompressible Euler equation Ot U + U 9 V x U ~- -- V x p ,
(2.4.25)
V x 9u = O,
which satisfies the estimates:
Ilu<0> IIw.~
1
and
~
~ <~ Ilu IIvr
(2.4.26)
(ii) There exist solutions of the periodic 3d incompressible Euler equation with a vorticity linearly increasing in time, according to the formula:
fly ~ ull~
O~
~ tllv ~ u<0>ll 2L O G
(2.4.27)
9
(iii) There exists no continuous smooth function, q~(t, s) independent of the Reynolds number v such that one has for smooth solution of the 3 space periodic NavierStokes equation the estimate:
Ilullw,.~).
(2.4.28)
PROOF. Let u ~ (X2) be a X2 dependent smooth periodic function and similarly let u ~(x l, X2) be a (x l, x2) dependent smooth periodic function. Then U(xl,x2,
x3)-
(Ul(Xl,X2, X3),u2(xl,x2,
_
(.o~). o..~
x3),u3(xl,x2,
_ ,.0~x~) ' ~1)
x3))
(2.4.29)
is a periodic smooth solution of the 3-dimensional Euler equation with constant pressure.
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519
Now introduce 0 E [0, 1] such that 1
0 g= ~,
1
1 ~< 1 - 0 '
2
1 ~< p < ~------------~' 0- 3(1
(2.4.30)
Pick two 6,-dependent families: u ~ (x2) uniformly smooth away from zero and behaving near 0 like (6`2 4- x2) 0/2 and
u~
uniformly smooth away 0 behaving near 0 like (6` + x 2 + x2) ~ Consider the corresponding solution of the Euler equation constructed according to the recipes (2.4.29)
V~(x,,~,x3) = (.~(x,,x~,~3),.~(x,,x~,~),.~(x,,x~,x~)).
(2.4.31)
Explicit computation show that on one hand U (0) is uniformly bounded in W l, p and that on the other hand, for any t E R, t g: 0 and uniformly for 6, small enough one has:
fR~/z310x~"S(t)I~ dx~dx2dx3 = __f~2/Z2 ]tOx!UO's (Xl -- tu O's(x2), X2) X (/gO'stl (X2) -- Ox2tt30's(Xl- tttO'S(X2), X2))] p dxl dx2
c,.
f03, 3 Io .
(x. x2)(uO
,x2
>/Ct p [ [Xl Ip [xz[ p Jt~ I~<3 (6`2 + ixl2)(3/2_0) p (6,2 + x2)(1_0/2) p dxl dx2.
(2.4.32)
Now when 6, goes to zero the right hand side of (2.4.32) behaves like the integral
dr fo ~ r3(l_O)~-------~r 1
and therefore goes to infinity for p ~> 23 (1-0) 1 " The same method can be used to prove the item (ii) with 6, fixed and t going to infinity. Finally if a function 4~(t, s) independent of the viscosity would satisfy (2.4.28) then letting the Reynolds number go to infinity in the Navier-Stokes equation one would contradict the item (i) and the proof is complete. The item (iii) gives some evidence to the difficulty of proving the existence of smooth solutions for the 3d Navier-Stokes equation and could introduce the next section. D
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C. Bardos and B. Nicolaenko
2.5. Existence a n d regularity results f o r the 3d N a v i e r - S t o k e s equation. The w e a k solution o f J. Leray
In this section it is assumed that the viscosity is non zero (finite Reynolds number) and therefore the equations of the motion in R 3 are Otu + u . V x u -- v A x u = - - V x p ,
(2.5.1)
Vx . u = O .
Assuming that the solution is smooth and multiplying (2.5.1) by u one obtains the local balance of energy
1 (a, + u . Vx - VZXx)-~lu
12 +
vlVxu
12 +
(2.5.2)
Vx " ( p u ) - - 0
which by integration from 0 to t and over 1K3 produces the a-priori estimate.
1
fo
3 ]U(X, t)] 2 d x -Jr-t)
fo'Ilull 2u'
'fo
(R3)(s) ds - -~
3 ]u(x, O)
12
dx.
(2.5.3)
The presence of the term
f0 t Ilu(s)ll 2HI(• 3)(s)ds plus some weak time regularity ensure enough compactness property to prove, with initial data uo(x) ~
L2(]R2),
Vx " uo --O,
the existence of a weak solution
u(.,,)
.,,=0.
(2.5.4)
In (2.5.4), Cw(•+; L2(R3)) denotes the space of function defined in R+ with value in L2(R3)) and continuous with respect to the L 2 weak topology. This is the basic result of Leray (1934). However, as known, even for smooth initial data it has not been possible to prove that the solution will be smooth for all time. Furthermore the class of solutions constructed by Leray is not regular enough to afford a proof of uniqueness (in the same class); also it is not regular enough to show that it satisfies the conservation of energy (2.5.3), a fortiori it is not known if it satisfies the local balance of energy (2.5.2). The following comments are usually made: (1) The instability theorem of P.L. Lions and R. DiPerna proven above shows that uniform estimates (with respect to v) are not available and suggests that the dependence of the regularity with respect to the viscosity may be difficult to control. (2) Some simple regularity results can easily be obtained, for instance:
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521
(i) If the initial data belongs to the space Hk(R 3) with k > 5/2 then the solution is smooth during a finite time ]0, T[ with T independent of v. (ii) If at the time tl the weak solution (which belongs to C~ Le(R3))) is in H i then it is smooth on a interval tl, T~, with T~ > tl > 0 depending on v and on the norm of u (t l) in the space H 1. (iii) The conjunction of the point (ii) with the fact that u is in Lz(Rt+; HI(R3)) implies that the solution will become eventually smooth for t large enough (how large, up to now, is an open problem). This also implies that the set of points where the weak solution may be singular is small. In fact this idea already was present in the original work of Leray and was refined by several authors. On its present form this refinement culminate with the work of Caffarelli et al. [33] where it is shown that if singularities exists for the Leray solution they should be contained in a set of Hausdorff measure smaller or equal to 1 in R3 • R+. Later on Sohr and von Wahl [ 156] proved for the pressure associated to the Leray solution, the estimate:
p E L 5/3 (if2 •
Z[)
which allowed Fanghua Lin [ 125] to produce a simpler proof of the result of [33]. (3) With the item (ii) the smoothness is realized if one shows that the weak solution is bounded in
Observe that with the divergence free condition one has
3 IVxu(x)]2 dx -- s
3
IV x ul 2 d x .
(2.5.5)
The right hand side of (2.5.5) is usually called the enstrophy and from the Navier-Stokes equation one deduces for co = V • u the equation: Otco + u 9 V x c o - co. V x u - v A x c o = O.
(2.5.6)
The energy estimates implies the estimate:
v
foTs
t)[ d x d t <~ C [ u o ( x ) ] 2L2(R3)" 3 [(co. Vru)(x, "
(2.5.7)
Introduce the direction ~ of the vorticity defined by
(x, t) =
co(x, t) leo(x, t)l
(x, t) = 0
if co(x, t) # 0, (2.5.8) if co(x, t) = 0.
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522
Multiplying Equation (2.5.6) by ~ (x, t) one obtains (formal computation is done first then a rigorous proof can be obtained by a regularization process): (Ot -+- u - V ~
-
vmx)l~l-+- vlo~l-31V~l 2
@ Z(CO2t~kl- cokcol)()icokOicol ~ ikl
(2.5.9)
[co" Vxb/[.
In agreement with the convexity of the function co ~ [col the last term of the left hand side of (2.5.9) is non negative and with the energy estimate this gives the bound:
f I~(~, ,)[d~ ~ c f~3 {luo(~)I~+
[coo(x)[} dx.
(2.5.10)
Observe that among the quantities which have been shown to be uniformly bounded are sup t>O
[ V x u ( x , t ) Idx
and
[ V x u ( x , t ) ] 2dx
3
dt
3
while the typical one which is missing for global regularity is sup fR IV x u(x, t)[ 2 dx ~ C. t>0
3
The gap is not that big but seems very difficult to fill. (4) It was already observed in by Serrin [ 150] and by Kaniel and Shinbrot [ 100] that in dimension 3 the supplementary hypothesis:
~(~, t) ~ L s (0, ~; (L r (a))~),
2 3 -+-~3 S
r
was sufficient to ensure the persistence of regularity and the uniqueness. The marginal case is s = 2 and r = cx~,result which has been recently improved by Kozono and Taniuchi [104]. Introducing the space BMO (cf. [117]) which contains L ~ they have shown, in the spirit of the Beale-Kato-Majda theorem (Theorem 2.4) that the condition
fo ~ Hl~(.,t)
dt <
was enough to ensure the persistence of regularity up to the time T. In the same spirit it should also be observed that another sufficient condition for the uniqueness and regularity has been obtained for solutions with value in L 3(R3). Since the transformation:
(u(x, t), p(x, t)) w-~ (~u ()~x, ,k2t), ~2p(~x, Xt))
Navier-Stokes equations and dynamical systems
523
preserves both the solution of the Navier-Stokes equation and the norm of u in L ~ ( 0 , ec, L3(R3)) this space seems to play a crucial role for the analysis of the problem. Such point of view introduced by Kato and Ponce [ 103] and Weissler [ 167] involved several contributions from Calderon [40], Cannone [41 ] and culminated with the work of Furioli et al. [87]. Once again the gap between uniform bound in L 3 and L 2 seems small. Eventually the derivation of the relation (2.5.10) let appears some relation between the regularity of the solution and the regularity of the direction of the vorticity and in fact, more precisely, one has: THEOREM 2.5 (Constantin and Fefferman). (i) For any Leray solution of the 3d Navier-Stokes equation one has:
/0T
dt x,t ); lco(x,t )l> S2}
IVx~(x,t)12dxdt
+ Iv • uo(x)l}dx.
(2.5.11)
(ii) Assume that the direction of the vorticity of the weak solution u is uniformly Lipschitz with respect to x when the modulus of this vorticity is large; this means that there exist two positive finite constants C and p such that one has
Y(x , y , t) 9 "(R3)2 •
R + {Ico(x, t) I > ~ and Io~(y,,)l
> [sin4~(x, y, t)[ ~<
Ix - yl
>
s2 ] (2.5.12)
P with ~b(x, y, t) denoting the angle of the two vectors co(x, t) and co(y, t). Then the vorticity, if bounded for t = 0 in Lz(IR3) remains bounded in the same space for t >~0 and therefore the solution is "regular".
REMARK 2.6. The significance of the relation (2.5.11) is: in regions of high vorticity the direction of the vorticity is regular in an averaged sense but uniformly with respect to the initial data and with a 1/v dependence with respect to the viscosity. The significance of the assertion (ii) is that singularities (or loss of control of the regularity) to appear need both large (in modulus) vorticity and large oscillations of the direction of this vorticity. PROOF. As above the proof is made with a priori estimates which are later used for rigorous proof with the introduction of some regularization process. Here the emphasis is put on the a-priori estimates. First, observe that the relation (2.5.11) is just a direct consequence of the estimate (2.5.10). Second, to make the proof simpler and to focus on the key point it is assumed that the estimate (2.5.12) is valid not only at points (x, y) where the vorticity is greater than S-2 but everywhere. Releasing this is hypothesis implies the introduction of terms which are quadratic with respect to the co instead of being cubic and which can be easily handled.
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524
Therefore one writes for the formal estimate:
OtfiR3]('O(X't)[2qL1)fil~3]Vx('O(X't)]2dx~ s ](wVxu,w)]dx.
(2.5.13)
The last term of the right hand side is cubic with respect to w. In fact it involves the strain matrix: S(x,
1 (Vxu + (Vxu)*) } (x, t) = S(w)(x, t),
t) -
where appears the direction
~(x,t) =
co(x, t) leo(x, t)l
of the vorticity. With
~.. Y y --
lyl
and
M 0",
1
co)- ~ [~" | ('~"x co)+ (~" x co)| )']
one has
dy 3 P.V. f M(~', co(x + y ' t))lY13 S(x, t) -- 4--7 and 3
/,
j~ (y, o,~x, ,)) (y, o,(x + y, ,) o,<x ,))dy 3
' t)12 ~-n
JR3
(~', ~x, ,)) (~-, ,o~x + y, ,) ~(x, ,))
dy lyl 3" (2.5.14)
It is in this last term that the hypothesis is used because
](7,~(x,t))(Y,o~(x + y,t),~(x,t))l ~< [o,(x + y,t) I ] sin(~(x + y,t),~(x,t))l <~Clyl
(2.5.15)
and therefore in Equation (2.5.14) the order of the singularity has been reduced. This equation becomes:
I(~oVxU~o>l~ clio<x, ,>12f~_~I~ocx+ y, n l ~dy
2
- cl~ocx, ,)l i cx,,)
(2.5.16)
525
Navier-Stokes equations and dynamical systems for which the following estimate can be easily obtained:
Ilz(x,t)llL~ -
,Io,(x+y,t)ll- ~
, dx
V C([]eo(.,t)HL,)2/3(Heo(.,t)[[L2)I/3.
(2.5.17)
With the Cauchy-Schwartz relation and the Gagliardo-Nirenberg inequality which is presently used in the following form:
(fir .,[o,(x)l4dx )'/2 .< c (fR ~lvxo,(x) 12dx )3/4(s ,lo,(x)l-dx .~ )
1/4
one obtains:
s I(~ovxu~o)I dx ~ c s Loo(x,1)12 l ( x , t ) d x <~
IIz(x,,)ll~2
~lo)(x)l 4dx
(f~)~,4(j~ 31Vxeo(x)12dx
• (llo,(.,,)ll~
~lo,(x)l 2
dx)1/4
1)2/3([l~,(.,,)ll~) 1/39
(2.5.18)
Eventually the uniform estimate on the L l of the curl is used (cf. (2.5.10)), giving:
31(o,v.,-~oo)ldx .<
3Io)(x)12dx
3[V~o,(x)l~dx
= c
3IVx~
12dx
(llo,(.,,)[l~) '/~
3[~176 12dx
s
V ~V 3IV~-eo(X)[2 dx + Cv -3
3 le~
t)12 dx
. (2.5.19)
The term fR3[Vxw(X)[ 2 dx is in Equation (2.5.13) balanced by the viscosity and therefore one obtains for the enstrophy the relation:
s
1s ) ( f2 dx~ .< c ( f3~ Io,(x,,)l
2 dt
3
-2
3
2 "
~ Io,(x,,)l ~ dx )2/3
(2.5.20)
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C. Bardos a n d B. Nicolaenko
Since one has
fo (fo
I,o(x, t)] 2 dx
)2,3
dt <~ T 1/3
(/0
~Io~(x, t)l 2 dx dt
)
2/3
one concludes the proof with the estimate of energy and the Gronwall lemma.
(2.5.21) D
REMARK 2.7. In fact the uniform Lipschitz condition can be relaxed what really matter is an estimate of the type
I(Y,o)(x § y,t),~(x,t))l <~cIo)(x + y,t)llyl
(2.5.22)
which is much weaker (valid for two vectors of opposite direction). The above results should be compared with the analysis done by Babin et al. [6,7] who, motivated by the geophysical applications, consider the Navier-Stokes equation with a large Coriolis force" Otu + u . V x u + 2I2oe3 x u - v A x u
+ V x p = F,
Vx . u --O.
(2.5.23)
In (2.5.23) e3 is the vertical unit vector and S20 is the frequency of the background rotation which introduces a Coriolis force which is assumed to be large compared to the other parameters of the flow. A detailed analysis is done in the case of a periodic domain or stress-free boundary conditions. In fact the linearized version of Equation (2.5.23) was studied (cf. Arnold and Khezin [3]) by Sobolev who started from an analysis done by Poincar6 and who applied it to the description of the behaviour of fuel tanks of rotating projectiles. The work of Sobolev was done in Kazan around 1942 and by that time classified. It was declassified in [ 155]. In the extension of this analysis to the genuine nonlinear equation (2.5.23) Babin, Mahalov and Nicolaenko used a sharp Fourier analysis involving small denominators and a Diophantine condition on the incommensurability of the geometrical parameters of the domain. It is shown in [6-8] that the solutions of the 3-D Euler or Navier-Stokes equation of uniformly rotating fluids can be decomposed into the sum of the following terms: a solution of the 2-D Euler (or Navier-Stokes) system with vertically averaged initial data, a vector field explicitly expressed in terms of phase and a small remainder term. In the course of the proof [6,7] have obtained the following stability-regularity results: (i) Whatever the size of the smooth initial data the life span of the corresponding regular solution of the Euler equation is ensured to go to infinity when s goes to infinity. (ii) For non zero but fixed viscosity v, whatever the size of the smooth initial data, the corresponding classical Leray solution of the 3-D Navier-Stokes system becomes smooth (for T ~< t < cx~) for s large enough. This is true for all domain geometrical parameters. Specifically [ 11 ]"
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527
THEOREM 2.8. Let v > 0, c~ > 1/2, IlU(0)ll~ <. M~ and T+ 1
sup T
fT
IIFII~- l dt <~M G 2
(2.5.24)
YT.
Let 1-2o ~ $2~ (M~, M G , v). Then solutions of rotating Navier-Stokes equations for any periodic domain parameters are regular for all t and
Ilu{,)ll
M; v,
(2.5.25)
o
A
THEOREM 2.9. Let v > 0 and conditions of Theorem 2.6 hold. Let Ilu(0)ll0 ~ M0, T -Ilu(O)ll2/v. Then, for everyfixed I2o >/S-2~ with a'2~ - 12~(MG) and for any weak solution u(t) of rotating Navier-Stokes equations which is defined on [0, T] and satisfies the classical energy inequalityon [0, T], the following is true: u(t) can be extended to 0 < t < +ec and it is regularfor r <~ t < +oc" it belongs to H1 and Ilu(t)lll ~< CI(MF1, v) for every t>~r. A
1 In particular, Theorem 2.8 relies on the global regularity of a ,, 2~-dimensional" limit nonlinear Navier-Stokes equation as 120 --+ oo, [7,11 ]. These results are not conditional, in contrast to the work of Constantine et al., with the following remark. In the rotating equation (2.5.23) when the vorticity 120 is large, but bounded, the highly oscillatory (in time) solution is regular. One can show that (2.5.23) is equivalent to a Navier-Stokes equation without Coriolis term, with a base steady flow (-I20y, +S20x, 0) of vorticity 2120e3, plus spatially periodic perturbations of vorticity O91 . Then if 1o911 is not too large with respect to 11201, with lI201 >> 1, one proves [13] that the corresponding Navier-Stokes system stays smooth for all times (COl is not a small perturbation). Note that the perturbed col-flow is genuinely 3-dimensional. The technique of bootstrapping regularity of solutions of 3-dimensional Navier-Stokes equations by perturbation from limit equations has been done in various contexts: thin domains [146], helical flows [131]. In these previous works, limit equations are 2-D Navier-Stokes equations for which global regularity is well known. In [ 11 ], for the first time, the limit equations are genuinely 3-dimensional, but with restricted wave-number interactions in B(u, u) ("2 89 Their global existence is nontrivial and requires dyadic decomposition methods and Littlewood-Paley theory [ 117]. Similar results for more general Boussinesq equations of geophysics can be found in [9-12,14].
3. H i e r a r c h y of e q u a t i o n s
3.1. Introduction As said in the general introduction one shows that, with the Boltzmann equation, the Navier-Stokes and Euler equations can be derived from a genuine Hamiltonian system of N interacting particles.
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528
This Hamiltonian system is the beginning of the hierarchy. The end of the hierarchy is the introduction of turbulent models which are in some cases constructed with a statistic description of the fluid. In this case some basic properties of dynamical systems like the ergodic hypothesis are involved and some "magic" numbers like the Kolmogorov exponent appears. Eventually one should observe that the different steps of the hierarchy share in common several features like the evolution of the notion of entropy and the recurrent use of moments or averages.
3.2. The B o l t z m a n n - G r a d limit The purpose of this section is to provide a rapid overview of the derivation of the Boltzmann equation from molecular dynamics. The points that should be emphasized are the following: (i) In this problem there are two natural parameters N the number of particles and a the radius of these particles. N is a very large number and a (expressed in common units, such as centimeters) is very small; Consider a rarefied gas in a box whose volume is 1 cm 3 at room temperature and atmospheric pressure. Then N "~ 10 2~ a -- 10 -8 cm and N a 2 -- 1 m 2 is a sizable quantity. Therefore it is natural to consider situations where lim
No -2
N---->o~, a--~O
is strictly positive and finite. This gives the mean free path and the Knudsen number. (ii) The transition from a reversible problem to a irreversible problem is made by an averaging process which takes in account the self interaction of the particles of the media. The direction of the time appears because one obtains an equation for averaged quantities at time t > 0 only keeping information on averaged quantities at time t -- 0. The same construction should be possible for negative time but would lead to a Boltzmann equation with negative term that would therefore increase the mathematical entropy. (iii) On one hand the nonlinearity helps because at variance with diffusion approximation of reversible linear systems the entropy at the level of the Boltzmann equation (a quantity which naturally decays) is the limit of a quantity which, due to the nonlinearity, is not conserved by the molecular dynamics. Therefore this does not contradict strong convergence results. (iv) On the other hand non-linearity creates also a limitation on the obtaining of rigorous results based on strong convergence because such results would be, when the kinetic limit is involved, in contradiction with the instabilities of the compressible and incompressible Euler equations described in the previous section. At present there are two types of rigorous results. Both involve regular quantities and therefore they should be kept away from the limits leading to singular solutions of the compressible Euler equation and there are two ways to do so.
Navier-Stokes equations and dynamical systems
529
The first one (Lanford [118]) is to start from very regular initial data and prove the results for a very small time that would avoid the time where the compressible Euler equation may present singularities. The second one is to consider a dilute gas (Illner and Pulvirenti [95]) in an infinite media which will never behave like the solution of the compressible Euler equation and therefore global-in-time convergence proof can be obtained in this case. However since this regime does not lead to the Fluid dynamics equations it should not be considered as a pertinent step for our hierarchy. Below only a formal proof is given following the Section 2.2 of Cercignani, Illner and Pulvirenti [43]. The convergence proofs quoted above can also be found in this book. In any case for the derivation it is both natural and compulsory to invoke the B B G K Y hierarchy named after Bogoliubov, Born, Green, Kirkwood and Yvon. This point of view was discovered by Yvon in 1935 and rediscovered independently eleven years later by Kirkwood, Born and Green on one side and by Bogoliubov on the other. It is in the construction of the BBGKY hierarchy that the Boltzmann entropy (which as explained in the previous section is simply related to the macroscopic entropy) appears as a limit process which is not in contradiction to the derivation of a reversible system from an irreversible one. The starting point is the consideration of a family of N particles of radius a which evolve freely in the whole space and interact through elastic collisions. More precisely if two particles with incoming velocity (~, ~,) and centers x and x, collide (i.e., Ix - x , ] - or) then the outgoing velocities (~', ~,) compatible with the conservation of mass momentum and energy are given (in term of the incoming velocities) by the formula:
with X mY*
co - Ix - x, I'
~ - ~* + co[co" (~ - ~* )]"
(3.2.1)
For obvious reason the above problem is called the hard sphere model. Other models are based on mass points interacting with central forces. However in this case the Boltzmann Grad limit is more difficult to obtain and to the best of our knowledge it is up to now only done with the introduction of a ad hoc cutt off (cf. [42, p. 59]). N is the Avogadro number and it is of the order of 102~ and cr is of the order of 10 -8 meters while N o 2 is of the order of 1 square meter. This means that the Boltzmann equation should be derived from the molecular dynamics by letting N go to infinity, o- to zero and letting No -2 go to a positive finite constant which is the inverse of the Knudsen number. It is convenient to denote by z
=
.....
.....
•
the variables of the s dimensional phase space and by ps (z l, Z2 . . . . . Zs) the probability of having jointly the s particles at the point {xl, x2 . . . . . xs} with velocity {~l, ~2 . . . . , ~ }.
C. Bardos and B. Nicolaenko
530
It is assumed that these functions are symmetric with respect to their arguments (observe that if such a property is true at some time it is conserved for all time by the collision rule given by (3.2.1)) and are equal to zero on one hand for s > N and on the other hand for
z ~ A s,
A s = {z s such that Vl <~ i, j <~ s, (Xi r Xj)Ixi - xjl > a }.
(3.2.2)
The meaning of Equation (3.2.2) is that it is assumed that the particles do not penetrate one into the others. Furthermore one has
PS(zS) -- f(R 3xR3)N-, p N
( Z l , Z2 . . . . .
Zs, Zs+l
.....
ZN) (s+I)<~j<~N 17 dzj. (3.2.3)
Since the particles evolve freely in AN one has N
Ot p N (X, V, t) + Z
~i Oxi p N (x, V, t) = O.
(3.2.4)
1
This equation has to be accompanied by suitable initial and boundary conditions. To take into account the fact that the shocks between the particles are elastic it is assumed that the distribution P N is invariant under the transformation induced by such shocks, i.e., p N (Z) = p N (Z ~) for any pair (z, z') defined by the following relations: (i) There exists a pair (i, j), i r j such that [ x i - x j [ =or (i.e., z 6 0 A N ) . (ii) Z~ is given in terms of Z by the formula !
Z --{Zl,Z2
.....
(Xi,~i--O)ij((.Oij" Vij))
.....
(xi, ~j -~-o)ij(o)ij " Vij)) . . . . . ZN}, Xi --Xj Vij -- ~i -- ~j, o)ij = ix i _ xJ I ,
(3.2.5)
which is simply the translation at the level of probability distribution of the formula (3.2.1). For the initial data it is assumed that p N (Zl, Z2 . . . . . ZN, 0) is a probability density which is invariant under any permutations of the variables zi. Even better it is assumed that this probability is factorized:
p N (Z l, Z2 . . . . . ZN, O) --
VI
I~j~N
f (zj),
f (zj) >
0, ~3 •
f (z) dz - 1. (3.2.6)
Following [43] Equation (3.2.4) is integrated over A N with respect to the variables
Zj,
(s+l)<~j<,N,
Navier-Stokes equations and dynamical systems
531
and one obtains"
a t P ~ + f ~ ~ i a x i Pu
I-I
dzj
(s+l)~j~f
i=l N
+f
Pu(s+l)~j~N I-I dzj - 0 .
k--(s+l)
(3.2.7)
Since the boundary of the integration domain is characterized by the relation ]Xi -- X j] m O" it depends (even for i <~ s) upon xi. Therefore one obtains for the second term of the left hand side of (3.2.7)
N
f ~iOxiPN
l-I dzjm~iOxiPS-- ~ f P(s+l)~i'~~176 (s+l)~j~N k--s+l ~ (3.2.8)
with OOikdenoting the outer normal to the sphere of radius o- and center Xk and daik being the surface element on the same sphere. The second integral term of (3.2.8) is easier to compute because it involves the integration of a derivative taken with respect to one of the integration variables. One obtains:
dzj=~fP(s+l)~k.COikd~ikd~k H (s+I)~j~N i=1
f ~kOxkpN N
+
Z f P(S+2)~k "COikdo'ik d~k dxi d~i. i--s+l, ir '~
(3.2.9)
Therefore with (3.2.8) and (3.2.9) the following relation is deduced from (3.2.7):
OtP s +
~iOxi Ps-~ i--I
i--1
~ P(S+l)Vik'COikdCrikd~k k=s+l ~
N
1 -]- -~
~ i----s+l,
f P (s+2) Vki "Ogik d oi k d~kdxi d~i dxi .
(3.2 . 10)
i~k ~
In the above equation Vik = ~i - ~k is the relative velocity of the ith particle with respect to kth particle. The relations O)ik -- --09ki have been used to replace ~k .COik by 89Vik .COik Observe that with the boundary condition pN (Z) -- pN (Z') with z' given by (3.1.4), the last term of the right hand side of (3.2.10) is identically zero.
532
C. B a r d o s a n d B. N i c o l a e n k o
Observe also that the first integral in the right hand side of (3.2.10) is the same no matter what the value of the dummy index k is. This index can be abolished; x,, ~, is written in place of xk, ~k and (3.2.10) is transformed into:
O,P"+ ~-]~iO~c~PS-(N-s) ~ f P('+~)V,..o~idaid~.. i=1
(3.2.11)
i=1
In (3.2.11) the arguments of p(s+l) defined by the relation:
a r e ( Z l , z2 . . . . .
Zs,
z, = (x,, ~,)) while Vi and
ni
are
Vi - - ~ i - - ~ * , Xi
(3.2.12)
-- X,
O)i
We separate the contribution of the two hemispheres S~_ and S / , respectively, defined by V/ 9 o)i > 0 and V/-wi < 0. In addition we remark the relation dai - - o "2 dwi where dwi is the surface element on the unit sphere described by O) i and eventually we obtain
Otps-+-~~iOxiPS
=(N-s)~
i=1
f i=1
--fIRfs3 _ P(S+l)IVi.wiIdwid~, ) where
P (s+l)t
means that in
p(s+l)
P(S+l)tlVi'wildwid~*
+
(3.2.13)
the argument ~i and ~ * are replaced by the following
ones: ~[ -- ~i -- O)i (O)i" Vi),
(3.2.14)
~ ; -- ~ , .qt_o) i ( o) i . Vi ) . At this point a choice in the direction of the time has been made because the velocities after the shock have been express in term of the velocities before the shock. The above integrals can be changed into an a single integral by changing O)i into - w i in the second integral. The index i in 09i can be dropped provided the argument x, in the second integral of the i th term is replaced by X , m Xi m (DO"
while x, is replaced by Xi -Jr-0){9"in the first integral. These computation leads to a system of N equations for N unknown ps.
Navier-Stokes equations and dynamical systems Ot PN -+-
533
~i Oxi PN
i=1
(PN(s+l) t ]Vi.(.oili=1
(s+l)
P~
]Vi.o)il)do)id~,
(3.2.15)
•
which is called the BBGKY hierarchy. The Boltzmann limit is obtained by letting N go to infinity and a to zero with the condition: lim Na 2 -- e-1 and the initial data:
/.
pN __ H
f (zj),
f (zj) >/O,
I f (zj) dzj -- 1.
(3.2.16)
d
If we assume (this is one of the main object of the contributions of Landford and Illner and Pulvirenti) that the convergence (for s fixed and N going to infinity) lim P~v- ps N--+oo holds for any s in a convenient topology, we can deduce from (3.2.15) an infinite set of equations (for 1 ~< s < oo)
Otps + ~ ~i OXi ps i=1 1~ s 6i= 1
(p(s+l)t p(s+l) 3 • 2 \" N I gi .(.oi] - - . N [gi .o)i ]) dcoi d ~ ,
(3.2.17)
which is called the Boltzmann hierarchy. This derivation is completed by the three following statements. "(i)" Introduce the Boltzmann equation for hard spheres which will also be considered in the next section:
1
OtF + v. V~-F- -C(F),
(3.2.18)
F(O, x, v) -- F in (x, v) >~O,
(3.2.19)
8
where the collision operator C (F) is quadratic and given (with an abuse of language) by
C(F) --C(F, F ) - f f ( F ( F ' - FiF)l(Vl -- V).wldwdvl,
(3.2.20)
C. Bardos and B. Nicolaenko
534
!
with the F, F1, F' and F 1 appearing in the integrand understood to mean F ( t , x , .) evaluated at the velocities v, Vl , v ! and v !1 respectively, where the primed velocities are defined by v' =
v + ~oco.
(Vl
-
!
v1-
v),
vi -
oJoJ.
(vi
-
v),
(3.2.21)
for any given (v, Vl, o9) 6 R D x R D x ~D-l. Use the fact that for smooth initial data close to an absolute Maxwellian (p, and 0, are constant)
M--
tO:~
(2st0,)3/2
e x p ( - 1 / 2 l vl2/O, ),
(3.2.22)
the Boltzmann equation (3.2.18) has a unique smooth solution (defined at least during a finite time). "(ii)" For smooth initial data the Boltzmann hierarchy has a unique solution (defined at least for a small time). "(iii)" If F(z, t) = F(x, ~, t) is the corresponding solution of the Boltzmann equation then the unique solution of the Boltzmann hierarchy with initial data given by
ps
(Zl, Z2, Z3 . . . . . Zs) --
17
f (zi)
(3.2.23)
l <<.i <<.s is given by the same factorization: P S ( z l , z 2 , z3 . . . . . Z s , t ) - -
U
(3.2.24)
f(zi,t).
l<~i<~s
The item (i) is now classical in theory of Boltzmann equation (Ukai [162] or Nishida and Imai [ 141 ]). The item (ii) has been solved by Landford and Illner and Pulvirenti under weaker hypothesis than the one needed to prove the convergence of the P~. The item (iii) is obtained by direct inspection. As a consequence under convenient hypothesis the function P~ (zi . . . . . z,,, t) converge to a function
PS(z 1. . . . . Z n , t ) - -
I--I f ( z i , t ) i <~i<<.s
with f (z, t) solution of the Boltzmann equation. This implies in particular that the function p i (z, t) converges to the solution of the Boltzmann equation and that the P)~ factorize at the limit. Such a property is called propagation of chaos. It is important to notice that the above convergence does not contradict the appearance of the decay of entropy for solution of the Boltzmann equation. On one hand it should be observed that for any solution of the Liouville equation and for any function F one has
dt
3•
1<~i<~N
Navier-Stokes equations and dynamical systems
535
On the other hand (this is just obtained by Fubini Theorem and change of variables) for any solution of the Boltzmann equation one has
d(flR
dt
3xIR3 )
flogf(z,t)dz)<~0
(3.2.26)
with a strict inequality whenever f is not an absolute Maxwellian. Eventually with (3.2.25) one has
f(~
3X~3)
N1
f (z, t) log f
fro
3•
(z, t) dz <~f(R 3Xk3) f (pN(zI ~ Z2,
9. o
(z, O) log f (z, O)dz
, ZN, 0))
H
dzi
H
dzi.
x l o g ( P u (Zl, z2 . . . . . ZU, 0))
l~i<~N N
3•
(PN(zl,Z2 . . . . . ZN,t)) • l o g ( P N(zl, Z2 . . . . .
ZN, t))
(3.2.27)
l <~i <<.N However,
f ~ 3 • ]~3
f(z,t)logf(z,t)dz
is not the limit of
H N(PN)(t) -- -~
3•
N (PN(zl
. . . . . ZN, t))
x l o g ( p N ( z l . . . . . ZN, t))
H
dzi
l~i~N but of H1 (P1)(t) = ffR3x~3)N
(P1N(Z, t)) log(p1
(Z, t))
dz.
And according to the Proposition 3.1 (below) one has H l(Pl)(t) <~H N (PN)(t) with strict inequality unless P l(t) is factorized. Therefore the strong convergence of p1 (t) does not contradict the decay of entropy of the solution of the Boltzmann equation. In fact one observes also that the factorization property given for P N at time t = 0 is immediately lost for t > 0 but is recovered for the limit of P~r (t) (s, t > 0 fixed and N going to infinity).
C. Bardos and B. Nicolaenko
536
PROPOSITION 3.1. Suppose that pN is a symmetric probability density on the phase space and that ps are the s particles distribution functions associated with p N. Then
Hi(p1) ~ H N ( p N) with equality if and only if
N pN (Z) -- I-I p l (Zi ) i=1
for almost all z. PROOF. Note that for x, y ~> 0 one has (with the right hand side equal to zero for y - - 0 and to - o o for y > 0 and x - 0 ) : x-y
X
~> y l o g - . Y
Therefore one has
o
(3.2.28)
and this relation is equivalent to the relation:
f pN (Z) log pN (Z) dz > / N . H 1(PI) with equality only when the middle term of (3.2.28) is zero, i.e., when factorization occurs. R REMARK 3.2. Rigorous proofs contain many more ingredients however two points should be stressed. (1) Since the Boltzmann equation is quadratic it involves only binary collision therefore an important step is to prove that other events can be excluded and this is a direct consequence of the following theorem (cf. [43, p. 65]): "The set of points that are led into a multiple collision under forward or backward evolution of the dynamical system and the set of points such that where there is a cluster of collision instants under forward and backward evolution are of measure zero in the phase space". (2) The solution of the BBGKY hierarchy can be written in an integrated form (or weak form) leading first to a formal series expansion:
P~v(z)(zS,t) -- ~_~ n=O
dtl
fo'
dt2..,
a
fo
dtnSa(t -- tl)Osa+l
Ds+n ( zs' O)
• Scr(tl - tz)...Qs+nS~r(tn). U
(3.2.29)
Navier-Stokes equations and dynamical systems
537
with PNS -- 0 for s > N, with S~ and Qs~+l operators describing the advection and the collisions. It is in the uniform estimate of the right hand side of (3.2.29) that the hypothesis on the data or on the small time intervals of validity does appear.
3.3. The fluid dynamics limits Continuing the hierarchy we return to Equation (3.2.18):
1
(3.3.1)
OrFc + v . Vx F~ = - C ( F s ) 6
with C (F) denoting the collision operator given by (3.2.20) and e-
lim
No -2 .
N--+ oc, or--+0
Equation (3.3.1) (cf. [91] for details in the following discussion) shows that
'if
-
FI[(Vl
-
v).
co I do) dvl
8
is homogeneous to a frequency. If the variation of F in t and in x are not too fast, this frequency is the reciprocal of the averaged time between two successive collisions undergone by the same typical particle under the distribution F moving with speed v. However this frequency depends on the particle distribution itself, which makes it difficult to use this expression as a tool to discriminate between various qualitative behavior of this particle distribution. Rather, pick an averaged macroscopic density p, an averaged temperature 0, and choose a macroscopic length scale )~, (for instance the size of the domain where the flow takes place or the average size of Ox F / F at t = 0). Then rewrite Equation (3.3.1) for the dimensionless number density ~, 02/2 F-F P, in term of the dimensionless time, space and velocity variable"
i"
t VrO-~,
~"
v
ft.,
zk,
~-,
and obtain:
1 C(F') ' OiF + ~'VyFA -- K,---~
(3.3~.1)
where Kn the (dimensionless) Knudsen number is the ration of the collision mean free time to the macroscopic time scale.
C. Bardos and B. Nicolaenko
538
All hydrodynamic limits of the Boltzmann equation consist in considering this dimensionless form and in discussing the limit as Kn (and possibly other parameters) tend to 0. Physically, this means that a great number of collision take place in the gas per unit of (macroscopic) observation time. Observe that this is not in contradiction with the usual phrase about the Boltzmann equation which applies to gases in a state of low density because one has: Nff 2
Kn The analysis done below applies also to other collision operators which are introduced either for physical or numerical reason. Therefore it is interesting to select the basic properties of the operator which are used at different steps: (i) conservation properties and an entropy relation that implies that the equilibria are Maxwellian distributions for the zeroth order limit; (ii) the derivative of C ( F ) satisfies a formal Fredholm alternative with a kernel related to the conservation properties of (i). Properties (i) are sufficient to derive the compressible Euler equations from Equation (3.2.1) (Theorem 3.3). The compressible Euler equations also arise as the leading order dynamics from a systematic expansion of F in e. Properties (ii) are used to obtain the Navier-Stokes equations; they depend on a more detailed knowledge of the collision operator. The compressible Navier-Stokes equations arise as corrections to those of Euler at the next order in the Chapman-Enskog expansion. This expansion shows that in a compressible gas the Knudsen and Reynold number are of the same order. To recover directly from the Boltzmann equation the incompressible Navier-Stokes equation one also introduces the Mach number Ma which is the ratio of the bulk velocity to the sound speed and the Reynolds number Re which is a dimensionless reciprocal viscosity of the fluid. These numbers (cf. [ 116]) satisfy the relation
ma - ~. Re
(3.3.2)
Therefore when e goes to zero, to obtain a fluid dynamical limit with a finite Reynolds number, the Mach number must vanish too.
The compressible Euler limit. The integral of any scalar or vector valued function f (v) with respect to the variable v is denoted by (f); ( f ) - ~ 3 f ( v ) dr.
(3.3.3)
Use of Fubini theorem and change of variable to show that the operator C satisfies the conservation properties
(C(F)) - O,
(vC(F)) - O,
(Ivl2C(F)) -- O,
(3.3.4)
539
Navier-Stokes equations and dynamical systems
which are the simple translation at the level of the function F of the corresponding properties for the system of particles of the previous section. As a consequence one has the following conservation laws:
Or(F) + Vx . (vF) =0, (3.3.5)
Ot(vF) -+- Vx 9 (v | v F) -----0,
(1)
(1)
Ot ~lvl2F + V x - v-~lvl2F --0. Similarly it has been observed in the previous section that (C(F)log F) is non-positive, this implies the local entropy inequality
3t(FlogF) + Vx . (vFlogF) --(C(F)logF)<~ O.
(3.3.6)
A more detailed analysis shows that
(C(F) log F ) = 0
(3.3.7)
implies that F is a Maxwellian: P ( F = (2Jr0)3/2 exp
l l v - u l 2) 2 0 "
(3.3.8)
The parameters p, u and 0 introduced at the right side of (3.3.8) are related to the fluid dynamics moments giving the mass, momentum and energy densities:
(F)=p,
(vF)=pu,
(~ Ivl 2F = p
lu12+2
"
They are called respectively the (mass) density, velocity and temperature of the fluid. In the compressible Euler limit these variables are shown to satisfy the system of compressible Euler equations (3.3.11 below). The main obstruction to proving the validity of this fluid dynamical limit is the fact that, as said in Section 2.4, the solutions of the compressible Euler equations generally become singular after a finite time. Therefore any global (in time) convergence proof cannot rely on uniform regularity estimates. A reasonable assumption would be that the limiting distribution exists and that the relevant moments converge pointwise. THEOREM 3.3. Let F~(t, x, v) be a sequence of nonnegative solutions of the equation 1
O,G + v . Vx G = - C ( G ) ,
(3.3.9)
540
C. Bardos and B. Nicolaenko
such that as e goes to zero, Fe converges almost everywhere to a nonnegative function F. Moreover, assume that the moments
(Fe),
(vF~),
(w | wg~),
(wlwl2g~),
converge in the sense of distributions to the corresponding moments (F),
(vF),
(v | vF),
(vlvl2F);
the entropy densities and fluxes converge in the sense of distributions according to
lim ( Fe log Fe ) -- ( F log F), 8--+0
lim (vFe log FE) = (vFlogF); e---->0
while the entropy dissipation rates satisfy
lim sup(C ( Fe ) log Fe ) ~< (C (F) log F}. e---->0
Then the limit F(t, x, v) is a Maxwellian distribution, p(t,x) ( F ( t , x , v ) = (2zrO(t~x))3/2 exp
l lv - u(t,x)[ 2)
2
0~x)
'
(3.3.10)
where the functions p, u and 0 solve the compressible Euler equations, Otp + v~ 9 (pu) - O , at(pu) + vx 9(pu | u) + Vx(pO) - 0 ,
Ot(p(~lul2 +23 0 )) + Vx "(PU(21 lu'2
5))
(3.3.11) -o
and satisfy the entropy inequality,
(3.3.12) PROOF. Multiplying (3.3.9) by e(1 + log Fe) and integrating over v gives the entropy relation s(Ot(Fs log Fs) + Vx . (vFs log Fs)) --(C(Fs)log Fs).
(3.3.13)
Letting e go to zero in (3.3.13) and using the convergence assumptions of the theorem regarding the entropic quantities shows that the limiting distribution F must satisfy
0 ~< lim sup(C(Fe)log 8--+0
Fe) <~(C(F)log F}.
(3.3.14)
Navier-Stokes equations and dynamical systems
541
But the entropy dissipation rate of C(F) is non-positive by assumption, so (3.3.14) implies (C (F) log F) -- 0. The characterization of equilibria (3.3.8) then gives that for almost every (t, x) the distribution F is a solution of the equation C(F) = 0 and is a Maxwellian distribution with the form (3.3.10). The system of local conservation laws a , ( & ) + v x . (vF~} = O,
(3.3.15)
a,(vF~) + W . (v | vF~} = O,
at =Iv f~ + gx . v-~lvlR f~
-- O,
is not closed. Each of these equations for the determination of the time derivative of a moment involves the knowledge of a higher order moment. However, if the convergence assumptions of the theorem regarding these moments are used, one can pass to the limit of s going to zero and replace Fs by F, as given by (3.3.10), in these equations. A system of five equations for the five unknowns {p, u l, u2, u3, 0 } is obtained which is the compressible Euler system (3.3.11). Finally, utilizing the entropy dissipation property
(C(Fe) log Fe} ~< 0,
(3.3.16)
Equation (3.3.9) leads to the inequality
Ot(F~ log Fc) + Vx 9(v Fs log Fs) ~< 0.
(3.3.17)
Once again using the convergence hypothesis of the theorem regarding the entropy densities and fluxes along with the form of F given by (3.3.10), this inequality gives the classical entropy inequality (3.3.12). [3
The compressible Navier-Stokes limit. In the derivation of the compressible Euler limit the main ingredient turned out to be the identification of the equilibrium points of the collision operator. Such points, as observed, are the Maxwellian: P
M(p.,,o)- (27r0)3/2 exp
(
llv-ul 2 0
2) "
(3.3.18)
As said above the Navier-Stokes equation is derived as an higher order approximation, either as it is done in the present section as a second order approximation for the solution of the Boltzmann equation or as it will be done in the next section for fluctuations near an absolute Maxwellian. Therefore the properties of the Frechet derivative (involved in higher order expansion) appear in the present section. Denote by M the absolute Maxwellian p = 1, u = 0, 0 = 0) and consider perturbations of the form:
F = M(I + f )
C. Bardos and B. Nicolaenko
542
with f in the Hilbert space L 2 defined by the scalar product
(flg)M -- (fg)m --
f
(3.3.19)
f(v)g(v)M(v)dr.
The linear and quadratic operators L and Q are defined according to the formula:
1
nC(M(1 + f)) -
2
M
C ( M f , M) +
1
C ( M f , My) (3.3.20)
-- L f + O(f, f ) . Direct computation shows that L is given by the expression:
L f -- f f
M,(f; + f'-
fl +
f) I(v, -
v).~old~odv,.
(3.3.21)
It is a self adjoint Fredholm operator in the space L 2 . Its kernel is the 5-dimensional space spanned by the functions {1, Vl, v2, v3, Iol2}. Furthermore it is a non positive operator. The vector or tensor valued functions
A(v)--
1
1 12 - - ~5 ) v, -~lv
B(v) -- v Q v - :-Ivl2I 55
(3.3.22)
are orthogonal to the kernel of L" therefore the equations
L(A') -- A,
L(B') -- B,
(3.3.23)
have unique solutions in Ker(L) • The rotationally invariance of the collision operator implies that these solutions are given by the formula:
A'(v) -- -ot(lvl)A(v)
and
B'(v) = - f l ( l v l ) B ( v )
(3.3.24)
with ot and 13 denoting two positive functions (cf. Chapman and Cowling [45] for their explicit computation) and the formulas" 1
2)
(3.3.25)
define two numbers which in some sense are the "universal" viscosity and heat conductivity. A function H~ (t, x, v) is said to be an approximate solution of order p to the kinetic equation (3.3.1) if 1 Ot He + 1). Vx He -- - C ( H e ) + O ( s P ) , 8
(3.3.26)
Navier-Stokes equations and dynamical systems
543
where O(s p) denotes a term bounded by sp in some convenient norm. An approximate solution of order 2 will be constructed in the form
Hs = Ms(1-t- sgs -+-s2ws),
(3.3.27)
where (Ps, us, 0s) solve the compressible Navier-Stokes equations with dissipation of the order s (denoted CNSEs): (3.3.28)
at p~ + vz 9 (p~ u ~) = O,
(3.3.29) 3 -~ps(Ot + us " Vx)Os + psOsVx " us 1 1 1/2 -=-8-~V, Os/20"(Us) " ff(lts) @ SVx " [K,O s VxOs].
(3.3.30)
The Chapman-Enskog derivation can be formulated according to the following theorem. THEOREM 3.4. Assume that (Ps, us, Os) solve the CNSEs. Then there exist gs and we in Ker(L(p~.u~,O~)) • such that He, given by (3.3.27), is an approximate solution of order 2 to Equation (3.3.1). Moreover, gs is given by the formula 1 _10~1/2
gs = --~Ps
~(IVI)B(V)
' ~(u~)
A ( V ) . VxOs
(3.3.31)
PROOF. In the computation below the subscript s is omitted in the notation of the local Maxwellian M(p~,.~,o~), in the variable V and in the linearized collision operator L(p~,u~,O~). Setting the form (3.3.27) for an approximate solution of order two into (3.3.26) yields the formula
(Ot + v. V x ) M (Or + v. Vx)(Mg) -+-8 M M
(
1
)
-- L(g) + s L(w) + -~ Q(g, g) . A direct derivation of (3.3.18) gives the formulas 1
OuM -- V - - ~ M ,
OoM_(lWl2 3) 1
(3.3.32)
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C. Bardos and B. Nicolaenko
utilizing these shows that the contribution of the first term on the left side of (3.3.32) is given by the formula (Or + v . V x ) M
M
(Or + v . V x ) p
=
p
1
+
2
+V.
(Or + v . Vx)u
4"0 3 ) (Or + v . Vx)O
~lWl - ~
0
"
(3.3.33)
The CNSE~ are used to replace the time derivatives of the functions p, u and 0 by expressions involving only spatial derivatives. This introduces terms of order e, corresponding to the right side of Equations (3.3.29) and (3.3.30), into (3.3.33)" (Or + v . V x ) M ----
VxO + A(V) . ~ + eR,
1B(V) "
(3.3.34)
M
with
R=V.
(vx . [v, cr(u)])
p,,/O
(1
+ 5 Ivl2
l / 2 v , o'(u) " cr(u) + Vx 9 [K, Vx0] -
1
pO
(3.3.35) From (3.3.32) and (3.3.34) it follows that the term of order one with respect to e has to be given by the formula (3.3.31). To complete the proof one must show the existence of a function w that cancels the term of order one in (3.3.32). This amounts to proving the existence of a solution to the equation L(w)-
R +
(Or + v . V x ) ( M g ) M
1 - - Q ( g , g). 2
(3.3.36)
Such a solution exists if and only if the right side of (3.3.36) is orthogonal to the kernel of L and this (details of the computation are omitted (cf. [19])) turns out to be realized when (p~, u~, 0~) are solution of the CNSE~. D REMARK 3.5. Analysis of the above computation shows that the existence of an expansion of the form M e ( l + ege + e2we)
for a solution of the Boltzmann equation with (pe, ue, 0~) solution of "some" compressible Navier-Stokes equation is possible if and only the viscosity v and the thermal diffusivity tc are given by the formulas" v - v, eO~/2,
and
x - x, eO~/2.
(3.3.37)
Navier-Stokes equations and dynamical systems
545
From the formula (3.3.37) one deduces two important facts. First the ratio of the viscosity and the thermal diffusivity is an "absolute" number (independent of the Knudsen number and of the temperature) given by: Pr ~ K,
and therefore defined by the collision operator. Second when the Knudsen number goes to zero the viscosity goes also to zero at the same rate. Therefore it does not seem possible to derive directly a Navier-Stokes equation with a finite Reynolds number from the Boltzmann equation. The way to do it is to consider that the Mach number is also of the order of e and then according to the relation c = Ma/Re one obtains at the limit the incompressible Navier-Stokes equation and this is the object of the next section:
The incompressible Navier-Stokes limit. From formula e = Ma/Re, one deduces that in order to obtain a fluid dynamics regime (corresponding to a vanishing Knudsen number) with a finite Reynolds number, the Mach number must vanish and to realize distributions with a small Mach number it is natural to consider them as perturbations about a given absolute Maxwellian (constant in space and time). By the proper choice of Galilean frame and dimensional units this absolute Maxwellian can be taken to have velocity equal to 0, and density and temperature equal to 1; it will be denoted by M. The initial data Fs (0, x, v) is assumed to be close to M where the order of the distance will be measured with the Knudsen number. Furthermore, if the flow is to be incompressible, the kinetic energy of the flow in the acoustics modes must be smaller than that in the rotational modes. Since the acoustics modes vary on a faster time scale than rotational modes, they may be suppressed by assuming that the initial data is consistent with motion on a slow time scale; this scale separation will also be measured with the Knudsen number. Thus, solutions Fe to the equation 1
(3.3.38)
eOt Fe -q- v . Vx Fc -- - C ( F c ) ,
E
are sought in the form Fe -- M(1 + ege)
(3.3.39)
and one has the THEOREM 3.6. Let F~(t, x, v) be a sequence of nonnegative solutions to the scaled kinetic equation (3.3.38) such that, when it is written according to formula (3.3.39), the sequence g~ converges in the sense of distributions and almost everywhere to a function g as e goes to zero. Furthermore, assume that the moments
(ge)M, (L-'(a(v))|
(vg~)M,
(v C) vg~)M,
(vlvl2g~)M ,
(L-'(a(v))e(g~,g~))M, (L-'(B(v))Q(g~,g~))M
546
C. Bardos and B. Nicolaenko
converge in D' (R + x R3x) to the corresponding moments (g)M, (L-l(a(v))|
(L-l
(v|
(Vg)M, ,
(VIvI2g)M ,
(L-l(a(v))O(g,g))M ,
|
(L-'(B(v))Q(g,g))M.
Then the limiting g has the form g -- p + v " u + (~lv[2 - ~ ) O ,
(3.3.40)
where the velocity u is divergence free and the density and temperature fluctuations, p and O, satisfy the Boussinesq relation
Vx .u = 0 ,
Vx(p + 0) = 0.
(3.3.41)
Moreover, the functions p, u and 0 are weak solutions of the equations
5 (o,o + u. v o) = K, a o
Otu + u 9Vxu + V x p = v, Au,
(3.3.42)
with v,, x, given by (3.3.25) and p denoting the pressure which as usual in the incompressible case is the Lagrange multiplier of the constrain Vx 9u = O.
PROOF OF THEOREM 3.6. Setting (3.3.39) into (3.3.38) gives 1
1
eOtge + v . Vxge = - L ( g e ) + Q(ge, ge) e -2 "
(3.3.43)
Multiplying this by e, letting e go to zero, and using the moment convergence assumption, yields the relation L(g) --0.
(3.3.44)
This implies that g belongs to the kernel of L and thus can be written according to the formula (3.3.40). The derivation of (3.3.41) starts from the equations for conservation of mass and momentum associated with (3.3.43): 60t(ge)M + Vx . (vge)M --0,
(3.3.45)
eOt(vge)m + Vx 9 (v | vge)m = 0 .
(3.3.46)
Letting e go to zero above (understanding the limit to be in the sense of distributions) gives the relations v~ . (vg)M = 0 ,
v~ . (v | vg)M = 0 .
547
Navier-Stokes equations and dynamical systems
When g is replaced by the right side of (3.3.40) these become (3.3.41). The limiting momentum equation is obtained from 1
Ot(vge)m + -V,c 9 (v | vge)M - - 0
(3.3.47)
8
by first separating the flux tensor into its tracefree and diagonal parts:
1 ((
O t ( v g c ) M -+- - V r
"
e
V |
V -
1
5 Iv[
ge M + -eT x
)
5 11)12ge M--0"
(3.3.48)
This is best thought of as being in the form 1
Ot(vge)M + - V ~ " 8
(B(v)gc)M + Vxpe --0,
(3.3.49)
where the pressure is given by p~ -- e - l ( 89 In the same spirit, the limiting temperature equation is obtained by combining the density and energy equations for (3.3.38) as
0,
((l~lvl 2 -
1
+ - V x " (A(v)ge)M - 0 .
ge M
(3.3.50)
8
Utilization of the moment convergence assumption and the limiting form of g given by (3.3.40) provides the evaluation of the distribution limits lim
Ot(vge)M
--
Ot(Vg)M
OtU,
--
8-+0
limOt
~ -+ 0
-2
Iv
-
(3.3.51) -2
ge
M
ivl2
-- Ot
--
5 -~ g
5
M - - -~ Ot O .
As is classical (since the contribution of Leray) in most treatments of the incompressible Navier-Stokes equations, the pressure term that appears on the right side of (3.3.49) will be eliminated upon integrating the equation against a divergence free test function. To complete the proof of the Theorem 3.6, the limits of the moments 8 -1 ( B ( v ) g e ) m in (3.3.49) and ~-l = (A(v)g~)M in (3.3.50) have to be estimated. Start from the identities (recall that L is self-adjoint) (A(v)ge) g - - ( L - ' (A(v))L(g~))M,
and eliminate L(g~) using Equation (3.3.43), 1
e0tg~+v.V~g~--L(g~)+
1
Q(g~ ge)
C. Bardos and B. Nicolaenko
548
The convergence assumptions of the theorem then imply that the limiting moments may be evaluated by 1
s__+0 ~(Alim (v)gs}M -- s--+olim8 V x 9 {V(~) L - 1 (A(v))&}M 1
- lim -{L -1 (A(v))Q(gs, qs))M, s~o 2 1
s~o -(8B(v)gs)M
-
-
(3.3.52)
soolimVx 9{v | L-l (B(v))gs}m
1
- lim - { L -1
~0 2
(B(v))Q(gs, gs)}M"
The limiting form (3.3.40) and the Boussinesq relation (3.3.41) imply that
(
(12
Vx . (v | L-' (A(v))g}M -- L-' (A(v)) | v -~lv - -~ M = --(o~(]vl)A(v)| A(v)} M 9VxO.
.0 (3.3.53)
This expression gives the thermal diffusion term appearing in the second equation of systems (3.3.42). Even more directly, the limiting form (3.3.40) implies
(3.3.54) After applying a divergence, this expression gives the viscous term appearing in the first equation of systems (3.3.42). Next, consider the moments (L -1 (A(v))Q(g, g))M and (L -1 (B(v))Q(g, g))M; these may be evaluated by using the fact that C (F) vanishes for all Maxwellians. The first and second differentials of M(p,,,o) computed at the point (1,0, 1) are
dM _ M(dp _q_v . du + (~lvl 2 --~3) d O),
(3.3.55)
deM _ M ( d p + v.du + (~lv12_ _23)dO)e + M(dep + v . deu + (~lvl 2
3 ) d 2 o ).
(3.3.56)
Comparison of (3.3.55) with the limiting form (3.3.40) shows that a correct choice of parametrization leads to dM -- Mg and d2M -- Mg 2. Twice deriving the formula that states Maxwellians are equilibria for the collision operator then gives
0 = d2C(M) = D2C(M) : (dM v dM) + DC(M). d2M = O 2 C ( m ) ' ( m g v Mg) + DC(M). (mg2).
(3.3.57)
Navier-Stokes equations and dynamical systems
549
Applying the definitions of L and Q, this becomes simply
Q(g,g) - -L(g2).
(3.3.58)
Using relation (3.3.58) and the self-adjointness of L, the desired moments are found to be
1 -1 )) _ -~(L ( A ( v ) ) Q ( g , g M -_-1
-~(L
-1
(B(v))Q(g, g))M =
=
I ( L - I (A(v))L(g2))M 2
I (A(v)g2)M = --5uO ' 2 2
(3.3.59)
2
I(B(v)g2)M - - - B ( u ) .
2
(3.3.60)
Formula (3.3.59) gives the term u. Vx0 while (3.3.60) gives the term u. Vxu. The proof of Theorem 3.6 is now complete. E] REMARK 3.7. Any proof concerning the fluid dynamical limit for a kinetic model will, as a by-product, give an existence proof for the corresponding macroscopic equation. However, up to now no new result has been obtained by this type of method. Uniform regularity estimates would likely be needed for obtaining the limit of the nonlinear term. These estimates, if they exist, must be sharp because, as explained in the previous chapter, the solutions of the compressible nonlinear Euler equations become singular after a finite time and the solutions of the incompressible Euler equation (if not singular) may exhibit serious instabilities. In agreement with these observations and in the absence of boundary layers (full space or periodic domain), the following theorems are proved and were quoted in the previous chapter: (i) Existence and uniqueness of the solution to the compressible, or incompressible Navier-Stokes equation, for a finite time that depends on the size of the initial data, provided this initial data is smooth enough (say in H s with s > 3/2). This time of existence is in both cases independent of ~ and when e goes to zero the solutions converge respectively to the solution of the compressible Euler equations or to the solution of the incompressible Navier-Stokes equation. (ii) Global (in time) existence of a smooth solution to the compressible or the incompressible Navier-Stokes equations provided the initial data is small enough (in a convenient norm) with respect to the viscosity. These points have their counterparts at the level of the Boltzmann equation and at the level of the macroscopic limit of the corresponding solutions: (i) Existence and uniqueness (under stringent regularity assumptions) during a finite time independent of the Knudsen number, was proved by Nishida [140] (cf. also Caflisch [34]). When the Knudsen number goes to zero this solution converges to a local thermodynamics equilibrium solution governed by the compressible Euler equations.
550
C. Bardos and B. Nicolaenko
(ii) Existence of a global in time smooth solution of the Boltzmann equation provided the fluctuation (with respect to an absolute Maxwellian) of the initial data is small enough compared to the inverse of the Knudsen number. The above consideration can be adapted to the rescaled equations
1 eOtFe + v. VxFe = -Ce(Fe)
(3.3.61)
with initial data of the form Fe -- M(1 + erge).
(3.3.62)
It is easy to adapt the result of Nishida and to prove that with an initial data which is a smooth fluctuation of an absolute Maxwellian there will exist a finite time say, T = T* such that on the interval (0, T*) the statement of Theorem 3.6 (which corresponds to r = 1 in (3.3.62)) can be rigorously proven, furthermore in this case if g~ is small enough at t - 0 the above results holds for T* = cx~ (cf. [25]). Similarly for r > 1 it is possible to show (cf. Bellouquid [27]) that the solution of the Boltzmann equation will be smooth for all time and will converge to the solution of the Stokes equation. Using a method with many similarities to Leray's, DiPerna and Lions [63] have proved the global existence of a weak solution to a class of normalized Boltzmann equations, their so-called renormalized solution. This solution exists without assumptions concerning the size of the initial data with respect to the Knudsen number. In this case it was natural to conjecture that the DiPerna-Lions renormalized solutions of the Boltzmann equation converge (for all time and with no restriction on the size of the initial data) to a Leray solution of the incompressible Navier-Stokes equations. This statement was the object of a program initiated in [21] and [22]. This program has been recently fully completed by the contribution of Golse and Saint Raymond [92].
4. Turbulence and turbulence modelling 4.1. Introduction and the example of the k-e model Phenomena described by the Navier-Stokes equation, may become, in particular for very large Reynolds numbers extremely complicated (as said in the introduction the world "turbulence" which is never completely defined is used in these situations). In the mean time the persistence of the divergence free condition and the fact that the energy remains bounded implies that it is the vorticity which becomes in some places very important both in size and in variation of its direction. This gives to the trajectories of the fluid some important averaging effects which correspond in the case of finite dimensional system to a complex system (notes, written by Leonardo da Vinci, quoted by several authors seem to indicate that he had already an intuition of this complexity).
Navier-Stokes equations and dynamical systems
551
Therefore a first natural approach is the assumption that what we observe can be described by a statistical turbulence. Namely it is assumed that the velocity of the fluid is a random variable given by the formula:
(4.1.1)
u (x, t, co) -- U (x, t) + fi (x, t, co)
with fi (x, t, o~) denoting a random variable of mean value zero and that it is only the knowledge of the averaged value that will be important for applications. Equation for this averaged value would be some super Navier-Stokes equation and would play for the NavierStokes equation the role played by the Navier-Stokes equation itself for the Boltzmann equation or by the Boltzmann equation for the equation of a system of N molecules. A theoretical reason for the study of such equation would be the idea that serious mathematical progress is obtained first in coarser descriptions, because this level contains as a limit the more detailed one; for instance no progress on the proof of the regularity of the 3d Navier-Stokes equations was ever derived from the mathematical analysis of the Boltzmann equation and at variance the results on the Boltzmann equation can be viewed as adaptation (even if some of them are highly non trivial) of known results on the NavierStokes equations. Therefore one may think that progress in the understanding of turbulence may be a compulsory step in solving the classical open problems for the 3d Navier-Stokes equations like the existence of smooth solution in the large. Inserting the right hand side of (4.4.1) in the Navier and denoting by (.) the average with respect to the random variable co gives the equation: OtU + U . V,cU - v A U + U V x U + V.r(fi |
= -V,: P,
(4.1.2)
which contains a "closure" problem because (fi | fi) which is called the R e y n o l d s stress t e n s o r is not expressed in term of U. However by a change in the pressure the Reynolds stress tensor can be always chosen to be traceless and therefore if one assumes (i) That this stress tensor depends only on Vx U, and (ii) That the mapping VxU ~ (fi | fi) is invariant under Galilean transformations (isotropy assumption) one finds out that this Reynolds tensor is indeed proportional to VxU +t VxU, i.e.,
|
+' VxU).
(4.1.3)
The scalar a ( x , t) depends on the time and the position and is hopefully positive. This correspond to the introduction of a turbulent viscosity. In spite of the absence of complete rigorous derivation some rules are used for practical computations. The most common one being probably the k - e model introduced by Landauer and Spalding in 1972 [117] and widely used in numerical simulations. The basic idea is that the turbulent diffusion depends only on the fluctuation of energy (at variance with other part of the subject and other section of this monograph here the turbulent energy is denoted k and not e) and on the fluctuation of enstrophy
1
2)
u
2
C. Bardos and B. Nicolaenko
552
then a dimension analysis gives for vr(x, t) an expression of the form k2 V T ~ Cv m . 8
To determine the functions k and s one introduces an equation for t7 by subtracting Equation (4.1.2) from the basic Navier-Stokes equation with solution U + fi this gives the equations:
0,~ + ~. v x u + (u + ~ ) v ~ - ~ / ~ - WR - -Vx~,
(4.1.4)
and for co = V x u
o,~ + ~. vx(v • u) + (u + ~ ) v ~ - (v x v + ~)v~
-~v~v
- vA,~--V
x V~ R,
(4.1.5)
Equation (4.1.4) is multiplied by t7 and Equation (4.1.5) is multiplied by &. Basic assumptions (with up to now no rigorous justifications) are made concerning the approximation of the terms
by terms of the form
0, (O) + uvx (0) - vr zx(0) according to the convection of a passive scalar by a random field. Eventually an ergodicity hypothesis is used to replace random averages by spatial averages when needed and the following system is obtained.
OtU + U V x U + VxP - v A x U - Vx cv - - ( V x U + t V x u ) Otk-k-UVxk - c v k2s2 IVxU
o,~+uv~
+ (VxU)tl2
-
Vx
.
-- 0,
[ c~--~-Vxk k2 ] + s = O,
c,k ~ - I v ~ u + ( v ~ u ) ' I2 - v ~ . [ ck32- ~V x e
]
+c2
7_ ~2
-
(4.1.6)
o.
with Ci denoting several positive constants which are determined either by experiment or by phenomenological considerations. It is known that there exist some cases where the above derivation is not valid (in particular near the walls). Even if the most convenient hypothesis are assumed, many important gaps remains in the proof of the validity of the k-e model. (i) It is assumed that the velocity of the fluid is a random variable u (x, t, co). This seems reasonable keeping in mind a generalization of the Birkhoff ergodic theorem to the NavierStokes flow. However this ergodic theorem (to be stated) requires the existence of a nontrivial invariant probabilistic measure and the definition of such a measure is (for many
Navier-Stokes equations and dynamical systems
553
reasons) a widely open problem, some partial results having been obtained by Foias [72, 73] Foias and Prodi [81] and by Fursikov, Vishik et al. (cf. [163-165] and [166]). Some of these results will appear in the next chapter. (ii) There is no universal parameter like the Knudsen number or the Mach number, and since macroscopic phenomena are involved the Reynolds number which would be the best candidate to measure the relaxation to turbulence is a local quantity. In fact it is the fluid self interaction which is responsible of the relaxation to "turbulence". (iii) There no evident rigorous formulation that would play the role of the thermodynamical equilibrium and no trend to relaxation like the decay of entropy at the level of the Boltzman equation.
4.2. Wigner transform and defect measures for the Reynolds tensor Since there is no complete mathematical theory that even in some particular cases would produce an expression for the turbulent Reynolds tensor, several ideas may be used; many of them do have in common the introduction of the two points spatial or temporal correlation function: ( (
r
)
(
r
fi x + -~, t | fi x - -~, t
)) or
such quantities are the object of many experimental measurements which do involve Fourier transform, which for instance in the spatial configuration is:
(R(x,k,t))--
~
( (
r
3e - ' k gt x + - ~ , t
)
|
(
r
x--~,t
))
dr.
(4.2.1)
With the inverse Fourier transform one deduces the relation:
(fi(x,t)|
~
3('R(x,k,t))dk.
(4.2.2)
In fact the two above formulas turn out to be the Wigner transform and its reciprocal. Along this line it is important to keep in mind the fact that the Wigner transform provides (for the energy) some type of local high frequency expansion. The tensor valued function R(x, k, t), or its average plays for the Navier-Stokes equation the role assumed by the thermal equilibrium (Maxwellian for instance) at other level of the hierarchy. Since it involves only the fluctuation one may assume that it is invariant under Galilean transformation and this would lead for instance in 3 dimensions, to the formula: A
A
R (x k, t) = '
E(Ikl x t ) ( k ' 4Jrl I~
I
k| Ikl 2
where E (]k], x, t) is a scalar function called the energy spectra of turbulence.
(4.2.3)
C. Bardos and B. Nicolaenko
554
Of course with the assumptions that R(x, k, t) is invariant under Galilean transformations and depends only on
1
V
and on the tensor (7x U -t- t Vx U) one recovers the formula: k2
R(x, t) ~ - - ( 7 x U + tVxU). 6
(4.2.4)
In spite of its long history, the statistical approach does not seem to be compulsory to introduce turbulent effect and it is important to observe that all the issues raised for statistic family of solutions do have their counterpart when one considers the family of defect measure of a sequence Un of deterministic solutions of Euler or Navier-Stokes equations which are uniformly bounded in energy sup t'l Iun (X, t>0, n Js?
t)']2 dx ~< C < cx~
(4.2.5)
and which do not uniformly satisfy other "regularity" estimates. More precisely only the energy estimate
5
]Un(Xt)
I d x + v n /o'f
IlVxu,,r
l f ~ lu(x, O) I2 d x 2 dx ds <~ -~ (4.2.6)
remains valid and it is assumed that the viscosity 1)n is either zero or goes to zero. In this case, modulo the extraction of a subsequence, Un converges "weakly" to a function U (x, t) which satisfies also the estimate (4.2.5). However due to the likely lack of compactness it may happen that lim Un(X, t) @ Un(X, t) 7~ U(x, t) @ U(x, t)
(4.2.7)
n-----~oo
and the difference
R(x, t) = lim un(x, t) | Un(X, t) - U(x, t) | U(x, t) n-----~oo
=
lim (Un(X, t) - U(x, t)) | (Un(X, t) - U(x, t))
(4.2.8)
n-----~oo
is a measure valued positive tensor which is zero only in case of strong convergence. It is called the defect measure and has been introduced for related purpose by several authors (cf. for instance [90,159]). The limit U (x, t) is the solution of the "turbulent" equation:
OtU + U. VxU + VxR + VxP = 0 , Vx. U = 0 with R being like in the random case the Reynolds stress tensor.
(4.2.9)
Navier-Stokes equations and dynamical systems
555
This tensor may be present: When u,, (x, t) is a sequence of Leray solutions of the Navier-Stokes equations in R 3 with the same given regular initial data and a sequence of viscosities vn which goes to zero with n. Because in this situation the global existence of a smooth solutions of the Euler equations remains an open problem. It is almost surely present in the following situations: "(i)" When u,, (x, t) is a sequence of solutions of the Euler equations (or of the NavierStokes equations with viscosity going to zero) with initial data Un(0, t) uniformly bounded in L2 but not in a more regular space (for instance when the initial data exhibit large oscillations). "(ii)" When both large time behavior and zero viscosity limit are simultaneously considered. "(iii)" Even for finite time, for the solutions un(x, t) of the Navier-Stokes equations in a bounded domain C2 when the viscosity v,, goes to zero and when a viscous boundary condition
u,,(x, t) -- 0
for x 6 0C2
(4.2.10)
is prescribed. It this situation a boundary layer appears near the boundary but due to the non linearity of the problem this boundary layer (at variance with what happens for linear problems) may propagate inside the domain. It has been recently proven by Asano and Caflisch and Sammartino (cf. [37]) that such a phenomena is not present, but only for small time and analytic initial data. Assuming that in all of the above cases the weak limit U (x, t) of the sequence Un(x, t) is a smooth function, the following "conjectures", which are the deterministic counterpart of the "folklore" of statistical turbulence, should be studied. One introduces the sequence of functions /)n -- un - U
which converges weakly to zero and which plays the role of the fluctuation and its Wigner transform
A r) Rn(x,k,t) -- fR 3e -~k ( ( (fin x+-~,t A
R(x, k, t) -
| ( ( (fin x - - ~r,)t )
(4.2.11)
A
lim
R,,(x, k, t)
I1---~ r
which is the analogous of (4.2.1). The local turbulent energy and turbulent enstrophy could be defined as e---
1
lim l u , , - U I 2, 2 ,,~ 1 lim
2
11 ----~ 0 0 ,
v[9' • (un-
dr,
U)[ 2
C. Bardos and B. Nicolaenko
556
It this context exactly as in the case of the random solution one may assume (and may in the future in some case prove) a Galilean invariance hypothesis which gives:
A E(Ikl,x,t)( e(x, k, t) -- --~(~1~ I
kQk) ikl2
(4.2.12)
leading as above to the introduction of the turbulence spectrum. Furthermore the Galilean invariance implies that the Reynolds tensor itself can be as above written as
R{x, t) = -vT{x, t)(v. u + 'WU). The next step in this analysis should be to prove that the scalar vr (x, t) is non negative. This does not result from the positivity of the defect measure R itself. Explicit model of this weak convergence can be constructed for the dispersive limit of the KdV equation or for the non linear Schr6dinger equation and show that the appearance of analogous phenomena (positive diffusion) are possible but not systematic.
4.3. The Kolmogorov-Kraichnan theory Another approach to the closure problem is the direct analysis of the turbulence spectra E (x, t, k) introduced above under the Galilean invariance hypothesis. This program was initiated by Kolmogorov in 1941 and stimulated many further researches. With the following assumption: There exists a region (called the "inertial range" 0 ~< kl ~< Ikl ~< kl) where E(lkl,x, t) depends only on Ikl and on
d ]z~(x, t)l 2)
e=~-7{
a dimensional analysis gives the formula
E(k) = C(x, t)e2/3[k1-5/3
(4.3.1)
with C (x, t) an adimensional number. This is the famous Kolmogorov law. It is independent of the equation of motion; no mechanical explanation for its validity in three dimension has yet been offered here. It is well verified both by physical and numerical experiments (quoted from Chorin [47, p. 52]) and furthermore it will lead to an analysis of the degree of freedom of the fluid. In fact the intrinsic nature of the turbulent spectra should be present in the case where the action of the macroscopic part of the fluid U on the stress tensor (fi | fi) is replaced by the action of an external force f on random fluctuations of mean value zero. It turns out that in this configuration, with a fluid evolving in a D = 2 or D = 3 periodic domain the analysis is both relevant and simpler. For formal and rigorous results it will use the dynamical aspect of the Navier-Stokes flow and this is the object of the next section.
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5. Invariant measures, attractors, and evaluation of the number of degree of freedom of the flow
5.1. Introduction and formal derivations In this section the solutions of the Navier-Stokes equations in a bounded domain with a time independent forcing term are considered. This forcing term can be either distributed in the domain or located on the boundary. However, for sake of simplicity, only the case of an internal force acting on a fluid defined in a periodic domain Y2 -- [0, L] D is described. Therefore, the equations are
Otu + u . Vxu - v A u + Vxp -- f,
Vr " u - O ,
u(x, O) - uo.
(5.1.1)
Using the Galilean invariance one can assume without loss of generality that
fs2 f (x) dx - O,
and
fs2 u(x) dx - O"
(5.1.2)
The phase space H is defined as the L2-completion of smooth periodic divergence free functions satisfying (5.1.2) and P denotes the orthogonal projection of (L2([0, L])D) D) onto H (Leray Projection). The following standard notations are used:
Au -- - A u , B(u, v ) - P[(u. Vxv)], (u, v) -
lfo
--s
,LI, t u ( x ) . v ( x ) d x ,
(5.1.3) ((u,v))=
lul-
~ 1 f 0 ,L~; V~u(x) ~ " V~v(x) dx,
(u,u)l/2
[lull-((u 1)))1/2
The operator A is selfadjoint positive and the domain of A 1/2 coincides with the space V - H A HI (~2) d, Ilull2 -[A1/2ul 2. The quantities llu[2 and Ilu]l2 represent the kinetic energy and the enstrophy per unit mass of the flow described by u. Eventually the following identities are recalled:
(B(u, v), v) -- O,
if D - - 2 , 3 , (5.1.4)
(B(u, u), av) + (B(u, v) + B(v, u), au) -- 0
if D - - 2 .
Ignoring in the present section the difficulties related to our incomplete knowledge of the regularity and uniqueness of the solutions of the Navier-Stokes equation and making the
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558
convenient regularity hypothesis, one describes the solutions of (5.1.1) with the introduction of a non linear semiflow
u(t, x) = S(t)uo(x)
(5.1.5)
and defines the global attractor ,A as follows: the global attractor A for the semiflow {S(t)}t>~o is a compact set in the space H, ~A c H, such that
S(t)A = A
gt ) 0
(5.1.6)
and A attracts all bounded sets of H, i.e., for all 13 C H bounded, for all e > O, there exists 7'1 = Tl (e, 13) such that, for t ~ T1 (e, B), S(t)B is included in an e-neighborhood of A. When the viscosity is large enough (small Reynolds numbers) A is reduced to the unique solution of the time independent equation u. V~u-- vAu + V ~ p = f,
(5.1.7)
V~.u=0.
However, as is the case in finite dimensional models (like the Lorentz attractor [128]), which is constructed as the simplest Galerkin approximation of the Boussinesq equation), the structure of the complexity of A increases with the Reynolds number. The first steps in this process are described by adaptation to the Navier-Stokes equation of the standard bifurcation theory and could be found for instance (with other references) in Chossat and Iooss [46]. Then the trend toward the complexity of the attractor should be understood by the introduction of a cascade of bifurcations. However rigorous construction of bifurcations after the second order seem to be out of the scope of our present knowledge and are in any case very different from the analysis in a turbulent regime which is the goal of the present section. Observe that the complexity depends on the viscosity v, the size of the box L and the magnitude of the driving force f ; therefore, it should be described in terms of a dimensional number depending on these three quantities. Such a number is called the generalized Grashoff number. In dimension 2 it is given by the formula:
G-
- -j
(5.1.8)
I/(x)l 2dx
and in dimension 3 it is convenient to replace the above definition by the formula:
a-
L2IA_I/2f[
v2
= Lv22 ( f s 2
( - - A - I f ( x ) 9f ( x ) ) d x
)1/2 .
(5.1.9)
For large Grashoff number the fluid should become ergodic and define an intrinsic probability measure. More precisely, one assumes the following ergodicity hypothesis Herg:
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559
There exists a unique probability measure # on the phase space H invariant under the action of the Navier-Stokes semiflow such that, for almost any initial data u0 ~ H and any functional @ (u) representing some physical quantity associated with the fluid flow u,
{'(UO))-- T--~o~li(1 m fo T O(S(t)uo)dt)
-- fH O(uo)d#(uo).
(5.1.10)
Using the Fourier series representation:
u(x)-- Z kcZ
ak(u)e2Jr/Lkx
(5.1.11)
D
one can define, for any 0 < tel < K2, UKI.z2
UKI.K2(X) --
Z
ak(u)e2:r/Lkx
(5.1.12)
KI~
and assume Herg-is (ergodicity and isotropy hypothesis) the existence of a positive function E(x)(>~ 0) such that one has:
lim- lf0T T--~oo T
]S(t)uoK~K~]2dt- fH luoK~,K~I2d#(uo)- fK 'c2 E ( x ) d x . ""
"
t
(5.1.13)
As in the previous chapter, the function E (x) is called the energy spectrum of the turbulent flow produced by f. It gives an intrinsic (if not rigorous, see below) definition of an object which adapts to the present context the definition given in (4.2.3). The fact that Fourier series representation is used instead of Fourier transform creates no problem, and the fact that the function E can be defined in term of the modulus of the wave number corresponds to the isotropy hypothesis made in Section 4.2. Therefore, the question raised there can be addressed in the present context and leads to formal and in some cases rigorous results. Observe that one has immediately:
lim
T--~ o o
7'fowIs(t)uoK, ,K2II dt- L
IIu0K~,K2II2 d#(uo) - f l 2 tc2E (x) dx
(5.].14)
and
1LT
lim -T~ee T
]AS(t)uoKI,K2 12dt -
IA UOKI,K2]2 d#(uo) --
f
x4E(x)dK. I
(5.1.151
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C. Bardos and B. Nicolaenko
5.2. Kolmogorov and Kraichnan inertial range This section is devoted to the construction of the inertial range. The argument is inspired by the classical ideas of Kolmogorov and Kraichnan and we follow the exposition done in [74]. It is assumed that f = f0,K0 where x0 is of the order of 2rc/L (the lowest wave number). For tci and x2 given, the following notations are used: U< "- b/0,Kl ,
U -- b/KI,K2,
U> = uK2,~.
(5.2.1)
First the case D -- 2 is considered; therefore, with (5.1.4) one deduces from the energy balance equation (for tel > x0) the relation:
ld
---IIvll 2 - -vlAvl 2 - (B(v<, v<), Av) 2dt + (B(v, v) + (B(v>, v) + B(v, v>)), Av<)
+ (B(v, v) + (B(v<, v)+ B(v, v<)), Av>) -(B(v>,v>),Av).
(5.2.2)
Taking the average in the sense of (5.1.10) one obtains:
0 -- -v(IAv] 2 ) - (((B(v<, v<), Av)) + (B(v + v>,v + v>), Av<)) + ((B(v + v<, v + v<), Av>) -((B(v>, v>), A(v< + v))).
(5.2.3)
Now assume that at these wave numbers x and 2x the enstrophy is in average carried only from low wave numbers to high wave numbers (Kraichnan's cascading scenario), then one has
((B(v+v>,v+v>),Av<))~O
and
((B(v>,v>),A(v< +v)))~'O (5.2.4)
so that
v(lAv[ 2) "~ - ( ( ( B ( v < , v<), Av))) + ((B(v + v<, v + v<), Av>)).
(5.2.5)
Thus, as long as v(lAol2) - v fx to ~4 E(~)d~ "~ uxSE(x)
(5.2.6)
is small compared to the two terms of the right hand side of (5.2.5), one has (((B(v<, v<), A v ) ) ) ~ ((B(v + v<, v + v<), Av>)).
(5.2.7)
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561
The left-hand side represents the mean enstrophy/mass passed per unit time from the component with wave number less than i( to wave numbers living in [2x, 4x). Let rl denote the constant dissipation of enstrophy resulting from Kraichnan scenario. Let also tel be the smallest wave number from which the Kraichnan scenario is valid. Then writing (5.2.6) for x = 2 jK1 (j = 1,2 . . . . ) and summing up, one obtains:
v(lAuK,.oQI 2) ~ --((B(uo.K,, U0,KI), AUKI.2K,))~ q
(5.2.8)
so with the assumptions that tel --~ x0 --~ 27r/L, L >> 1,
v ( l A u z , , ~ l 2) ~
v
~4E(~) d~ - v(IAul2).
~4E(~) d~ ~ v 2~
1
(5.2.9)
This last quantity is the dissipation (due to viscosity) of the enstrophy/mass per unit time of the whole fluid flow. As long as vKSE(x) << q,
(5.2.10)
the component with wave number in [x, 2x) is just transferring enstrophy with the constant rate _~ 7; from lower wave numbers to higher wave numbers. Following Kraichnan [109] and Frisch et al. [89] we inject into the above formulas a phenomenological description of turbulence. To start, observe that the wave number tc has the physical dimension of (length)-1 and the component with wave number x is considered to represent eddies of linear size about ~-. I Thus, as a function of x the component ux.2K is thought to represent the system of eddies of linear size 6 (1/2K, 1/x]. The transfer of enstrophy is considered to be produced by the breaking of the eddies into eddies of linear size ~< 1/2x. This breaking is assumed to occur after the eddy travels a distance comparable to its linear size. Since the energy/mass of the eddies with linear size E (1/2x, 1/K] is in average about
f
2K E ( ~ ) d ~ "~ x E ( x ) ,
(5.2.11)
the average velocity of those eddies is about VK "~ (xE(x)) 1/2.
(5.2.12)
Therefore the time necessary for the eddies to travel their linear size is about
tK ~ ( 1 / K ) / V z -- 1/(KVK)~
1/(K3/2E(K)l/2).
(5.2.13)
On the other hand, the enstrophy/mass of the eddies with linear size E (1/2x, 1/K] is
~2E(~)d~z3E(x).
fx K
(5.2.14)
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C. Bardos and B. Nicolaenko
According to the breaking mechanism the mean dissipation of the enstrophy/mass per unit of time should thus be K 3 E(K) _ K3 (K E(tr tx
3/2
(5.2.15)
which gives
r/2/3 E(x) ~ x 3 .
(5.2.16)
According to the previous arguments, one expects (5.2.16) as long has i( > x0 and (5.2.10) hold. Using (5.2.16) one finds that (5.2.10) is equivalent to x 2 << rll/3/v, that is" (5.2.17) Equation (5.2.17) defines the Kraichnan dissipation wave number [109]. For larger wave numbers the viscosity forces become dominant. In the three-dimensional case, D = 3, the role played by the enstropy in the preceding argument is taken over by the energy. Consequently, one starts with f2K ~2E(~) d~ ~ - ( ( B ( v < , v<), v)}+((B(v< + v, v< + v), vv)}
(5.2.18)
and then writes"
{(8~<. ~<). ~ ) } ~ - { ( 8 ~ < + ~. ~< + ~). ~))-___~- ~{tl. ii~}
(5.2.19)
provided that
vx3E(x) << e.
(5.2.20)
Here e represents the mean dissipation of energy/mass per unit of time. The Kraichnan mechanism now leads to the estimate KE(K) tx
(KE(K))3/2K,
(5.2.21)
that is, $2/3
E(x) ~ x5/3
(5.2.22)
for 2 3
VK4/36 - << 6,
(5.2.23)
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563
that is,
6 ) 1/4 x<<x~-
7
"
(5.2.24)
This number tee is called the Kolmogorov wave number. The spectra given by (5.2.23) and (5.2.16) are, respectively, the Kolmogorov spectrum for turbulence and the Kraichnan
spectrum for 2D turbulence. The wave number where the mechanism described above holds (in or 2D is the inertial range of turbulence). The empirical evidence for the existence of the Kolmogorov inertial range of turbulence is much stronger than that for the existence of the Kraichnan inertial range of turbulence; this may be due to the fact that we have at our disposal more experiments in D = 3 than in D = 2. Furthermore both the phenomenological theory and the rigorous mathematical analysis described below indicate for D = 2 the existence of a logarithmic correction.
5.3. Kolmogorov-Kraichnan waves numbers and asymptotic degrees of freedom In almost all cases the use of the primitive Navier-Stokes equation for the computation of a flow direct Navier-Stokes Simulation DNS introduces a discretization which involves a finite number of degrees of freedom. Evaluation of the order of magnitude of this number is the first step of the computation. It turns out that there are several approaches for this evaluation, some based on the heuristic argument as a continuation of the above discussion and others based on some more mathematically tractable objects like the notion of attractors. At the present, to the best of the knowledge of the writers, no formal mathematical derivation of the relation between the different approaches exists; derivations should come from a better understanding of the ergodic aspect of the theory. However surprisingly (or not surprisingly?) the different approaches lead to very similar estimates. The heuristic approach is discussed below as a continuation of the previous section, and the more mathematical approach will be one of the main the objects of Section 5.4. The mean dissipation of energy/mass per unit of time which appears in the evaluation of the Kolmogorov scaling law and of the Kolmogorov dissipation wave number can be evaluated with the following heuristic argument: If
ll0
e - -~
E(K)dx
(5.3.1)
is the average of the energy/mass in a turbulent fluid flow with an average dissipation rate
e
-
-
v
f0 ~
K2E(K) dK,
(5.3.2)
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C. Bardos and B. Nicolaenko
then te -- e / e should represent the characteristic time for the dissipation of energy and the characteristic mean velocity should be U -- x / ~ . The corresponding length 1 -- U te can be viewed as the average distance travelled by the turbulent eddies until they dissipate. So U2
U3
e. . . . te
l
.
(5.3.3)
This is the Kolmogorov estimate for energy dissipation in a turbulent fluid flow. Since U and L are the characteristic velocity and length for the flow one introduces the Reynolds number R e --
UL
(5.3.4)
and obtains with (5.2.24), (5.3.3) and (5.3.4) the following formula for the Kolmogorov wave number: (5.3.5)
LKe "~ (Re) 3/4.
Above it has been observed and used that in D -- 2 the enstrophy dissipation v - (IAul e)
(5.3.6)
has to be considered instead of the energy dissipation. However the same analysis leads in this case to the same formula: (5.3.7)
Ltco "-" (Re) 3/4.
The dissipation length is therefore given by lo = le -
1 KO
1 Ke
for D -- 2, (5.3.8)
for D = 3
(cf. Foias [74]). Since structures of size less than l0 (respectively le) correspond to wave numbers which are in the dissipative range, they are rapidly annihilated by viscous effects and therefore are of no dynamical consequence. On the other hand, any eddy of size l, 7 (respectively l~) will be tracked at some grid point. One expects that the degrees of freedom of a 2D, respectively 3 D, flow should be at most about for D - - 2, (5.3.9) -
(c~)
3
for D - - 3
or with (5.3.5) and (5.3.7) (Re) 3/2 for D -- 2 and (Re) 9/4 for D -- 3.
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565
Assuming that the non linear semiflow has a compact global attractor A one could use the fractal dimension of this attractor as an alternate definition of the number of freedom of the turbulent flow. Recall that the fractal dimension of a compact subset A of a Hilbert space H is defined by the formula: lim sup dM(.A) - 8--+0+
log N~ (A) log 1
(5.3.10)
where, for 8 > 0, N~ is the smallest number of balls of radii equal to 8 needed to cover A. The fractal dimension can be oo even if its Hausdorff dimension is 0 (cf. [29]). Moreover (cf. [67,80,29]), if din(A) < oo there is a dense set of orthogonal projections P (Marie's projection) in H of rank ~< 2dM(A) + 1 with a H61der continuous pseudo inverse P - I : A - - + H,
(5.3.11)
P o p - 1 = IA"
Therefore, the fractal dimension is a better indicator than the Hausdorff dimension of the number of parameters necessary to describe a set as well as the dynamics it may carry. This observation is particularly important for the exponential attractor which is introduced in Section 5.5. Furthermore it is appropriate to mention that exponential attractors are probably more relevant than the global attractor A itself. These are outgrowths of ,A, still with fractal dimension but attracting all solutions at an exponential rate. Moreover, the estimates for their fractal dimension are as sharp as the one for dM(A). The notion of the number of determining nodes should be halfway between the concept of degrees of freedom according to Landau and Lipschitz and the dimension of the attractor. The points of a fixed finite set ~ in the domain of the fluid are called determining nodes whenever, for any two solutions u, v of the Navier-Stokes equations, the convergence on /7 implies the global convergence of these solutions, i.e.:
lim(u(t,a)-v(t,a))--0(inR t---~O0
D) va6f"
=:~ l i m l u ( t , . ) - - v ( t , . ) ] H = 0 . t---~OG
(5.3.12) Eventually it is worth mentioning that a theorem of Takens [ 158] asserts that generically one node should suffice, but it is not known if the Navier-Stokes is generic in the sense of Takens. For dimension of the attractor and for the number of determining modes, rigorous results are available. They are not as precise for determining modes as for the dimension of the attractor (cf. [98,50]) but in any case they are more tractable than the heuristic estimates of Kolmogorov and Kraichnan. Finite fractal dimension for the attractor raises the following deeper questions, which are partially answered in Section 5.5: (i) Can we imbed the attractor in a smooth finite-dimensional manifold; (ii) Are the dynamics on the attractor equivalent to the dynamics of a finite differential dynamical system (also called "inertial dynamical system") on such a finitedimensional manifold.
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5.4. Mathematical tools for rigorous results In this section the dynamical system point of view is systematically used to produce some estimates on the number of degrees of freedom. As said above the approach differs from the historical approach of Kolmogorov and Kraichnan but the results are in extremely good agreement. At first glance a complete justification of the above analysis should include at least (i) The use of regular semi flow. (ii) The existence and uniqueness of the probability measure # on the phase flow satisfying the hypothesis Herg (cf. formula (5.1.10)). However: (i) As discussed at length in previous sections the existence of a "better" solution than the Leray weak solution is for D = 3 still an open problem. (ii) Furthermore, nobody has ever come close in proving even for D = 2 the existence of a probability measure/z satisfying the hypothesis Herg and even more the hypothesis Herg-is. To partly overcome these difficulties one proceeds as follows: (1) As usual, weak solutions of (5.1.1) are considered for D -- 3. However, classical energy estimates show the existence of an absorbing ball B bounded in H such that for any solution u(t) of (5.1.1) there exist a to which depends on the solution such that:
t > to =~ u(t) E B.
(5.4.1)
All these solutions always converge in the weak topology of H to a maximal weakly compact set .At C B with the property that if u0 e .A then there exists a (weak) solution u defined for - o e < t < cx~ bounded in H such that u(0) = u0. By definition this set is the global attractor of the weak semiflow associated to Equation (5.1.1). It contains a dense open (for the weak topology of H ) subset Areg with the property that for u0 e .Areg the solution u with this initial condition (at t = 0) is unique and analytic on a time interval 0 e (tl, t2). In the case D = 2 B is a bounded subset of V and therefore is compact in H, .,4 coincides with .Areg and is the usual global attractor of the dissipative evolution equation. For precise results in dimension 3 the extra hypothesis .,4 -- Areg or equivalently B C V will be necessary. (2) "Weak stationary statistical solutions of the Navier-Stokes equation" are defined as probability measures # on H which satisfy the following relations:
f
llull2#(du) < ~ ,
f
[v((..
#(.A)-
1
(5.4.2)
and +
- i.
= o
(5.4.3)
for all test functionals q~ : H ~ IR which are Gateaux differentiable in H at any point u 6 V with derivative ~/,' bounded on subsets of V. In dimension 2 stationary statistical solutions
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coincide with probability (Borel) measures which are invariant for the non linear semiflow, i.e.,
f
f
Vt>0.
Rigorous mathematical treatment of this notion appeared in Foias [74] and were developed by several authors; in particular it was shown first by Foias [72,73] that such measures are solutions of the Hopf equation. The latter was also studied by Fursikov-Vishik [ 163-165] and Fursikov-Ehmanuilov [86]. Finally one observes that the support of the measure is the global attractor A; for a comprehensive survey of statistical solutions and their connection with turbulence see also [75]. (3) Eventually the notion of a generalized limit is used and denoted L i m T ~ . With this notion the following result is a soft hybrid of both Birkhoff ergodic theorem and Krylov-Bogoliubov theory: THEOREM 5.1. For any (weak) solution u defined on (0, oo) with initial data at t - O , there exists a stationary statistical solution # such that Lim --
'f0
cb(u(t)] dt --
,
T--+~ T
,
f
~(u) d#.
(5.4.4)
With this statement one can easily and rigorously prove still with energy estimates the following THEOREM 5.2. In dimension D -- 2 the set Range(r/) --
{ 'fo vlAS(t)uol2dr, uo ~ H} Lim -:r~e~ T
(5.4.5)
coincides with the set
f ~lA(u)12~(du) where lZ runs over all probability (Borel) measures invariant f o r the semiflow S(t), t > 0.
Since the driving force f is assumed to be localized to the low frequency modes, the only quantities which characterize the flow are the size of the "box" L, the viscosity v and the L 2 norm of f , and this leads to the introduction of a dimensional number called the generalized Grashoffnumbers constructed with these quantities:
G -
132
=
132
L2lA-l/efl G, =
v2
,L]2
f(x). f(x)dx
2(f 0
= V2
for D - - 2 ,
,L]z(--A)-I f (x) " f (x) d x
),,2
(5.4.6) for D - - 3
C. Bardos and B. Nicolaenko
568 and rigorous estimates on
-
T
-
T -
lfo
~lA(S(t)uo)., v II(S(t)uo)
K,~
12dt, [I2 dt
'
forD--2,
and
for D = 3
leads (cf. [76]) to almost rigorous evaluation of Kraichnan and Kolmogorov wave numbers in term of the Grashoff number:
CoG1/6L -1 <~ too ~ C1G1/3L -1D = 2, (5.4.7)
CoGI,/aL -1 <~ tCrl<~C1GI,/ZL -1D = 3. In (5.4.7) Co and C1 denote universal constants; for D = 2 the ergodicity hypothesis Herg is assumed and in dimension D -- 3 a regularity hypothesis is added. For a large enough wave number (after Kolmogorov or Kraichnan dissipation wave number) the dissipation effect dominates, and this should imply an exponential decay for the turbulent spectra (of course with the assumptions that f = f0,K0 and that G >> xoL). Then (lluK,~l] 2) for x >> too and (lu.,~l 2) for tc >> tc~ are very small due mainly to the viscous dissipations, and these averages should behave like the Fourier components of the linear equation
du +vAu=O dt
fort>0
(5.4.8)
leading to an expression of the form:
E(K) ~ C1 e x p - C 2
for tc >> to,, for D = 2, (5.4.9)
E ( x ) ~ C1 e x p - C 2
for x >> tee, for D = 3.
Such results are not proven (and in some cases may be false) (cf. [154,134]). However it is important to notice that weaker (not too weak) forms of (5.4.9) can be obtained at least for D = 2 with full mathematical rigor. Phenomenological analysis and numerical experiments lead to the idea that (5.4.9) should be replaced by estimates of the form
E ( K ) ~ C1E(Ko)(K/K~)~exp--fl(-~~)
for x >> xo, for D = 2, (5.4.10)
E ( x ) ~ C1E(Ke)(K/Ke)~exp--fl(-~E )
for K >> K~, for D = 3.
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569
For the case D = 2 recall the "phenomenological relation" (cf. (5.2.16))
/]2/3 _
E(~)~
~
-~,~
v2
.
The proof of (5.4.10) with D -- 2 is equivalent to the obtention of the estimate:
f IA~/2uK,~I 2tX(du) ~, CG2/3(1+~) +~ L 2(v21
e -~KLG -'/3 "
(5.4.11 )
The approach is based on Gevrey spaces (it follows [85] and it seems that the potential of the method is not yet fully exploited). It starts from the relation:
f
12#(du) <~e_eOtK f [A~/2eOLAI/2ul2ix(du)
IA~
with 0 > 0. (5.4.12)
A slight improvement of the proof in [44] leads to the estimate:
[I
eG(l~
'
5L Aj2 U 112 ~< C1
Vu 6 ,A
(5.4.13)
under the assumption that CIG>~I
and
G)Lx0.
(5.4.14)
By integrating (5.4.13) with respect to #, using (5.4.12) one obtains for oe = 1
f
lA l/2uK.~cl 2Ix(du) <~co G 2 ~--~e v2 - G''~ '
(5.4.15)
)]
Observe that except for the constants (they should have a uniform dependence on the Grashoff number) an estimate of the type (5.4.10) for D = 2 has been proven. Eventually the weakest form of (5.4.10) is the analysis of the power spectrum of the velocity at a given point, namely the expression:
P(co) -
1
lim T~ T
e -~' (s(t)uo)j(xo)dt
j = 1, 2, 3.
(5.4.16)
Once again, for D = 2, it can be rigorously proven that m (do)) -- P (co)do) defines a positive Borel measure on R such that
f
v2
oc e-~~176 (do)) <~ CG 2-~ (log + C G + 1), L2 ~0 - - C - -
1)
(5.4.17) G -2 (log + cG
+ 1)-1.
C. Bardos and B. Nicolaenko
570
The Kraichan-Kolmogorov approach gave, in Section 5.2, an estimate of the number of degrees of freedom in terms of the Reynolds number as (Re) 3/2 for D = 2 and (Re) 9/4 for D = 3. The dimension of the attractor gives an alternative way of measuring the number of degrees of freedom of the flow. Even if this approach is completely different, what is striking is the fact that it leads to the same type of estimates if one observes that the Reynolds number which appears in this derivation is bounded by the Grashoff number:
Re <~CG 2/3. First results for the attractor in dimension 2 were obtained by Ladyzhenskaya [ 112-115] and by Foias and Temam [84]. More precisely, in dimension 2 the global attractor is perfectly defined (with no extra hypothesis), is compact in H and its fractal dimension can be estimated in terms of the Grashoff number according to the formula:
dM(A) <~coG2/3(log(clG) -k- 1) 1/3.
(5.4.18)
This estimate was obtained by P. Constantin et al. [59] in 1988. Observe that (5.4.18) differs from the Kraichnan estimate by a logarithmic term which cannot be present when directly derived by the arguments of Section 5.3. However, remarkably, in a follow up of [58,59], Okhitani has shown [ 142] that a more careful analysis within Kraichnan heuristic framework does yield the logarithmic corrective term. More recently, Liu [129] presented a proof that dM (A) >~cl G 2/3 with, for the driving force, a well chosen eigenvector of A. For D = 3 one assumes that the attractor A is a bounded subset of V (consequence of conjectured but non proven regularity results). Then one defines the quantities:
g-- s u p ( l i m
I 1 four vL-3
fs2(VxU " VxU(X, t))5/4 dx dt] 4/5)
(5.4.19)
and
With a slight improvement (made possible by A. Eden et al. [68]) of the result in [58,59] one shows, that
dM(,A) <~co(LKe),
(5.4.20)
however with ~ larger than the one given by the Kolmogorov scaling law. The proofs of (5.4.18) or (5.4.20) use some of the basic tools of dynamical systems extended to infinite dimensional spaces, and therefore this justifies that a short description of the proof of (5.4.18) be given below.
Navier-Stokes equations and dynamical systems
571
Sketch of proof of the estimate of the fractal dimension of the D = 2 attractor. Inspired by methods of finite dimensional dynamical systems, one first introduces the derivative DS(t, uo) of the flow S(t) as the solution of the equation:
-
+
(s(t).0) + (s(t).0)vx
+ Vxp - 0 .
Vx -4~ = 0 , (5.4.21)
ok(O, x) = ~ E H,
(DS(t, uo)~) = dp(t, x).
The operator DS(t, uo) is compact in V and one can introduce the infinite sequence or1 (t, u0) ~> ot2(t, u0) ~>-.. ~> ot,,(t, uo) >~... ~ 0 of eigenvalues for the self adjoint positive operator (DS* (t, u0) o DS(t, uo))1/2. Classical Lyapunov numbers would be defined as: ~.,,(u0) -- lim {ot,,(t, t--+ oo
u0)}l/t
and
#,,(u0) - log)~,,(u0).
(5.4.22)
However such pointwise Lyapunov numbers may not exist (since we do not know the existence of a canonical ergodic measure # on A), therefore one uses a topological version of uniform (global) Lyapunov numbers introduced in [53] and expanded in [68]. Let:
Pk(t, uo) = otl (t, u0)ot2 (t, u 0 ) ' ' "~k(t, uo),
(5.4.23)
rCk(t) -- sup{ Pk(t, uo)" uo ~ A},
(5.4.24)
because of the subexponential identity rCk(t + s) <<.rCk(t)rrk(s), it can be shown that the following limit exists:
Hk -- lim (rCk(t)) l/t.
(5.4.25)
I---+OQ
One can then define recursively the uniform Lyapunov numbers Ak, k = 1, 2 . . . . . A1----171, A I A 2 = 1 7 2
.....
Al...Ak=17k
....
(5.4.26)
and the uniform (global) Lyapunov exponents are defined by: #m = l o g A m ,
m ~> 1,
(5.4.27)
equivalently: # 1 G - / 2 2 -q-"""-~- # k =
1 lim -logzrk(t).
t--+ e~ t
(5.4.28)
These exponents converge to -cx~ as k --+ oo. Then, with some further hypotheses on the uniform differentiability of S(t) with respect to u0 in A, one uses classical fractal geometry arguments [51,64,68] to cover A by iterations of increasingly refined families of balls; each ball of radius e centered at some u0 is deformed by DS(t; uo) into an ellipsoid whose principal axes are oil (t, uo)e . . . . . ~,~ (t, uo)e . . . . . The scaling laws of such coverings yield estimates on the Hausdorff and fractal dimensions of A according to:
572
C. Bardos and B. Nicolaenko
THEOREM 5.3 [53,160,68]. If for some n ~ 1 (5.4.29)
# l + 11,2 -+"""" + # n + l < O,
then
#n+l < O,
(#1 -4-"#2 -+-""" + / Z n )
I#~+11
(5.4.30)
< 1
and (i) The Hausdorff dimension of .A is less than or equal to
n+
(#1 -+- #2 -+-"""-+- # n ) +
(5.4.31)
I#,,§ I
(ii) The fractal dimension of.4 is less than or equal to
max (j_k_(/z'+/z2+'"+/zJ)+) I/2n+l I
l~j~n
(5.4.32)
'
where
/2,,+1 = limsup
sup Oen+ 1 (t; uo) ] 9 ' log rt-u0~. A
(5.4.33)
t--+ ~x~ t
Next introduce an m-dimensional volume element in V spanned by m independent elements ~l, ~2 . . . . . ~m" denote Uj (t) = DS(t, uo)~j and observe that a variant of the classical Liouville theorem gives" IUI(t) A ' ' ' A U m ( t ) [ A m v
~< I~,(t) A..-A~m(t)lAmveXp
(Yo'
Tr(DS(r, uo) o O m ( r ) ) d r
)
.
(5.4.34)
In (5.4.34) Qm(r) denotes the projector on the space spanned by U1 (r) A ' " A Um (r). For the Lyapunov exponent describing the evolution of the "volume element" one shows the relation: lZ l nt- tl,2 -+'''"-+- lZm ~ qm
= lim sup sup
sup
t-+cxz UoEA ~iEV, II~ll~
('f0' 7
Tr(OS(r, uo) o Qm(r)) d r
)
.
(5.4.35) To estimate the quantity for Tr(DS(r, uo) o Qm (r)) dr
Navier-Stokes equations and dynamical systems one introduces an orthonormal (in V) basis of Tr(DS(r, uo)o Ore(r)) -
Z
573
Qm(z-)V,{q~j(z-)}and uses the relation:
(DS(r, .o)~bj, Adgj)
l <~j<~m
Z
{ -vlAdpjl2 -+-(B(~j,
~j),
Au)}.
(5.4.36)
l ~ j <~m
With the properties of the quadratic advection operator B in dimension 2 one has:
1/2
1/2
(B(dpj, di)j), Au) <~IPlg~(n)la]L2(S_2)lAUlL3/4(n)
(5.4.37)
with
I j<x)l 2,
p<x)-
~<x)- ~
l <~j<~m
Iv,j(x)l 2.
(5.4.38)
l <~j<~rn
The proof is completed with the two following estimates:
Z
]Agpj(x)I2 >~CXlm2
(5.4.39)
l <~j <~m
where X1 is the first non zero eigenvalue of Stokes operator A and
(
IPIL~(n) ~< C 1 + l o g
(1~
Z
Iacj(x)l
2))
(5.4.40)
l <~j<~m
which comes from the log-singularity of the Green function in two space variables and which in a weaker form is due to Brezis and Gallouet [32] (alternative proofs of (5.4.40) can be found in Lieb [124] and Constantin [48]). This is the very estimate which is responsible for the log correction in (5.4.18)
5.5. Exponential attractors As observed in the previous section, the viscous effects make the fluid dependent on a finite number of degrees of freedom and therefore there are good reasons to develop the analogy with finite dimensional dynamical systems and even to try to reduce the Navier-Stokes flow to a flow on a finite dimensional manifold. According to this idea several authors [82,58,59] proposed the notion of inertial manifold closely related to the eigenmode decomposition of the linear operator A. Consider a semi flow S(t), in a Hilbert space H, generated by an evolution equation of the form
ut + vA(u) + B(u) = F
(5.5.1)
C. Bardos and B. Nicolaenko
574
where A is (as is the case for the Navier-Stokes equation) a linear self adjoint operator and B a lower order non linear operator. Introduce an orthogonal decomposition of the Hilbert space H into the space spanned by the first N eigenvectors of A and its orthogonal complement. Denote by PN and QN the two corresponding projections and observe that Equation (5.5.1) is then decomposed into a system of two equations according to the formula:
p = PNu,
q = QNu,
Pt + va(p) + PN(B(u)) -- PNf,
(5.5.2)
qt + va(q) + QN(B(u)) = QNf. Then one says that the above spectral decomposition defines an inertial manifold .M if there exists a Lipschitz map q~ : PN H -+ QN H with the following properties: .M --Graph(q~)- {(p, ~(p)), p ~ PNH},
(5.5.3)
S(t)A/[ C .M
(5.5.4)
u > 0
which attracts uniformly exponentially all solutions of Equation (5.5.1). The existence of the inertial manifold would imply not only that the dynamics are finite dimensional but also that it is completely described by the evolution of the first N eigenmodes (in periodic configuration of the first N Fourier modes). The existence of such an invariant manifold has been proved for several equations like the Kuramoto-Sivashinsky [79], the Ginzburg-Landau equations [58,59] or a hyperdissipative version of the Navier-Stokes equation (cf. [54,133]). However, it has never been established for the genuine Navier-Stokes equation even in two space variables (published claims are incorrect) and besides technical difficulties this fact can explained as follows: A consequence of the existence of an inertial manifold is that the higher order modes (of order greater than N) are completely driven by the lower order modes; in the "folklore" of the field they are "slaved modes" and this property seems to be in contradiction with current phenomenological theories of turbulent intermittencies. Recalling that such theories rely on averaged properties: spectral modes of arbitrarily large frequency and non small amplitude may appear intermittently in physical space with a small probability and such occurrence makes impossible a rigorous description of the dynamics of infinite dimensional system by a N modes dynamical system. Therefore, Eden, Foias et al. [67] have proposed a more physical and more robust (under perturbations) notion which is the exponential attractor: DEFINITION 5.4. Let {S(t)}t>>o be a Lipschitz continuous semiflow with a positively invariant compact set X, X C H, S(t)X C X for every t/> 0. A compact set A40 is called an exponential attractor for S(t) if (i) ,4 = At>o(S(t)X) c .A/[o, (ii) S(t)(A4o) c Ado Vt ~ O, (5.5.5) (iii) A//0 has finite fractal dimension dF(A4o) (iv) dist (S(t)Xo, A4o) <<.Ce -fit for convenient constants C and/3.
Navier-Stokes equations and dynamical systems
575
Exponential attractors are fractal objects which not only contain the ultimate attractors but capture important slow scale transient dynamics. Clearly exponential attractors are not unique; by definition any two exponential attractors are exponentially attracted to each other. The major difference between exponential attractors and the global attractor is that the latter may only attract at an algebraically slow rate (there are examples to that effect, [108]). The major difference between inertial manifolds and exponential attractors is that the latter do not assume any global slaving of small scales. As a consequence, the exponential attractors can deal with cases where an exponential convergence is not restricted within a smooth manifold structure. As far as the theory goes, it might well be a fractal set. The physical relevance of exponential attractors for Navier-Stokes turbulence is discussed in [66]. The existence of an exponential attractor, its dimension and the value of the constants C and 13 appearing in (5.5.5(iv)) can be obtained by an iterative covering process (cf. [67]) from a dichotomy principle, called the squeezing property. Because of its importance, we recall its definition. DEFINITION 5.5. Discrete squeezing property (DSP). In the context of Definition 5.1 one will say that a semiflow S(t) satisfies the weak discrete squeezing property if there exists an orthogonal projection PN of rank N and a positive time t, such that the relation (5.5.6)
<
implies the relation
[
<
iuo-
with 0 ~< 3 < 1.
(5.5.7)
The condition (5.5.6) can be rephrased as "when the difference between two solutions is mainly concentrated in small scale modes" and the consequence (5.5.7) means that the difference is contracted in time during some past (from t = 0 to t,). Strong versions of the squeezing property go back to [83] and Ladyzhenskaya [113]. For the proof of the (DSP) property a convenient tool is the quantities:
)~(t, u, v) -
I l u ( t ) - v(t) II2 Ilw(t) l[2 ]u(t) - v(t)] 2 -- ]w(t)l 2 '
(5.5.8)
and #(t, u, v) -
I A [ u ( t ) - v(t)][ 2 Ilu(t) - v(t) II2
=
[A[w(t)][ 2
(5.5.9)
IIw(t)ll 2
defined for two solutions u(t), v(t) with w(t) -- u(t) - v(t). Now, for the D -- 2 NavierStokes equation, the squeezing property is deduced from the energy estimate: d
2
d-71"-'(tl +
,
12
G2p
<. c--sl (t l
2
(5.5.10)
C. Bardos and B. Nicolaenko
576 which leads to: [w(t,)l 2 ~< exp
- C v U + C1 L2 I t ,
x Iw(0)l
(5.5.11)
The same estimates can also be obtained for the enstrophy. Exponential attractors constructed with the DSP have fractal dimensions higher (as a function of the Grashoff number) than the estimates of dF(A) for the global attractor which relies on Lyapunov number techniques. Eden et al. [65,67] give an alternative construction of exponential attractors based on the concept of outer Lyapunov exponents and outer Lyapunov dimension. The outer Lyapunov exponents are defined as in the beginning of Section 5.5, but with the "sup over u0 E A" replaced by "sup over u0 6 X" in Equations (5.4.24) and (5.4.33), where X is the positively invariant compact set of the semiflow S(t). The outer Lyapunov dimension d0L of.A40 is given by a formula identical to (5.4.32), with the ~i replaced by outer Lyapunov exponents. In principle,
dF(.A) <~doL (.A,4o),
(5.5.12)
but in terms of the practical estimates which both use the trace operator formulas (5.4.35), the two dimensions above are indistinguishable. In that sense, such exponential attractors have optimal outer Lyapunov dimension. Eventually one recovers also for the fractal dimension of the exponential attractor (which contains the global attractor) an estimate in
G2/3(logG + 1) 1/3
(5.5.13)
in agreement with (5.4.18). Recently, Le Dung and Nicolaenko [ 119] have demonstrated that exponential attractors are objects as universal as global attractors for dissipative infinite dimensional dynamical systems: no squeezing properties, nor fine structure of Lyapunov exponents are required. They extend the theory of exponential attractors from the Hilbert space setting to the Banach space setting. The only requirements are for the semiflow to be C 1 in some absorbing ball and for the linearized semiflow at every point inside the absorbing ball to split into the sum of a compact operator plus a contraction. In some sense, [119] establish a global exponential dichotomy for infinite-dimensional dissipative dynamical systems; however, the exponential attractor .A40 is not in general a smooth manifold. Let E be a Banach space, U C E an open set and S" U -+ E a C 1 map. We consider the discrete dynamical system {Sn}n~=l generated by S. We start with the assumption that there is a compact connected subset X C U and S ' X --+ X and S possesses a universal (global) topological attractor A which is a compact, connected set given by oo
A--- N sn(x)" n=l
(5.5.14)
Navier-Stokes equations and dynamical systems
577
We denote by s the space of bounded linear maps from E into itself. For a given positive real )~ we denote by/2x (E) the set of maps L E s such that L can be decomposed as L = K + C with K compact and IIC II < ~-- Here IlC II denotes the norm of the operator C. The main result of [ 119] is the following THEOREM 5.6. If there exists ~. E (0, 1) such that DxS(x) E s f o r a l l x E X, then the discrete dynamical system {S n }c~ possesses an exponential attractor. n=l Define S as the map induced by Poincar6 sections of a Lipschitz continuous semiflow S(t), t >~ O, at the time t = T* for some T* > 0; that is, S := S(T*). We consider the discrete semigroup {S n }n~>0 generated by S. Once the existence of exponential attractors for the discrete case is proved, the result for the continuous case follows in a standard manner (e.g., see [67]). We have THEOREM 5.7. Let X be an absorbing set for a continuous semiflow S(t). Suppose that there is a T* > 0 such that S = S(T*) satisfies the condition of Theorem 5.3. Assume further that the map F(x, t) = S(t)x is Lipschitz from [0, t] x X into X for any T > O. Then the flow {S(t) }t)o admits an exponential attractor Ad. An immediate consequence of the above is the existence of exponential attractors for the fast-rotating 3D Navier-Stokes equations (2.5.23) investigated in [6,7]. This is the only known rigorous result of its kind for genuinely 3D Navier-Stokes-like equations. In the absence of an inertial manifold one would like to address the following question: Is there a natural way of reconstructing the dynamical system without recourse to the underlying equation? Once the existence of an exponential attractor of an infinite dimensional dynamical system is established, the next stage is to unravel the dynamics on this set. A natural way is to show that the infinite dimensional dynamical system is inertially equivalent to some finite dimensional one: DEFINITION 5.8 [67, Chapter 10]. Two dynamical systems are inertially equivalent if: (i) they have a common exponential attractor; (ii) the dynamics on that exponential attractor coincide. First, one can imbed the fractal exponential attractor AA into an Euclidean manifold with a Marl6 projection P which admits a continuous pseudo-inverse when restricted to PAd; note that Mafi6's projections are dense: THEOREM 5.9 (Modified Mafi6's theorem, [29]). Let H be a separable Hilbert space, Y a fractal compact subset of H such that dF(Y) -- D. If Po is an orthogonal projection with r a n k N ~> [2D + 1], then for every 8 E (0, 1) there exists an orthogonal projection P - P(8) such that lIP - Poll ~< 8,
(Ker P) A Y - {0}.
(5.5.15)
C. Bardos and B. Nicolaenko
578
The procedure of constructing a finite dynamical system which is inertially equivalent to an infinite dimensional one can be roughly described as follows [67, Chapter 10]. First we start out with a dissipative dynamical system associated to a PDE written in the evolution form d u / d t -- F(u), u(O) = uo, and project the evolution equation on .M via Mafi6's projection P onto a system of ODE's on an Euclidean space of dimension N - [2D + 1]. On PAd, this dynamical system is well defined by" dx
=PF{(PIM)-lx(t)}, dt x ( O ) - Pu(O).
(5.5.16)
The next step is to extend that dynamical system to a generalized dynamical system defined
everywhere in R ~r. The solutions of the generalized system of ODE's so obtained may not be unique and differentiable (for the definition and construction of such a generalized system, see [67, Chapter 10]). However, one can show that the solutions exist globally in time and are attracted exponentially to PA,4. It is possible to show that the projected system of ODE's generates a generalized dynamical system on the Euclidean space; the continuity points of that system form a dense G~ subset of R ~r. The next step is to lift the generalized dynamical system back to the infinite dimensional space by the lifting p - 1 . Unfortunately, without further properties on the inverse of the Marl6 projection, we cannot proceed with such a lifting to obtain a dynamical system which admits the set .M as an exponential attractor in a Banach space context. Remarkably, this is true in a Hilbert space. It was shown in [67, Chapter 10] that this lifting is possible if a H61der-Mafi6 projection theorem can be established; that is, if one can show that there is a Marl6 projection P whose inverse is H61der continuous on P.M. Recently, in [80] such a theorem is proven by Foias and Olsen for the case of infinite dimensional Hilbert spaces. We remark here (see [67, Appendix A]) that there are counterexamples where p - 1 cannot be Lipschitz so that the best result can only be where P-1 is H61der continuous. Theorem 1.1 in [80] considers a real Hilbert space H and X C H such that dF(X) < m/2, where dF denotes the fractal dimension. Then~ for any orthogonal projection P of rank m and 8 > 0 there is an orthogonal projection P such that IIP - P II < 8 and P Ix has H61der inverse. Combining the discussion in [57, Chapter 10], [80, Theorem 1.1] and [119, Theorem 5.4], we can conclude that: THEOREM 5.10. For a Hilbert space H, let the semiflow S(t) satisfy the conditions of Theorem 5.3 and .All be an exponential attractor for S(t). Then S(t) admits an inertially equivalent generalized dynamical system in H of dimension [2dF(M) + 1]. Here, [dF(.Ad)] denotes the largest integer which is less than or equal to dF(.Ad). There is reasonable hope to extend the above result to inertially equivalent dynamical systems which are locally of the Caratheodory-type on local pieces of smooth manifolds.
Navier-Stokes equations and dynamical systems
579
6. Coherent structures in two space variables Physical observations, illustrate by Figure 1 and numerical simulations illustrated by Figure 2 and studied for for instance in Farge et al. [ 102,69-71 ] and Marcus [ 137], show the
Fig. 1. The Jupiter Red Spot by courtesy of the Jet Propulsion Laboratory Pasadena.
Fig. 2. A Coherent structure generated by a numerical simulation by courtesy of Marie Farge and Nicholas Kevlahan.
580
C. Bardos and B. Nicolaenko
persistence in two spaces variables of coherent structures the most classical one being the Jupiter red spot (cf. Ingersol and Ingersol [96], and Cuong [97]) or the anticyclone of the Acores. In both cases the problems are 3d but due to the smallness of the thickness of the atmosphere they are mostly driven by two dimensional dynamic as said in Section 2.5 (cf. Equation (2.5.23) and related works [6] for instance). These coherent structures present an alternative for what would the thermodynamical equilibrium for turbulent flow. The mathematical construction of these structures relies up to now on the minimization of some type of entropy which would be related to conserved quantities unless the flow becomes turbulent. In this sense these structures are related to the theory of turbulence. However there are by themselves very regular. It is the transient regime and not the structures themselves which is related to turbulence. Eventually question of a mathematical relation between these objects and the one described in the previous sections (global attractors, turbulent energy spectra etc.) seems to be completely open. Consider the solutions of the 2d Euler equations in an open set 12 with boundary 01-2 and impermeability boundary condition u.~-O
on 012
(6.1.1)
with ~ denoting the outward normal. Of course the condition (6.1.1) is omitted when there is no boundary or in particular when S2 -- R 2 and in the periodic case S2 - R2\Z 2. Finally for the sake of clarity S-2 is assumed to be simply connected (even if some interesting examples for the theory do appear in non simply connected domains like the annulus
cf. [38]).
6.1. Stability o f stationary solutions
The geometry of 2-dimensional incompressible Euler equation is characterized by two following facts. (i) As already said in Section 2.3 the vorticity is conserved along the trajectories of the flow: Otto + uVxco - O;
(6.1.2)
(ii) With the divergence free condition and the impermeability boundary condition (when some boundary 0 ~2 is present) the existence of a scalar stream function q/ such that one has u(x, t) -- v •
t).
(6.1.3)
The current and vorticity corresponding to a vector field u will be denoted in this section q/u and COu and if there is no risk of confusion the indices u will be omitted.
Navier-Stokes equations and dynamical systems
581
The first consequence of (6.1.2) and (6.1.3) is that the stationary solutions are characterized by the fact that the gradient of their current and their vorticity are everywhere colinear and that gives (no proof is needed) the: THEOREM 6.1. A divergence free vector field u* (Xl, X2) is a stationary (time independent) solution of the 2d Euler equation if and only if there exists a real (in general multivalued valued)function Gt,, which relates the current tp* and the vorticity co* according to the formula:
o(o,*)-
(6.1.4)
REMARK 6.2. The above theorem gives a criteria used in numerical codes or physical experiments to detect if a solution of the Euler equation comes close to a stationary state: The plot of the points qJ (x, t), V x u(x, t) for t fixed and x E s should form a graph. The second consequence of the Euler equation itself and of the relation (6.1.2) is the PROPOSITION 6.3. For any real valued function dp the quantity
H(u) --
lu(x, t)l- dx + fo 4,(v •
t))dx
(6.1.5)
is conserved whenever u is a smooth solution of the Euler equation. In particular if u* is a stationary solution the quantity: H ( u ) - H(u*) -- -~ 1 ~ (]u(x,t)[
+s
2 -[u*(x)] 2) dx
•
(6.1.6)
is also conserved. REMARK 6.4. As shown by a basic example due to Scheffer [149] and Shnirelman [151] the conservation properties are not always true they require some regularity which are in particular ensured when the vorticity belongs to L~ This will be assumed in all this section. The energy which appears in (6.1.5) is given in term of the vorticity by the formula:
E-
-~1fs? ]u(x, t)] 2 dx - ~1] ~ ((_ A)_ 1co(.,t))(x)co(.,t)dx
(6.1.7)
with (--A) -1 denoting the inverse of the Laplacian with Dirichlet boundary condition, or in terms of Green function:
((--A)-lco)(X)-
f~ g(x, y)co(y)dy.
(6.1.8)
C. Bardos and B. Nicolaenko
582 Observe that one has:
- ]; G(co*)(m - co*) dx.
(6.1.9)
Eventually with 4) in (6.1.6) such that
ok'(s) = - G ( s ) ,
(6.1.10)
H(u)-H(u*)--lf~ -~
lu(x,t) - u*(x) 12dx -
+s -
-~
(~o*)(~o-~o*)dx
,)) -,(~o,)) dx lu(x,t)-u*(x)12dx
+ ~ 1 f o 49"(~(x , t))(co-o)*)2dx
(6.1 . 11)
,
where ~ (x, t) is a real number which depends on the values of co(x, t) and of o9" (x). The right hand side of (6.1.1 0) plays the role of a Liapounov functional and one has COROLLARY 6.5. Any stationary solution of the Euler equation is stable both for positive and negative time for the Hi(s2) norm, under perturbations with uniformly bounded vorticity, if one of the two conditions are satisfied: (i) The function G' with G appearing in (6.1.4) is strictly convex. (ii) There exists a large enough constant C such that - G ' ( s ) / > C. PROOF. The result is a consequence of the existence of two strictly positive constants c~ and 13 such that one has
o~(llu
,
,
~(llu(,, )-u*()ll
-~
L2(,.Q)) ~
)H
2,2(~) + lifo(t,.)- o~, (.)1 2~=(~)).
(6.1.12)
The existence of fl is always ensured by the conservation of the L ~ norm of the vorticity. The existence of ot is trivial in the case (i). In the second case consider the quantity H (u*) - H (u) and use the Poincar6 inequality to bound
IIu (,, .) - ,*(.)112L2 (S2)
Navier-Stokes equations and dynamical systems
583
by 21fs2 ~b"(~(x, t)) (O9 -- O9") 2 d x
.
The above theorem due to Arnold extends a series of results on linear stability obtained already in the last century by Rayleigh and others. On the other hand it is important to observe that any stationary state which satisfies the hypothesis of the above corollary is stable in the H l(,(2) norm both in the future and in the past. This implies that such a solution cannot be an attractor in the future for this norm; However this observation does not prevent the same solution to be an attractor in a weaker norm. And eventually the notion used in this chapter may differ from the one introduced before. One could try to find a stationary (may be unstable) solution with the property that "most" (in a convenient way) solutions would come very often in an arbitrarily small neighborhood of this solution. Therefore the criteria proposed here will differ from the one given in previous section. E]
6.2.
Criteriafor attractor
In this section are described some classical criteria for the co limit set of a family of solution of the 2d Euler equation. First acting as mathematician we give the recipes and then try to justify them. As in the theory of turbulence the reader should keep in mind the fact there are no up to now dynamical proof of the validity of these recipes. The arguments given are borrowed from other fields of physics, mostly statistical mechanic. Considered here are families of solutions u~ with initial data, current and vorticity: 0 ~o co~~and the limit points of the sequence co~(x, t), for t --+ oo, and for co~~converging US~ to co~ in L ~ (~2) weak* are analyzed. Observe that: (i) Weak* L ~ ( ~ ) convergence to a stationary state (u*, oJ*) satisfying the hypothesis of the Corollary 6.5 does not contradict the fact that this stationary solution is stable both in the past and in the future (the topologies are different). (ii) Even when the initial data converge in a very strong norm no uniform (with respect to time) estimate are available (cf. Remark 2.2 of Section 2.4) the only thing which is sure is that the curl remains uniformly in time bounded in L ~ (~2). Let 8i ---+ 0
and
ti ~
oQ
such that u~(., ti) converges to a stationary solution u* in Weak* L ~ (~2) then the following identities are true (the index i being omitted in what follow):
lim fs-2]u~(x'ti) 12dx - fs2 I]uO(x) ]2dx
(6.2.1)
lim f~2 c~
(6.2.2)
and
t) dx - fs2 c~176 dx
C. BardosandB. Nicolaenko
584
On the other hand for a genuinely non linear function F, due to the lack of compactness, one may have:
lim fs2 F(coe(x,t))dx # fs2 F(co~
dx.
(6.2.3)
However for a convex function F the relation limfs 2 F(coe(x,t))dx
>~~ F(co~
(6.2.4)
remains always valid. Therefore according to the intuition one should introduce the "entropy" and defines as a "good guess" the natural stationary solution as the one which minimizes the quantity
dx
fs21 (w*(x))I l~ (c~ under the constraints:
Ilu*(x)12 dx -
IIu~
2
dx
(6.2.5)
and (6.2.6)+
f oJ*~(x) dx - fs2 ~176 (x) dx.
A simple variational computation which uses in particular the formula (6.1.7): shows that any solution of this minimization problem should satisfy the equation: --A~p(x) -- c+e -/~g' -- c_e/~v,
~ -- 0
on 0f2.
(6.2.7)
In (6.2.7) c+ are the two Lagrange multipliers of the two constraints (6.2.6)• while/3 is the Lagrange multiplier of the constraint (6.2.5). The above equation is called (according to the scientists who introduced it in the field) the Joyce Montgomery equation and it has been widely studied (Ref. [99]). In the special case where the vorticity is of constant sign it is reduced to the so called mean field equation: exp(-fl~) --A~p -- c f ~ exp(--fl7 r) dx"
(6.2.8)
In the absence of dynamical proofs, numerical simulations and experiments have been done producing excellent agreement with the "attractor" computed with the above recipe.
Navier-Stokes equations and dynamical systems
585
A more detailed construction has been proposed by Robert and Sommeria [147,148], Miller et al. [138] and others (the initial idea probably going back to Lynden-Bell [130]) with the purpose of preserving all conserved quantities of the form
f~2 f (co~(x, t)) dx
Mf
and therefore not to exclude in some cases strong convergence for ti --+ oo. It is described below. The starting point is the introduction of a family of solutions with initial vorticity coo uniformly bounded in L * (s 'v'x e s
- ~ < - q ~< coO(x) <~ q < cx~
(6.2.9)
converging to coo in weak* L ~ (s Up to the extraction of a subsequence such convergence is characterized by a Young measure dv(y)x and one has:
weak* lim f(co~(x))-~-+oc
f
q
f(y)dv(y)x.
(6.2.10)
The strong convergence being characterized by the points where dv(y)x = 6y(x). With (6.2.10) one can defined a measure of mass 1 with support on the interval ] - q, q [ according to the formula:
l
'L/o
f (y)dTco(y) - -~1
dx
f (y)dv(y)x
(6.2.11)
and introduce the "reference measure" measure d a = dx | dzr0:
L
f (x, y) dao --
dx
xR
f (x, y) dyro(y). q
Now the recipe goes as follow: Among the measures which are absolutely continuous with respect to d a :
d#(x, y) = p(x, y ) d g select the one which minimize the so called Kullback entropy:
K (dl~) -
L f' dx
p(x, y) log(p (x, y)) d o
q
under the followings constraints:
(6.2.12)
586
C. Bardos and B. Nicolaenko
(i) A consequence of the definition of p with Young measures:
dx-a.e,
Rp(x, y) dv(y)x - 1,
(6.2.13)
fsep(
x, y ) d x - I~1,
fs2 dTcodx-a.e.
(ii) The conservation of real valued functions of the vorticity: (6.2.14)
vf, f s ? f R f ( Y ) P ( x , y ) d ~ (iii) The constraint of conservation of energy: 12
((-A)-leo)(x).co(x)dx - ~ with co(x) --
/,
( ( - A ) - ' co~176
yp(x, y)dv(y)x.
(6.2.15)
q
As in the Joyce-Montgomery equation a variational computation is easily done and implies that the function p(x, y) is a solution of the following system:
e~(y)-/b'~P(x) p(x, y) -- f• e~(y)_~y~(x) dv(y)x ' fR Ye~(Y)-~Yr -Ar
=
fR e~(Y)-~Yr
x
(6.2.16)
dv(y)x
In the above system the function or(y) is the multiplier of the constraints (6.2.14) while the number fl is the Lagrange multiplier of the energy constraint (6.2.15). These equations are called the Miller-Robert equations. The second equation of (6.2.16) has to be complemented by the boundary condition r = 0 on 0 s and then, since it is on the form -A~ = G(~)
(6.2.17)
it defines the current of a stationary state. The Joyce-Montgomery equation, the mean field equation and the Miller-Robert equation can be written in the form --ATt = G~(Tt),
~Pl0f~= 0.
(6.2.18)
The number/3, being the Lagrange multiplier of the constraint on the energy E(/3), can be interpreted as a temperature and the role of/3 < 0 is enhanced by the fact that it corresponds to high energy. One has the following
Navier-Stokes equations and dynamical systems
587
THEOREM 6.6. For ~ > 0 both the mean field equation and the Miller-Robert equation do have a unique solution. When the domain is simply connected the mean field equation has a unique solution for > -87r and the mapping 13 E ]--8re, ~ [ w-~ E (13) E ]0, cx~[
(6.2.19)
is decreasing. If the domain is starshaped, for fl less than a critical value tic (in the case of a ball tic = -8re) the mean field equation has no solution.
The proof of this theorem is by now classical one could look at the book [ 136] and at the papers [38,39] (in particular Theorem 3.2 and Proposition 3.3) and [105] for details and references. Notice in particular that when the domain is connected but not simply connected the situation is more complex. When 13 > 0, the steady state given by (6.2.18) both for the mean field equation and for the Miller-Robert equation satisfies the stability criteria of Arnold (Corollary 5.4). These stability results, for the mean field equation, have been recently extended to all values of fl E ]--8zr, ~ [ when the domain is simply connected [23]. Since this criteria is valid both for positive and negative time this state cannot be an attractor for strong norms. On the other hand for solutions of the Miller-Robert equation the stability in the strong norm remains an open problem for fl < 0 and with the fact that this construction takes into account all the conserved quantities this suggests that such solutions may be the best candidate for an attractor.
6.3. Some heuristic justification for the construction of the attractors As said above there is no up to now mechanical justification of the introduction of the solutions defined by the equations of Joyce and Montgomery-Miller-Robert et al. The arguments given rely on the analysis of some special type of solution and some limit process. Along this line the construction of Miller and Robert can be related to a notion of "concentration" of stationary states and a construction starting with piecewise constant initial vorticity. At variance the initial construction for the mean field equation was initiated by Onsager with the introduction of point vortices and a limit process for the corresponding Gibbs measure [143]. Once again in relation with dynamical systems and for a rapid introduction of the notion of negative temperature we shortly review Onsager approach and its further extensions. The first idea is the introduction of solutions of the two-dimensional Euler equation as finite sum of say N vortex points located at the points:
x~(t) = (x~, (t), xi: with intensity oti.
(t))
C. Bardos and B. Nicolaenko
588
To do so the Green function of the Laplacian is decomposed into its smooth and singular part according to the formula: 1 V(x, y) -- - ~ log Ix - y] 4- ~'(x, y) 27r
(6.3.1)
and 89~(x, x) is denoted y (x). Next one introduces the Hamiltonian
H ( X l , X 2 . . . . . XN) --
N
1 47r
Z
N
aiajV(xi,xj) + Zaiy(xi)
i,j=l" iCj
(6.3.2)
i=I
and the corresponding Hamiltonian system defined in S2N d ai--d~Xil -
Oxi2H,
d ai--d-;Xi 3 - - - O x i l H.
(6.3.3)
The main difficulty in the analysis of the above systems comes from the log singularity of the Hamiltonian, this is the reason why in the definition of this Hamiltonian the constraint i r j is prescribed and that for t = 0 all the xi are assumed to be different. Then one can show that (6.3.3) has a local in time solution which remains in $2. However this system may collapse in a finite time if two points collides. But if all the intensities ai have the same sign the conservation of the Hamiltonian implies global existence for the solution of (6.3.3). The connection of the above system with the solutions of the Euler equation is therefore not easy to establish and it is illustrated at best by the following result due to Marchioro and Pulvirenti [136, p. 165] which is quoted with no proof. THEOREM 6.7. Denote by ~,~ (x) the characteristic function of the ball of center x and radius e, introduce N points xi ~ S-2, assume that e is small enough to ensure that all the balls of radius e and center xi are small enough and contained in $2 and consider the vorticity toe (x , t) of the uniquely defined solution of the Euler equation with initial vorticity N toe (X, 0) -- ~-2 ~ ai Ee (xi) i=1
(6.3.4)
then as long (with respect to time t) as the system (6.3.2) does not develop collapses one has, of course in the sense of distributions" N
limcoe(x,t)--~ai6(xi(t)).
~--+0
i=1
(6.3.5)
Navier-Stokes equations and dynamical systems
589
The justification of the mean field equation which correspond to non negative vorticity is done with the introduction of the Gibbs measure associated to the Hamiltonian system (6.3.3) which is formally an invariant measure for the Euler equation. # ot,fi, N (dxl d x 2 . . . d x N ) -- Z a , ~ ( N ) - I e -~a2H(xl'x2 ......~x) dxl d x 2 " "dxN.
(6.3.6) Since # is defined in term of the Hamiltonian H of the system it is invariant; Z , ~ ( N ) -1 is a normalizing constant which is given by: Zu'fi(N)-I -- fs2N e-t~u2H(xt'x2
. . . . .
XN) dxl
d x 2 " "dxN
(6.3.7)
and which has to be finite. Indeed one has: 8~r , c~). Moreover, in this range of LEMMA 6.8. Z a , ~ ( N ) - 1 < ~x) if and only if fl E (--~-~N "temperature", the following estimates hold: Za.~(N) -1 <<.C(fl, No~, I~]) N
(6.3.8)
with C(fi, N~, I~l) N a constant depending only on the product fl, Not and on I~1. This lemma is quoted from [38] (cf. also [105]) where the proof, obtained with standard estimates, can be found. As in the derivation of the Boltzmann equation in Section 3.2 the limit of ~ot,fi,N (dxl dx2 . . . d x N )
is considered when N --+ c~ and ot --+ 0 with the introduction of the "marginals": ot. fl, N
]~j
(dxl dx2 . . . d x N ) -- d x l dx2 . . . d x j
X f~-2X-J d x j + l d x j + 2 " . d x u Z a , ~ ( N ) - l e -~a2H(xl'x2 ..... XN)
(6.3.9)
and the relations --fiN,
fl(fixed)
and
1
c~---. N
(6.3.10)
From the above lemma one deduces (cf. also [38,105] for proofs and details) the following: THEOREM 6.9. Assume that Y2 is a simply connected domain, let fl ~ (-8Jr, c~), and assume that the equation
-A~
e-t~O -- fs2 e - ~ d x
(6.3.11)
590
C. Bardos and B. Nicolaenko
has a unique solution (condition automatically fulfilled f o r ~ > O) then in the sense o f measures one has
lim #J ~:/~N, or: 1, N--+cx~
J ,N (dxl dx2 " . d x N ) -- I - I l[f(Xi). i:l
(6.3.12)
Observe that as in the derivation of the Boltzmann equation a factorization process related to the minimization of some entropy appears in the proof. As a conclusion once again one should observe the following facts: (i) The above derivation contains no mechanics. (ii) On the other hand a justification of the relevance of Equation (6.3.11) may come from the following interpretation of Theorem 5.9 (quoted from [136, p. 262]). "What is expected to happen is the following. The vortices are distributed according to the Gibbs distribution. When N is large they fluctuate very little. With very large probability they arrange themselves to form the solution of the mean field equation". (iii) As shown by Majda and Holen [132] the two above constructions (Onsager-Joyce and Montgomery on one side and Miller-Robert-Sommeria on the other side) produce the same solution if and only if the density p ( x , y) given by (6.2.16) is statistically sharp, i.e., if one has: p ( x , y) d v ( y ) x = 6~o(x)
(6.3.13)
with co(x) = - A T t ( x ) given by the second equation of (6.2.16).
Acknowledgements First the authors wish to thank profusely Professor Ciprian Foias for his encouragements and suggestions. Furthermore, as said above, part of this presentation owes much to his review article "What do the Navier-Stokes equations tell us about turbulence". Several sections of this presentation result from joint work or long discussion with friends and colleagues. For instance Section 3 is an upshot of a long term project of C. Bardos with Francois Golse and Dave Levermore, results on rotating fluids are in the core of a project of B. Nicolaenko with Anatoli Babin and Alex Mahalov. The section on coherent structures in this presentation follows discussions with Marie Farge who introduced these concepts in our community. Finally we owe to Uriel Frisch a general approach on turbulence. It is a pleasure for us to thank all of them.
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CHAPTER 12
The Nonlinear Schr6dinger Equation as Both a PDE and a Dynamical System
David Cai* and David W. McLaughlin t Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Kenneth T.R. McLaughlin Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Pde properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The integrable NLS equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Periodic spatial boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Temporally chaotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
603 605 609 612 618 618
4.2. Persistent homoclinic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Chaotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Very recent work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Spatiotemporal chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. A definition of spatiotemporal chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Information propagation in linear stochastic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Numerical measurements of spatiotemporal chaos for NLS waves . . . . . . . . . . . . . . . . . . 6. Descriptions of the chaotic state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Equilibrium statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Weak-turbulence theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Effective stochastic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
621 629 630 631 631 632 634 637 638 638 640 646
6.4. Nonlinear localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Asymptotic long-time behavior of NLS waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Statement of the Riemann-Hilbert problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Long-time behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Semi-classical behavior
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* Sloan Foundation Grant #96-3-1. +Funded in part by NSF DMS 9600128, AFOSR-49620-98-1-0256, and Sloan Foundation Grant #96-3-1. H A N D B O O K OF D Y N A M I C A L S Y S T E M S , VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 599
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8.1. Sample numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Formal semi-classical asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. The weak limit in the defocusing case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. More on the modulation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. The focusing case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Note added in proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
661 663 664 665 666 667 668 669
Abstract Nonlinear dispersive wave equations provide excellent examples of infinite dimensional dynamical systems which possess diverse and fascinating phenomena including solitary waves and wave trains, the generation and propagation of oscillations, the formation of singularities, the persistence of homoclinic orbits, the existence of temporally chaotic waves in deterministic systems, dispersive turbulence and the propagation of spatiotemporal chaos. Nonlinear dispersive waves occur throughout physical and natural systems wherever dissipation is weak. Important applications include nonlinear optics and long distance communication devices such as transoceanic optical fibers, waves in the atmosphere and the ocean, and turbulence in plasmas. Examples of nonlinear dispersive partial differential equations include the Korteweg-de Vries equation, nonlinear Klein-Gordon equations, nonlinear Schr6dinger equations, and many others. In this survey article, we choose a class of nonlinear Schr6dinger equations (NLS) as prototypal examples, and we use members of this class to illustrate the qualitative phenomena described above. Our viewpoint is one of partial differential equations on the one hand, and infinite dimensional dynamical systems on the other. In particular, we will emphasize global qualitative information about the solutions of these nonlinear partial differential equations which can be obtained with the methods and geometric perspectives of dynamical systems theory. The article begins with a brief description of a spectacular success in pde of this dynamical systems v i e w p o i n t - the complete understanding of the remarkable properties of the soliton through the realization that certain nonlinear wave equations are completely integrable Hamiltonian systems. This complete integrability follows from a deep connection between certain special nonlinear wave equations (such as the NLS equation with cubic nonlinearity in one spatial dimension) and the linear spectral theory of certain differential operators (the "Zakharov-Shabat" or "Dirac" operator in the NLS case). From this connection the "inverse spectral transform" has been developed and used to represent integrable nonlinear waves. These representations have provided a full solution of the Cauchy initial value problem for several types of boundary conditions, a thorough understanding of the remarkable properties of the soliton, descriptions of quasi-periodic wave trains, and descriptions of the formation and propagation of oscillations as slowly varying nonlinear wavetrains. In addition, more recent developments are described, including: (i) the formation of singularities and their relationship to dispersive turbulence; (ii) weak turbulence theory; (iii) the persistence of periodic, quasi-periodic, and homoclinic solutions, by methods including normal forms for pde's, Melnikov measurements, and geometric singular perturbation theory; (iv) temporal and spatiotemporal chaos;
The nonlinear Schr6dinger equation
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(v) long-time and small dispersion behavior of integrable waves through Riemann-Hilbert spectral methods. For each topic, the description is necessarily brief; however, references will be selected which should enable the interested reader to obtain more mathematical detail.
The nonlinear Schr6dinger equation
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1. Introduction
Geometric viewpoints have proven to be extremely useful for understanding qualitative behavior of finite dimensional dynamical systems [ 153,85,191 ], particularly behavior over long (infinite) durations of time. We believe that viewpoints which combine geometry and dynamics will prove to be equally useful for understanding qualitative long-time results for evolutionary partial differential equations (pde's). Even though pde's are infinite dimensional dynamical systems, we also believe that their deep fundamental properties will not be understood solely through natural extensions of finite dimensional methods to abstract infinite dimensional settings. Pde, computational, and stochastic methods will be essential in this process. Our purpose here is to expose graduate students, as well as other researchers in partial differential equations, to this qualitative and geometric view of partial differential equations through a brief overview of one specific class of nonlinear wave equations. We will emphasize global qualitative behavior of solutions. We will try to select references which develop the material in an accessible, even tutorial, manner. Some of these will contain extensive references to the original work. Here, we will make no effort toward historical referencing- leaving that to other more detailed review articles. However, some sample general references include: [ 178,190] for nonlinear waves; [ 120,156,56] for introduction to solitons; [ 181] for nonlinear lattices; [64,11] for inverse scattering transform; [ 158,161] for periodic inverse spectrum transform; [72] for recent developments. (See also extensive annotated bibliography [44].) We intend this brief overview to provide an outline or "study guide" for a graduate course which develops this qualitative viewpoint for the analysis of nonlinear waves, with the references leading to more detailed study. This article will illustrate this viewpoint for one class of nonlinear wave equations nonlinear Schr6dinger (NLS) equations,
iqt = V2q T (qq)~ q ,
(1.1)
~r >~ 0, as well as some of its natural extensions. This class of equations can be used to illustrate many striking features of nonlinear waves, each of which has been understood by a combination of methods from scientific computation and from the theory of pde's and geometric dynamical systems. These features include solitary waves and solitons; response of solitons to external perturbations; periodic waves and quasi-periodic wavetrains; the slow modulation of wavetrains; long-time asymptotics, including a decomposition of the field into solitons and radiation; finite-time blow up; instabilities and representations of unstable manifolds; chaotic evolution in deterministic pde's; spatiotemporal chaos and dispersive turbulence; nonlinear localization in random environments. After touching upon the pde properties of solutions of NLS in Section 2, we begin the overview in Section 3 with a brief summary of a spectacular success of the dynamical systems viewpoint for pde's - the complete integrability of soliton equations as infinite dimensional Hamiltonian systems. Established through the deep connection between spectral theory of certain linear differential operators and specific nonlinear wave equations, this complete integrability unveils and demystifies the mysteries of solitons. It also provides representations of important classes of nonlinear waves - including N-solitons in
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interaction and multi-phase wave trains, as well as the full solution of Cauchy initial value problems. Long-time asymptotic descriptions of the nonlinear waves - including dispersive spreading, scattering, and a decomposition of the field into solitons and radiation- follow from these representations, as well as small dispersion (semi-classical) asymptotics. Under periodic spatial boundary conditions, Floquet theory of the linear differential operators provides a proof of almost-periodic behavior in time of the general solution to the periodic Cauchy problem. For NLS with focusing nonlinearity, this work under periodic boundary conditions culminates in the identification and complete classification of all instabilities, and in the complete representation of their associated unstable manifolds and homoclinic orbits. Such detailed information is unprecedented for finite dimensional dynamical systems, let alone for nonlinear pde's - and indicates the power of the connection between linear spectral theory and certain nonlinear wave equations. In Section 4 we consider perturbations of the integrable NLS equation - damped-driven perturbations under spatially periodic boundary conditions. The instabilities in the integrable focusing case can generate chaotic behavior when the system is perturbed. First, numerical experiments showing temporal chaos are summarized, which are then correlated with integrable instabilities. Then the persistence of homoclinic orbits under perturbations is established with mathematically rigorous analytical arguments. Finally, the connection of these persistent homoclinic orbits with long complex transients and with symbol dynamics is briefly summarized. In Section 5, spatiotemporal chaos is found for these same perturbations by breaking the even spatial symmetry of the system. The concept of spatiotemporal chaos is defined, characterized in terms of "mutual information" at two separated spatial locations, and studied numerically. Then, in Section 6, macroscopic descriptions of the spatiotemporal chaotic state are briefly summarized- including equilibrium statistical mechanics, weakturbulence theories, and "effective stochastic dynamics". Section 6 concludes with sample effects of random coefficients such as "nonlinear localization"- emphasizing distinctions between the linear case and those of focusing or defocusing nonlinearities. In Section 7, we return to the integrable case and describe a powerful analytic method which has recently been developed to extract asymptotic information from the linear spectral representations of integrable nonlinear equations - the Riemann-Hilbert method. First, we define the representations developed by this approach and indicate their use for longtime asymptotics. We then outline the asymptotic method, which exploits rapidly oscillating kernels in the Riemann-Hilbert integral equations. Finally we describe the success of this approach by stating a theorem which provides the complete long-time asymptotics of NLS waves. In Section 8, semiclassical behavior of the NLS wave is described- first by numerical experiments which illustrate the sharp distinctions between linear, defocusing, and focusing behavior, and then by a formal modulation theory. We then mention the use of the Riemann-Hilbert approach to obtain "semiclassical" asymptotic behavior in the defocusing case. The section concludes with a brief description of a beautiful representation of the resulting modulation equations in terms of Abelian differentials and its use in unifying other equivalent representations. In this article we illustrate the potential power of the combination of pde and numerical methods with those from geometric dynamical systems. But we also emphasize the impor-
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tance of a totally new and unexpected idea in the creative process of mathematical discove r y - in this case the deep connection between linear spectral theory and certain special nonlinear wave equations. And we emphasize that the full use of such key new mathematical ideas requires further new analytical developments- in this case the Riemann-Hilbert representations of inverse spectral theory and their use for asymptotics. Riemann-Hilbert methods realize the power and breadth of integrable methods for modern analysis of asymptotic limits - for nonlinear waves and far beyond. This point is elaborated upon in the Conclusion, where we also mention the many open problems for research, once rigid integrability is relaxed. It is in resolving these that we expect the interplay between pde and geometric-dynamical systems to play an essential role.
2. Pde properties Energy methods can be used to establish the following global existence result [36,178]: THEOREM 2.1. Consider the Cauchy problem f o r iq, = V2q - g(q~)~ q, g = + I, q(t -- O) = qo E H 1(~N).
(1) A s s u m e e i t h e r ( a ) g > O a n d O < . cr < 2 / ( N - 2 ) , or(b) g < O a n d O < . ~r < 2 / N . Then 'v'q0 ~ H1 (]RN), 3! q C C[R; HI(•N)] which solves the initial value problem f o r the NLS equation. (2) For g < 0 and cr = 2 I N , Vqo E H 1(]1~U), Ioi2 < [RI2 :=~ q! q ~ C[R; H l (~N)] which solves the initial value problem f o r the NLS, where R denotes the solitary wave solution (2.2) o f the critical NLS equation. (3) For g < 0 and cr = 2 I N , if the energy (2.1) H(qo) < O, then (a) 3 a finite time T* such that limtl, T* Iq(t)lH~ = + ~ . (b) 3 afinite time T** such that limtl, T** Iq(t)l~ = + ~ . Similar results hold for a larger, more general class of NLS equations and boundary conditions. Here the Sobolev space H 1(RN), consisting of functions which are square integrable with square integrable first derivatives, is natural because of the energy invariant of NLS,
H(q)=-fIIVql2+
o ' +g 1 I q l 2 ~ + l ) l d N x '
(2.1)
which can be used to provide global control. Recently, solutions with "rough data" have also become important- for example, for statistical solutions (see below). For recent existence results for data rougher than H l , see [19,21,26,27]. It is clear from this existence theorem that the sign of nonlinearity is important, with g = - 1 (+ 1) called focusing (defocusing) nonlinearity, respectively. In the defocusing case,
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the energy (2.1) is positive definite and can be used for global control of the existence estimates. In the focusing case, the energy (2.1) is indefinite, and need not provide sufficient control if the nonlinearity is too strong, i.e., tr ~> 2 / N . In this case, the focusing nonlinearity can cause the solution to blow-up in finite time. To see this, consider the following differential inequality which follows from the focusing NLS equation: 62
dt 2 V(q) <<,4H(q),
cr >~2 / N ,
where the variance V is defined by the functional
V(q) =
f
[l~121ql2]dNx,
and where q = q(Y', t) denotes any solution in H I [ R N] for which the variance V(q) is well defined. This differential inequality immediately shows that, for initial data with negative energy [H(qin) < 0], the positive definite variance must become negative in finite time. Clearly, this contradiction implies a breakdown in the solution. Sobolev arguments then show that the solution blows-up by leaving H 1, and L ~ , in finite time. Such matters are discussed with mathematical rigor in references [79,36,178]. This blow-up can be understood intuitively, as follows: The focusing nonlinear medium acts as a lens which focuses more and more strongly, the more intense and focused the wave; hence, a catastrophic blow-up of intensity of the wave results, accompanied by the collapse of its spatial extent. In applications such as the propagation of a laser beam, this produces the striking effect of extremely intense, very sharply focused, spots of light. (These spots of light are called "filaments" in the nonlinear optics literature, and the laser beam is said "to filament".) The NLS equation (1.1) is a conservative wave equation. In addition to the energy (2.1), the L 2 norm
l(q) =-
f
Iql 2 dNx
and linear momentum -, P(q) - f qVc) - ~Vq dNx 2i J are also invariants- associated to the symmetries of time translation, phase translation, and space translation. In addition, (1.1) admits the important Galilean and scaling symmetries: If q (x, t) denotes a solution, so does
Q(x,t; )~) = )~l/~q()~x,)~2t).
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The NLS equation is classified as a dispersive nonlinear wave equation [ 190] because its linearization about q = 0,
iqt = Aq, has Fourier solutions of the form A exPli[k. ~ + co(/:)t] },
'r 6 IRN,
with real dispersion relation
Thus, different Fourier components travel at different speeds - leading to dispersive spreading, algebraic (in t) decay, and the concept of a group velocity - which for this linear equation is given by
~gp ([:) = V ~ c o - 2[:. Focusing nonlinearity acts against this dispersive spreading mechanism and can completely overwhelm spreading and produce singularities in finite time, or it can exactly balance the spreading mechanism and produce persistent solitary waves which are localized in space. Which of these alternatives occurs depends upon details of the competition between nonlinearity and dispersion. For example, consider the case of cubic nonlinearity,
iqt - Aq + 2(qO)q, for which solitary waves exist of the form q(Y, t) = exp ( - i t ) R ( l Y l ) , where R(r) is defined as the positive solution of
AR+(2R2-1)R--O, (2.2)
Rr (0) = O,
R (r) --->0
a s r ---~ + e c .
In dimension N = 1, these localized waves are stable, while for dimension N >~ 2 they are unstable; in fact, severely unstable to blow-up in finite time. When combined with Galilean invariance, a four parameter representation of solitary waves results, which in dimension 1 takes the form [-; 1 ,:
-
-
-
-
-
+
(2.3)
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This wave is (exponentially) localized in space, and has many of the characteristics of a "particle". The parameters 0~, v, y, x0) represent its amplitude [inverse-width], velocity, phase, and spatial location, respectively. This particle-like wave travels at constant velocity v and is very stable to perturbations of both the initial data and the equation. The stability and properties of this solitary wave have been established with many numerical experiments in the physical literature, with formal asymptotics [ 101 ], and with rigorous pde analysis [83,84,187]. These solitary waves are the most striking and important component of the solution of the NLS e q u a t i o n - as is clearly seen in numerical simulations. Emerging from generic Cauchy data (which vanishes sufficiently rapidly as Ix l --+ ec) are a finite number of solitary waves (see Figure 1), traveling to both the left and right, ordered by the magnitude of their velocities - together with a finite number of "nonlinear bound states" of these solitary waves. In addition, an algebraically decaying (in t) component is present which resembles dispersive radiation of the linear Schr6dinger equation. These solitary waves are remarkably robust. They exist for a large class of nonlinearities, persist (but slowly deform) under small perturbations of the equation, and can survive collisions with other solitary waves. In the case of cubic nonlinearity, no radiation emerges from these direct collisions. Rather, the two solitary waves emerge from the collision unscathed, with the same velocities and with no generation of additional radiation. The only change the wave experiences as a result of collisions between solitary waves is a shift in their relative phases. In effect (see Figure 1), for the cubic nonlinearity, NLS solitary waves travel and interact as particles experiencing elastic collisions. For other nonlinearities, very stable solitary waves exist, but radiation (at times slight and at times significant) is generated by collisions. These remarkable stability properties for localized waves of NLS make them potentially important in many physical and technological applications- including laser beams [157,90] and transoceanic telephone communication [152]. Understanding these stability properties has also generated considerable mathematical research. Formal asymptotic methods can be used to study linearized stability of the solitary wave [101,186,187] and its response to perturbations [101,146]. For the latter, the NLS equation is perturbed by the addition of small O(e) terms such as dissipation. Then approximate solutions are constructed from the solitary waveform, with two modifications: (i) the replacement of its constant parameters (such as the velocity v) with (slowly varying)
Fig. 1. Left panel: solitons emerge out of an initial wave packet; Right panel: collision of two solitons.
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functions of time; and (ii) the addition of a small correction to the solitary wave. Demanding that this correction remains small over times of O(e -1) identifies the correct slow modulations of the parameters. For example, small dissipation causes the velocity v(et) to slowly decay as a function of time. Some of these formal calculations have been made rigorous mathematically: Nonlinear stability of the solitary wave has been established [83, 84,187]; while modulation theory has led to some of the most successful combinations of pde scattering theory with geometric dynamical systems [188,173-175] for the study of the interaction of solitary waves with radiation. In this work, the evolution equations for the nonlinear wave are decomposed into discrete solitary and continuous radiative components; and the equations are then analysed with a combination of methods from scattering theory for pde's with center manifold-like arguments from dynamical systems theory. A global characterization of the interaction of solitary with radiation results [ 188,173-175, 59,160]. But the understanding of the remarkable elastic collision properties of solitary waves for the 1-D cubic NLS equation required totally new mathematical ideas [77,78,151,199] ideas which are very different from classical pde and dynamical systems methods. One description of these begins from the realization that the NLS equation (1.1) is a Hamiltonian system, ~H iqt = ~ '
which for the 1-D cubic case is a very special Hamiltonian system. In the next section we will describe those new mathematical ideas which identify the very special nature of this 1-D cubic NLS equation, and the reasons for the remarkable elastic scattering properties of its solitary waves.
3. The integrable NLS equation The 1-D cubic NLS equation, iqt -- qxx zF 2(q~)q,
(3.1)
is equivalent to the following linear system [77,121,199] 9x = U(z)~P,
9t = V(Z)~0,
U (z) - iZcr3+i
(0 q)
(3.2)
where
:FCl 0
'
( u
-
i[2x 2 +
2+
(qO -
o2)] 3 +
o a=(-2i) O +
2i)~q + qx '~ 0.,-)
0
}
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and where o3 denotes the third Pauli matrix o3 - d i a g ( 1 , - 1). This equivalence follows from the integrability condition for the overdetermined linear system (3.2): Note that system (3.2) consists in two equations for only one unknown q). As such, it is overdetermined and will possess a solution iff q)t,x = q)x,t. Explicitly calculating this condition, using system (3.2), shows that the integrability condition is equivalent to the NLS equation (3.1). The linear system (3.2) is known as the "Lax pair for NLS" [121,199]. From it, the nonlinear Schr6dinger equation (3.1) inherits a "hidden linearity" which is the key to an explanation of the truly remarkable properties of 1-D NLS. And it is this relationship between linear equation (3.2) and nonlinear wave equation (3.1) which is the "new mathematical idea" to which we referred at the end of the last section. The primary way this equivalence has been used to study 1-D NLS begins from the "x-flow" of (3.2): A
Lq) -- 1,99, where L=-io-3~-
(3.3)
(0 q) _4_0 0
"
(3.4)
We view this linear "x-flow" as a Sturm-Liouville eigenvalue problem, with eigenvalue parameter X. For example, consider the 1-D NLS equation (3.1) on the whole line ( - e c < x < + e c ) , for smooth rapidly decaying functions of x; i.e., in Schwarz class. (Actually, in the defocusing caseAIq(x) I --+ c > 0, while in the focusing case, the limit c vanishes.) Consider the operator L, Equation (3.4), as an (unbounded) differential operator on L 2 (R), which is known as the "Zakharov-Shabat" operator. Denote its point spectra (eigenvalues with L2(R) eigenfunctions) by {Xl, 1.2 . . . . . ~ , N } - A s the coefficients q ( x , t) of this differential operator evolve in time t according to 1-D NLS equation (3.1), one expects the eigenvalues Xj (t) to change with time. But they do not! A simple calculation using the Lax pair (3.2) shows that the eigenvalues are constant in t. These eigenvalues provide N invariants for the 1-D NLS equation (3.1) - where the number N, as determined by the initial data, can be very large and often exceeds the number of classical invariants of L 2 norm, energy, and linear momentum. Thus, the 1-D NLS equation possesses some unusual invariants, in addition to the classical ones. These invariants arise after considering the eigenvalues as functionals of the coefficients q(-, t): Xj (t) -- Xj [q (., t)] . This viewpoint leads one to consider determining q(., t) from spectral data of the differential operator (3.4). Clearly a finite number N of eigenvalues will be insufficient data to determine the function { q ( x , t) Yx ~ ( - e c , +~x~)}, and the eigenvalues will have to be augmented with additional spectral data. But this is a well known problem in mathematical physics known as "inverse scattering t h e o r y " - particularly so for the Schr6dinger operator of nonrelativistic quantum mechanics, but also for the operator (3.4) which is a form of the Dirac operator of relativistic quantum mechanics.
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The appropriate spectral data (see, for example, [78,64,10,11]) is S ---- {~j, cj, j -- 1, 2 . . . . . N" r(s
V~ ~ ( - o c , +r
(3.5)
where )~j denotes the eigenvalues, Cj denotes certain norming constants of the associated L 2 eigenfunctions, and r()~) denotes the "reflection coefficient" defined through the asymptotic behavior (as Ixl --> oo) of the generalized eigenfunctions for the continuous spectrum: if 7s (+) are matrix solutions of Equation (3.3), normalized by ~(+) (x,)~) ~ e izx~r3 ,
x --* +cx~,
(3.6)
then from the transition matrix (which is independent of x)
[~+]-i ~_ = (a0~,b()~,)
)
bO~, t) a()~, t)
'
(3.7)
one defines the reflection coefficient as follows:
rO~)-- a(,k)
(3.8)
(see, for example, [64]). Each member of the spectral data S is viewed as a functional of q, and the data S uniquely determines q by the integral equations of inverse scattering theory [50,78]; that is, the correspondence between q and S is one-to-one and invertible [50]. There exist several equivalent formulations of the integral equations of inverse scattering theory, including the Gelfand-Levitan equations [78] and the equations of Riemann-Hilbert theory [52,54]. The latter have proven to be the most powerful for mathematical analysis of asymptotic behavior. (See Sections 7 and 8.) While these integral equations are difficult to solve analytically, they do provide explicit representations of special solutions which consist of N-solitary waves in interaction, in the absence of any radiation r0~) --= 0 V)~. These representations have the functional form of a "log-determinant", which leads to interesting analysis as N (the number of solitary waves) tends to infinity. (See Section 8.) As q (x, t) evolves in time according to 1-D NLS, one can use the Lax pair (3.2) calculate the time evolution of the spectral data S:
)<j (t) = )<j (0), c j ( t ) = exp[4i~.jt]cj(O),
(3.9)
aO~, t) = a()~, 0), bO~, t) -- exp[-4i)~Zt]bO~, 0).
Thus, r()~, t) -- e x p [ - 4 i ) ~ Z t ] r O ~, 0),
(3.1o)
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and we have the following (infinite!) number of constants of motion: {~.j(q) Yj -- 1. . . . . N;
]r(Z; q)[ u
E (-~,
+e~)].
(3.11)
Thus, 1-D NLS (3.1) is an infinite dimensional Hamiltonian system with an infinite number of constants of motion. Indeed, exactly one-half of the spectral data is invariant {)~j Vj = 1 . . . . . N; Ir(&)l V& ~ ( - ~ , + ~ ) } , while the other half evolves linearly with t: {log(cj) 'v'j = 1 . . . . . N; logr(~.) 'v'~. E ( - c ~ , +cx~)}. Thus, using the inverse spectral representation, one establishes that 1-D NLS (3.1) is a completely integrable Hamiltonian system. This infinite collection of constants of motion explains the remarkable stability and elastic collision properties of solitary waves: First, one must understand the connection between spectral data and solitary waves. The log-determinant formula for N-solitary waves, together with the invertibility of the map to scattering data [50], establishes that there is a one-to-one correspondence between the solitary waves in the spatial profile and the bound state eigenvalues in the spectral data. The N eigenvalues correspond to N solitary waves, with the amplitude and speed of each fixed by the real and imaginary part of the associated eigenvalue. Moreover, the reflection coefficient r(Z) fixes the amplitude of the )~th radiative component of the nonlinear wave. The temporal behavior of the spectral data (3.10) shows that the speeds and amplitudes of the solitary waves are invariant in time, and are not altered by "interactions of the solitary waves". And, since Ir()~, t)l = Ir()~, 0)1, no radiation can be generated by these interactions. In other words, the infinite number of invariants so rigidly constrain the solution that the elastic collision properties of 1-D NLS (3.1) result! Solitary waves which satisfy the elastic collision property are called solitons, to emphasize the remarkable particle-like properties of these nonlinear waves.
3.1. Periodic spatial boundary conditions In our overview, we have focused upon solutions of 1-D NLS on the whole line, which decay rapidly as Ixl -+ ~ . Now we turn to solutions of (3.1) under periodic boundary conditions of (spatial) period s q(x + s t) = q(x, t). NLS is still equivalent to the Lax pair (3.2), and it is still relevant to view the Zakharov-Shabat operator (3.4) as an (unbounded) operator on L2(R), even though its coefficients q(x, t) are e-periodic functions of x. Since its coefficients are periodic in x, Floquet theory can be used to understand the spectral theory of the differential operator (3.4). The well-known Floquet procedures for Hill's operator [135,161] readily extend to the Zakharov-Shabat operator (3.4). Note that this operator is self-adjoint in the defocusing case, but it is not self-adjoint in the case of focusing nonlinearity. This lack of self-adjointness is the only real difficulty for its Floquet theory, and is also the source of the most interesting phenomena of the NLS equation under periodic boundary conditions [130]. The spectrum of the Zakharov-Shabat operator (3.4) with periodic coefficients, when viewed as an unbounded operator on L2(R), consists entirely of continuous spectrum which resides on a countable number of curves in the complex plane, called "bands of spectrum". In the self-adjoint case, these bands lie on the real axis, while in the nonselfadjoint case, they are not so constrained. In both cases, the bands terminate at periodic
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and antiperiodic eigenvalues, which typically have multiplicity one. However, for certain special coefficients, these bands can join at eigenvalues of higher multiplicity. (Consider, for example, the simplest case, q(x) = 0, for which all bands join at eigenvalues with multiplicity 2, and the continuous spectrum consists of the entire real axis.) Again, calculations using the Lax pair (3.2) show that the eigenvalues provide a (countably infinite) collection of constants of motion. Moreover, inverse spectral theory [ 130] (although not as complete for the nonself-adjoint Zakharov-Shabat operator (3.4) as for the Hill's operator [ 161 ]) shows that 1-D NLS (3.1) under periodic boundary conditions is a completely integrable Hamiltonian system. Its integration is accomplished through "Louiville's method" [5,4], as realized by an Abel-Jacobi transformation and theta functions. This procedure amounts to a transformation from q (x) to action-angle variables [99, 98], a beautiful procedure which is most easily described for soliton equations in the case of the Toda lattice [69,71,43]. Generically, the level sets of this countable collection of eigenvalue invariants, {q 6 H l ] ~.j(q) -- Lj(qo) Vj}, are infinite dimensional tori. The solutions to the NLS equation under periodic boundary conditions wind around this torus, executing almost-periodic motion in time t. One should think of the nonlinear Schr6dinger wave as being decomposed into a countable number of oscillators (called "degrees of freedom"), one for each dimension of the torus. Each oscillator has both an amplitude and angle of oscillation, with the amplitudes fixed by the constants of motion and the angles providing coordinates on the infinite dimensional torus. As the values of the constants of motion change, the tori deform and fill out (or "stratify") the function space H 1. As noted above, for special choices of coefficients q,, bands of spectrum can join. As the coefficient approaches a special q,, two eigenvalues coalesce. As this occurs, the torus become degenerate in that its dimension decreases by one. (Intuitively, the amplitude for one of the oscillators vanishes, and the system loses one of its degrees of freedom.) In the self-adjoint case, the pinched torus which results is always stable, in the sense that nearby coefficients have tori for which the oscillatory degree of freedom that was "pinched away" at q, now executes small amplitude oscillations [ 140]. In this stable case the singular level set {q 6 H 1 ] )~j(q) -- Xj(q,) Vj}, consists only in the degenerate torus itself. On the other hand, in the nonself-adjoint case, the degenerate torus T, can be unstable. When it is unstable, the singular level set is larger than the torus itself, containing the unstable manifold W ~'(T,) as well as the torus T,. Intuitively, the circle which is "pinched" becomes one lobe of a "figure eight", rather than just a point. (See [63,130] for pictures.) A homoclinic orbit results which approaches the degenerate torus T, as t approaches infinity: qhom(X, t) --+ T,
as t --+ •
Inverse spectral theory establishes that these unstable tori cannot result from eigenvalues which coalesce on the real axis; hence, instabilities and homoclinic orbits must be associated with complex valued multiple eigenvalues, which must be finite in number. Hence, the dimension of the unstable manifold W u (T,) must be finite. Such spectral matters are discussed in detail in [ 130,149].
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An elementary example which illustrates these instabilities and their associated integrable geometry begins from the trivial x-independent solution of NLS equation (3.1)"
qc(x,t" c, y ) - - c e x p [ - i ( 2 c 2 t
+ y)],
(3.12)
where (c, y) denote two real parameters. For each value of c, this solution is a single circle. It has only one degree of freedom, with the remaining (countable number) of degrees of freedom all "pinched away". In other words, the torus T, -- S 1, is one-dimensional; hence, extremely degenerate. The linear stability of solution (3.12) is easy to study:
q ( x , t ) -- q c ( x , t ) + ~ f ( x , t ) e x p [ - i ( 2 c 2 t
+ y)],
i ft -- fxx -t- 2 c 2 f + 2 c 2 f + O(6), f (x, t) -- } ( k ) exp[i (kx - co(k)t)], ,0
-
k 2[k 2 - 4
2].
Thus, the wave (3.12) is unstable to fluctuations with wave numbers 0 < k 2 < 4c 2" while shorter wavelength fluctuations are neutrally stable according to linear stability theory. The "quantization condition" which ensures spatial periodicity,
kj=
27rj
j-...,-1,0,+l
.....
shows that the number of unstable Fourier modes scales linearly with the size ~ of the periodic spatial domain. This instability of the plane wave (3.12) to long-wave fluctuations is a special case of a famous instability in nonlinear dispersive wave theory, known as the "Benjamin-Feir instability" [ 12] in the context of water waves and as the "modulational instability" in the context of plasma physics [198]. It is now understood to be the fundamental cause of solitary wave formation, of self-focusing and filamentation of a laser beam, and, more generally, of blow-up in finite time. This calculation of the tangent space to the unstable manifold of the circle S = qc shows dim W u (S) - 2 N + 1, where the "2N" comes from the {cos (kjx), sin (kjx), j = 1. . . . . N} basis of the unstable tangent space, and the "1" is the dimension of the circle S itself. Using integrable theory, one can identify all unstable tori T, for 1-D NLS, and construct rather explicit representations of their unstable manifolds W" (T,), which for the integrable NLS equation equal their stable manifolds W s (T,). This beautiful representation results from B~icklund (Darboux) transformations- a construction which we now describe in its generality [60,130,149]. First, one [60,130,149] establishes that, for each instability, there is an associated complex eigenvalue of with multiplicity at least two. Thus, there is a correspondence between instabilities and complex multiple points in the periodic and antiperiodic spectrum of the
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Zakharov-Shabat operator (3.4) - a correspondence which enables us to classify the unstable tori. Fix a periodic solution of NLS which is quasiperiodic in t, unstable, and for which the instability is associated to a complex double point v with multiplicity 2, for the operator L(q). We denote two linearly independent Zakharov-Shabat eigenfunctions at (v, q) by (~+, ~ - ) . Thus, a general solution of the Zakharov-Shabat linear eigenvalue problem at (q, v) is given by $ ( x , t" ~" o+, o _ ) =
c+6 + + o-6-.
We use r to define a transformation matrix G by G-G(X;
o )
v" ~) =-- N ( A -
X-O
N-
,
(3.13)
where N--
_&]
r
(/)2
~1
"
Then we define Q and l/, by Q(x, t) = q(x, t) + 2(v - fi)
r
~1~2 -}- r
(3.14)
and
(x,,.
-
c(z;
z),
(3.15)
where ~p solves the Zakharov-Shabat linear system at (q, v). Formulas (3.14) and (3.15) are the B~icklund transformations of the potential and eigenfunctions, respectively. We have the following: THEOREM 3.1. Define Q(x, t) and q-'(x, t" X) by (3.14) and (3.15). Then" (i) Q(x, t) is a solution of NLS, with spatial period g~; (ii) The spectrum a( L(Q)) - a( L(q)); (iii) Q(x, t) is homoclinic to q(x, t) in the sense that Q(x, t) --+ qo+ (x, t), exponentially as exp(-a~ Itl) as t --+ i c ~ . Here qo• is a "torus translate" of q, av is the nonvanishing growth rate associated to the complex double point v, and explicit formulas can be developed for the growth rate av and for the translation parameters 0• (iv) q.' (x, t; A) solves the linear system (3.15) at (Q, x). This theorem is quite general, constructing homoclinic solutions from a wide class of starting solutions q (x, t). Its proof is one of direct verification, following the sine-Gordon
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model [60,139]. Periodicity in x is achieved by choosing the transformation parameter )~ = v to be a double point. Several qualitative features of these homoclinic orbits should be emphasized: (i) Q(x, t) is homoclinic to a torus which itself possesses rather complicated spatial and temporal structure, and is not just a fixed point; (ii) nevertheless, the homoclinic orbit typically has still more complicated spatial structure than its "target torus"; (iii) when there are several complex double points, each with nonvanishing growth rate, one can iterate the B~icklund transformations to generate more complicated homoclinic manifolds; (iv) the number of complex double points with nonvanishing growth rates counts the dimension of the unstable manifold of the critical torus in that two unstable directions are coordinatized by the complex ratio c+/c_. Under even symmetry only one real dimension satisfies the constraint of evenness, as will be clearly illustrated in the example below; (v) these B~icklund formulas provide coordinates for the stable and unstable manifolds of the critical tori; thus, they provide explicit representations of the critical level sets which consist in "whiskered tori" [3].
An example: The spatially uniform plane wave. As a concrete example, we return to the spatially uniform plane wave qc, Equation (3.12), for which the entire construction can be carried out explicitly: A single B~icklund transformation at one purely imaginary double point yields Q = QH(X, t; c, Y; k = Jr, c+/c_):
Q.-[
cos 2p - sin p sech r cos(2kx + 4~) - i sin 2p tanh r
ce-i(2c2t+v)
1 + sin p sech r cos(2kx + ~b)
eqZ2iPce -i(2c2t+V)
a s "g ~
Too
(3.16)
where c+/c_ =-- exp(p + ifl) and p is defined by 2cexp(ip) = (1 + icr), o = x / 4 c 2 - l, r = o't - p, 4~ = P - (fl + zr/2), and where the spatial period g -- 1 (see Figure 2). Several points about this homoclinic orbit need to be made: (i) The orbit depends only upon the ratio c+/c_, and not upon c+ and c_ individually. (ii) Q H is homoclinic to the plane wave orbit; however, a phase shift of - 4 p occurs when one compares the asymptotic behavior of the orbit as t ~ - o o with its behavior as t --+ + Oo. (iii) For small p, the formula for Q/4 becomes more transparent: Q/-/"~ [(cos 2 p -
i sin2p tanh r) - 2 sin p sech r cos(2kx
+
dp)]ce -i(2c2t+y) .
(iv) The complex transformation parameter c+/c_ = exp(p + ifl) can be thought of as S x R. In the formula an evenness constraint about x -- 0 can be enforced by restricting the phase 4~ to one of two values 4~ = 0, zr. In this manner, evenness reduces the formula for QI-/from S x R to two copies of R. In this manner, the even symmetry disconnects the level set. Each component constitutes one "whisker" of the "whiskered circles". While the target q is independent of x, each of these whiskers has x dependence through cos(2kx). One whisker has exactly this dependence and can be interpreted as a spatial excitation located near x = 0
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Fig. 2. Homoclinic orbits associated with one instability (left panel) and two instabilities (right panel).
(Figure 2 (left p a n e l ) ) - while the second whisker (not shown) has the dependence cos(2k(x - zr/2k)), which we interpret as spatial structure located near x -- 1/2. In this example, the disconnected nature of the level set is clearly related to distinct spatial structures on the individual whiskers. In this example the target is always the plane wave; hence, it is always a circle of dimension one, and here we are really constructing only whiskered circles. On the other hand, in this example the dimension of the whiskers need not be one, but is determined by the number of purely imaginary double points, which in turn is controlled by the amplitude c of the plane wave target and by the spatial period. (The dimension of the whiskers increases linearly with the spatial period.) When there are several complex double points, B~icklund transformations must be iterated to produce complete representations of the unstable manifold. While these iterated formulas are quite complicated, their parameterizations admit rather direct qualitative interpretations (see Figure 2 (right panel)). Thus, B~icklund transformations give global representations of the critical level sets. The level sets in the neighborhood of these of critical ones have fascinating topological structure [63,130]. The plane wave example under even symmetry and with only one instability provides the simplest case. Here, dim W '' (q = S) = 2 - the dimension of each homoclinic
F Fig. 3. Trouser diagram.
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orbit, plus the dimension of the target circle q = S. In addition, NLS also possesses a four-dimensional invariant manifold which contains W u (q = S), and which can be viewed as the result of "shutting-off" all degrees of freedom except for the spatial mean and the "first radiation mode". In this four-dimensional space, the fixed energy level sets topologically form a trouser diagram shown in Figure 3. (The "trouser • S l'' forms the threedimensional, fixed energy manifold.) Note in particular the symmetric pair of homoclinic orbits and their relationship to the two legs, one of which represents a (periodic) soliton located at the center of the periodic domain at x = 0, and the other a soliton located onehalf period away at x = ~/2. When all other radiation degrees of freedom are excited, each forms a small disc (a center for each additional radiation degree of freedom), and the full phase space can be represented topologically (locally, near the trouser) as the product of the trouser with a countable number of discs. More complex examples are described in [130].
4. Temporally chaotic behavior The existence of instabilities and their associated homoclinic orbits for the integrable NLS equation indicates that external perturbations could induce chaotic responses in near by perturbed deterministic pdes. Moreover, the trouser topology nearby critical level sets, together with the correlation of the two legs of the trouser with two distinct spatial locations for a soliton ("center" and "edge" of the periodic domain), indicates that chaotic behavior under deterministic perturbations might involve a "random jumping" of a solitary wave between these two spatial locations. An exciting possibility arises - Smale horseshoes [ 171 ] and chaotic symbol dynamics [191 ] in a pde setting. Moreover, this temporal chaos - involving interactions of solitary waves with each other, with radiation, and with external perturbations- should be easily observed in numerical simulations, and even in laboratory experiments. And indeed this type of chaos does appear to exist for certain near-integrable systems - temporal chaos resulting from competitions between, and instabilities of, spatially coherent solitary waves.
4.1. Numerical experiments As described in the references [15,149], we designed some numerical experiments to investigate this exciting possibility. For example, we considered a damped-driven perturbation of an NLS equation in the form: - 2 i q t + qxx +
(1 )
-~qgl - 1 q -- iotq - V ~ F e -i•
(4.1)
with periodic even boundary condition q ( x + g., t) = q ( x , t),
q ( - x , t) = q ( x , t),
(4.2)
and initial condition a periodic extension of the single soliton waveform q ( x , O) -- 4eme -2ierx sech(2emx),
(4.3)
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where e - er 4- ie,n, with er chosen so that e -2ie'x is periodic of period s (usually er = 0 and em = 1/2). These numerical experiments are described in detail in the survey [149], including: (i) the numerical algorithms and their validation, which are essential when studying longtime temporal integrations of chaotic behavior of unstable orbits; (ii) the collection of chaotic diagnostics with which we post-processed the numerical data; and (iii) a detailed discussion of our numerical observations. Here we only give a brief description of typical observations, for the simplest case where temporal 6haos was observed. We organized our numerical studies into bifurcation experiments in which all parameters were fixed (dissipation strength or, spatial period g), except for the amplitude of the driving force F , which was increased from experiment to experiment as a "bifurcation parameter". In the simplest case, we set ot ~ 0.1538 and choose the spatial period g ~ 6.12, for which only one instability is present. (For larger periods g more complicated behavior was observed.) Sample results are pictured in Figure 4. While the details of the bifurcation sequence are somewhat involved [149], the general pattern may be summarized as follows. As F increases, the long-time behavior of the wave undergoes the following sequence of changes in Iq(x, t)[: (i) spatially flat, time independent; (ii) "sech-like" in space, time independent; (iii) sech-like in space, but time periodic; (iv) sech-like in space, quasiperiodic in time; (v) chaotic in time, with the sech-like excitation jumping from center to edge of the periodic spatial domain. We used standard chaotic diagnostics to identify chaotic b e h a v i o r - including Poincar6 sections, power spectra, Lyapunov exponents, and information dimension. Each of these diagnostics is defined and discussed in detail in [149]. In Figure 4, we show four sample "cross s e c t i o n s " - for time-independent, periodic, quasi-periodic, and chaotic temporal behavior. We emphasize that this experiment, which is the simplest that we have found which has chaotic behavior, is extremely important for our theoretical studies. In it, the chaotic state contains only one spatially localized coherent structure. At times this solitary wave is located at the center, and at other times it is seen at the edges of the periodic spatial domain. These two locations are the only two allowed under even boundary conditions. We believe that one source of the chaotic behavior is an irregular (random?) jumping of the solitary wave between center and edge locations (see Figure 5). This center-edge jumping of the solitary wave through homoclinic transitions forms the basis for the simplest description and model of chaotic behavior.
Fig. 4. Perturbed solitonic dynamics. From top to bottom: (1) locked state, (2) periodic state, (3) quasiperiodic state, and (4) temporal chaotic state. Plotted here are Iq(x, t)I. The right panels are the corresponding surface cross sections {Req(0, t), Imq(0, t) Vt}. Note that for the case of the quasiperiodic and chaotic dynamics shown here, the values of the driving F differ only by 0.4%.
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The nonlinear Schrgdinger equation
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1.0
i
0.8-
i
L 0.6
i
! _In i i :
7 '-[ r ,
7 717: "7
i
0.4-
!
I,
ii
I
!
!, i
!
!i
i
i
I
i
ii '
'i 0.0 ~ - - 1000.0
i !
i
I
I
1
1
J J
:
I
I
1200.0
[
ii
I
0.2-
I
i
!
!
ii
I
;
i
.,i 1400.0
1600.0
1800.0
2000.0
time
Fig. 5. Center-edge jumps of soliton in the chaotic state. The dark line segments are the temporal traces of the maximum of [q (x, t)l.
4.2. P e r s i s t e n t h o m o c l i n i c o r b i t s The first step toward analytical descriptions of such chaotic behavior is to assess the persistence of homoclinic orbits. These can provide a "skeleton" for chaotic trajectories. That is, persistent stable and unstable manifolds, and their intersections, provide a framework with which chaotic behavior can be described. Procedures for this description are well known for finite dimensional dynamical systems [85,191], and have recently been developed for the NLS p d e [ 131]. See also [ 147] for a rather detailed overview of these mathematical arguments. Here we present a brief sketch of the arguments, taken from [ 147], and state the persistence theorem. Specifically, we study a perturbed nonlinear Schr6dinger equation (PNLS) of the form iqt - qxx +
2[q~ - co2]q + i e [ D q - 1],
(4.4)
where the constant oJ 6 (89 1), e is a small positive constant, and D is a b o u n d e d negative definite linear operator on the Sobolev space H e,p i of even, 2n periodic functions. Specific A
examples of the dissipation operator D include the discrete Laplacian and a "smoothed Laplacian" given by A
A
D q -- -o~q - ~ B q ,
(4.5)
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where the operator B has symbol given by
b(k) =
k Or
k
This pde is well posed in H e1, p , and the solution q(t; e ) - - Ft(qin) has several derivatives in qin, and in the parameters such as e, with the exact number of derivatives increasing with decreasing e. Our analysis of this equation begins with two observations: First, when e = 0, the unperturbed NLS equation is a completely integrable soliton equation. Second, the "plane of constants" He, He ---- {q(x, t) l Oxq(x, t) - - 0 } , is an invariant plane for PNLS. In each of these two cases, [e - - 0 or q E He], the behavior of solutions q(., t) can be described completely. In the first case, this description is described in Section 3.1; in the second case, it is accomplished through "phase plane analysis". In the jargon of the theory of dynamical systems, our methods will be a form of "local-global" analysis, where at times the term "local" will mean close to the plane He, and at other times "local" will mean close to integrable solutions. In any event, throughout our global arguments, control is achieved either because of proximity to (i) the plane 17c or (ii) e = 0. 4.2.1. Motion on the invariant plane. form
On the invariant plane He, the equation takes the
iqt = 2[q~ - co2]q - ie[otq + 1],
(4.6) A
where it is assumed that the dissipation operator D acts invariantly on He as A
Dq -- - a q , for ot a positive constant. Equivalently, in terms of polar coordinates q = ~/-[ exp{ i O}, these equations take the form It -- - 2 e [or I + ~ Ot -- - 2 ( I
cos 0 ],
6 -0)21-+- ~ sin 0.
(4.7)
The nonlinear Schr6dinger equation
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/ /
/ \
', /
I
\
Fig. 6. Phase Plane Diagram of the ODE.
When e - 0, the unperturbed orbits on Hc are nested circles, with S~o a circle of fixed points given by I = o92. For e > 0, the perturbed orbits on Flc are very different (see Figure 6). First, only three fixed points exist: O, which is a deformation of the origin; Q, a saddle which deforms from the circle So; and P, a spiral sink which also deforms from the circle S~o. While the circle of fixed points S~o for the unperturbed (e = 0) problem does not persist as a circle of fixed points, motion near So~ remains slow for small positive e. We introduce the variable J, J = l --092,
and, in order to describe the slow flow close to this circle S~o, we rescale the coordinates r
=
vt,
(4.8) J = vj,
where v = v/-d. In these scaled coordinates, Equations (4.8) on the plane of an O(v) perturbation of the conservative system
Flc take
the form
jr -- --2( oto92 + cocos0), 0r = --2j . Thus, we see that near the circle So~, the slow motion is approximated as a driven pendulum, with energy 1
E(j,O)- = j 2 z
o9 (sin0 + otog0).
(4.9)
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4.2.2. lntegrable homoclinic orbits. completely Hamiltonian system,
The unperturbed (e--0) focusing NLS equation is a
6 oq
- i q t -- -7=_H,
(4.10)
with the Hamiltonian H given by
H-
f0 7r[qxqx
- (qO) 2 + 2o92qq] dx.
Consider the two parameter family of plane wave solutions, independent of x"
q(x, t" c, y ) -
c e x p { - i [ 2 ( c 2 - o 9 2 ) t - g]}.
Linearizing NLS about q (x, t" c, y) shows that this plane wave is linearly unstable, with positive growth rate cr for the linearized "cos x mode" given by o- -- v/4c 2 - 1. As described in Section 3.1, a B/acklund transformation will produce an orbit homoclinic to this plane wave:
q~
[cos2p-isin2ptanhr• 1 q: sin p sech r cos x
q,
(4.11)
where r -- o-(t + to),
e lp
l+icr -
-
2c
Here + labels a symmetric pair of homoclinic orbits. (Recall that - c o s x = cos(x + Jr), which shows that one sign (+) represents an excitation centered at x = 0, while the other sign ( - ) represents an excitation centered at x = Jr.) If we specialize to c = o9, q lies on the circle of fixed points So~, and the orbit qh is homoclinic to this circle. Thus, from one point of view, (4.11) provides an explicit representation of a "whiskered circle"; while from another viewpoint, it provides an explicit representation of the unstable manifold
W u ( & ) - W s (S~o) -
U
q~(t; y, to, c).
y,to, +
4.2.3. Melnikov integrals. Next, we define the Melnikov integral which will be used to establish persistence. First, we write the perturbed NLS equation in the form
qt--iH'(q)+eG(q),
(4.12)
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where H ' ( q ) = - q x x - 2[qO - o ) 2 ] q and where G(q) denotes the perturbation: A
(4.13)
G(q) = - ~ q - flBq - 1. Let I denote a (real valued) constant of motion for the unperturbed (e --- 0) system. DEFINITION. The Melnikov function (based on I) is defined by
M , -=
F
(I'[qh(t)], G[qh(t)])dt G
I+~J
I
dt.
(4.14)
In this definition, we assume that the integrals converge (which they do for our choices of constant I). Melnikov integrals, together with geometric analysis, are used to assess the fate of the orbit qh(t) under the perturbation. As is clear from its definition, the Melnikov integral Mt provides an estimate of the change in the value of the constant of motion I over the perturbed orbit. Without an additional geometric setting, this change provides very little information about persistence. When this integral does not vanish, one can establish no persistence [42]. However, in particular geometric settings, a simple zero of the Melnikov function with respect to one of its parameters can insure the intersection of certain stable and unstable manifolds, and the persistence which follows as a consequence of this intersection. Next we specialize the orbit qh(t) to one homoclinic to the circle of fixed points So). Setting c = co produces orbits homoclinic to S~o, which we denote by q~o:
q~o(t) - [ c o s 2 p - i s i n 2 p t a n h r + sinp sech r cosx 1 - sin p sech r cos x
o)expi[0b - 2p],
(4.15) where tan p = ,,/4o) 2 - 1 and r = (tan p)(t + to). While the orbit qo~ approaches the circle S~o as t --+ -+-oc, it approaches the fixed point o)exp i Oh as t --+ - e c and, as t --+ + e c ,
qo)(t)
--+ o ) e i(01,-4p)
Thus, the (heteroclinic) orbit experiences a phase shift of - 4 p
e-4i p -- I I - i x/4o)2 - 1 ]
It will be sufficient to use the the Melnikov integral based on the energy H. With these ingredients, we assemble the final expression for this Melnikov integral:
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PROPOSITION. For the specific perturbation (4.13) and homoclinic orbit q~o, Equation (4.15), the Melnikov integral takes the form
MH = Mn(ot, fl, Oh) --
S
(H'(q~o(t)), G(q~o(t)))dt
OQ
= [~M~ + ~M~ + M(Ob~],
(4.16)
where M~ --
F
(H'(q~o(t)),qo)(t))dt,
oo
M~ -
(H'(qo~(t)), ~qo~(t))et,
M(Ob) --
(H'(q~o(t)), 1)dr. oo
More explicitly, f~c
M~ =
4zr O92 sin 2 Po sech r
/ o 2rr
ec dr Jo
dx
erA3
x [sech r + sin po tanh 2 r cosx - sin 2 po sech r ( 2 + cos 2 x) + 2 sin 3 po sech 2 r cos x ],
M~ --
f_X~fo2Zr47rco2sin2posechr dr dx A5 oc
O"
x [sin Po sech r cos x - sin 2 Po sech2r (1 + sin 2 x)] x [2 sech r - 2 sin po sech2r cos x - 2 sin 2 Po sech r + 2 sin 3 Po sech2r c o s x ] at- O(sin K-2 po),
M(Ob) -- cos(O/) - 2po)
dr
dx
A2
x [sech r - sin po cos x], where po = tan -1 ~,/4co2 - - 1, A = [1 - sin po sech r cosx] and where the O(sin K-2 po) term in the M/~ equation is due to the fact that we used - 0 2 instead of B" in our calculation. Thus, the final expression for the Melnikov integral is of the form M(oe, 13, 0b) = c~M~ +/3Mr
+ M1 cos(0/) - 2po),
(4.17)
where M~, M~, and Ml are functions of co only. Clearly, for small oe and 13, this Melnikov function has simple zeros as a function of Oh. At issue, of course, is the geometric meaning of these zeros.
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4.2.4. Persistent homoclinic orbits for PNLS. Simple zeros of the Melnikov function (4.17) enable us to prove the following persistence theorem for the pde: THEOREM 4.1. The perturbed NLS equation (4.4) possesses a symmetric pair of orbits which are homoclinic to the saddle fixed point Q, provided the parameters lie on a codimension 1 set in parameter space which is approximately described by ol-- E(co)~.
E (co) can be computed explicitly to leading order e. In addition, various properties of the persistent homoclinic orbits (such as "take-off" angles) can be precisely characterized. These two homoclinic orbits differ by the location of a transient spatial structure- a solitary wave which is located either at the center (x = 0) or the edge (x = Jr) of the periodic box. As such, this theorem provides a key step toward a symbol dynamics for the pde. The proof of this theorem is described in mathematical detail in [131], and a detailed overview of the argument is presented in [ 147]. It is organized with "local-global" analysis, and it involves normal forms for the perturbed NLS equation, invariant manifold theory for NLS and geometric singular perturbation theory, combined with integrable theory and Melnikov analysis. It is important to keep in mind that, throughout the proof, control is obtained in one of two ways - either the orbits are (i) close to the invariant plane Hc, or they are (ii) close to the integrable case. Also keep in mind that the arguments will be a form of "shooting", where the goal will be to force an orbit to "hit" target manifolds of high dimension, but in an infinite dimensional space. To make these manifolds easy targets, we make them very large in the sense that they will be codimension 1. The steps in the proof are as follows: (1) Preliminary set up including (i) motion on Hc, (ii) coordinates near Hc, (iii) linear stability and time scales, and (iv) a normal form. (2) Local arguments including (i) persistent invariant manifolds, (ii) fiber representations, and (iii) the height of the stable manifold W s (Q). (3) Global arguments which describe the first and second Melnikov measurements. Here we restrict ourselves to a few remarks about these steps: Remark 1. In the local arguments for persistence of stable and unstable manifolds, and for their representations with "fiber coordinates", we use integral equation methods rather than more geometric ones [9,133] from dynamical systems theory. These integral equation methods are natural for pde's. Their use is consistent with our view that intuition, taken from finite dimensional dynamical systems, should structure the arguments- which should then be implemented with methods which are natural for pde's.
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Remark 2. Fiber-representations are constructions from geometric singular perturbation theory which permit one to follow the long-time fate of the full motion with an orbit totally restricted to a slow manifold. These were developed by Fenichel [65-68] to provide a geometric understanding of singular perturbation methods, such as those of Howard and Kopell [ 110,92]. Recent descriptions of these fibrations, with explicit examples, may be found in [97,145,147]. Remark 3. The argument is a "shooting method", with the final target the stable manifold of the saddle Q, W s (Q). To ensure that this "target is hit", this stable manifold must have sufficient height above the plane 17e. For this estimate, the effects of quadratic nonlinearities must be controlled. An elegant normal form transformation for the pde is used to control these quadratic nonlinearities by transforming them into cubics. Remark 4. The shooting arguments make use of two distinct time scales in the problem - a slow scale for motion near the plane He, and a fast scale for motion away from Fie. These arguments are organized into two measurements, associated with these two time scales. This organization of the argument follows some earlier work [ 145], in which the pde was truncated to a four dimension model problem. A particularly clear description of these finite dimensional arguments may be found in [ 111 ]. Remark 5. Intuitively (see Figure 7), the persistent homoclinic orbit will "leave the saddle point Q, creep slowly near the plane 17e along the unperturbed circle of fixed points So~ to a location near the "take-off" angle, rapidly fly away and return along a global orbit which is close to one of the integrable homoclinic orbits q~o(t; y). Upon its return near He, it will slowly creep back to the saddle Q, again near the circle S~o". Remark 6. The first measurement determines the intersection of the unstable manifold W u (Q) with a persistent center-stable manifold W Cs, which is codim 1. This is accom-
Fig. 7. Schematic Diagram of the Homoclinic Orbit.
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plished with the Melnikov function MH -- MH(YT), viewed as a function of the "take-off" angle VT. This Melnikov function measures the distance between these manifolds,
d{w
(Q), w
}-
Remark 7. Within the persistent W cs, the stable manifold W s (Q) is codim 1. Thus, once the orbit is known to reside in W cs , a single measurement can ensure that it returns to the saddle Q. This "second measurement" is accomplished using the pendulum energy (4.9) and the fact that the stable manifold W s (Q) is tall enough, as established by the normal form transformation. Remark 8. Once the persistence of one homoclinic orbit is established, the symmetry x --+ x 4- yr shows the persistence of a second homoclinic orbit, and the ingredients for a "centeredge" symbol dynamics are in place.
4.3. Chaotic behavior The simplest chaotic behavior which was observed in the numerical experiments for the perturbed NLS equation consists of a single solitary spatial excitation which jumps, irregularly in time, between the two distinct spatial locations at x -- 0 and x = yr. These numerical experiments, together with the persistence of a symmetric pair of homoclinic orbits, suggests a "symbol dynamics" explanation of this phenomena. 4.3.1. Symbol dynamics. More precisely, the term "symbol dynamics" refers to the existence of an invariant set in the phase space which is topologically equivalent to a set of all 2-symbol valued sequences. In our setting, these sequences would take the values of C (center) or E (edge), and the dynamics would be represented as a shift on this sequence space. As such, the dynamics, when restricted to the invariant set, is as random as a sequence of "coin tosses". In finite (usually very small) dimensions, the existence of such an invariant set is established by constructing a "Smale horseshoe" [ 171,153,85,191 ]. Such constructions have been carried out for orbits homoclinic to the saddle Q for the four-dimensional truncation [145], for a (2N + 2)-dimensional truncation in [132], and most recently for an infinite dimensional model of the pde [129]. Symbol dynamics is very appealing because it demonstrates the existence of chaotic motions which last for all time. However, it has some drawbacks: First, it occurs on a very small set in phase space, which is not shown to be (and is likely not) a stable set. As such, this type of chaos may not be observable. Moreover, the behavior depends on parameters in a bifurcation fashion. Often the parameter values required to show the existence of the horseshoe are very far from the values of the parameters at which chaotic behavior is observed in numerical experiments. (For example, in our analytical results [ 131 ], an additional dissipation/3 > 0 is required which satisfies a codim 1 constraint. However, in the numerical experiments [ 149], chaotic behavior is observed for/3 = 0, over a range of ot values.) Finally, the construction of the horseshoe is almost always performed for generic
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abstract models, rather than for a fixed specific dynamical system. To us, this generic situation seems to be a severe limitation of the practice of the m e t h o d - and particularly so for the NLS pde with its singular, two-time scale, homoclinic orbits. 4.3.2. Long complex transients. Recently, Hailer has been developing an alternative perspective, which he has applied to finite dimensional discretizations of the perturbed NLS equation [88,87], and to the pde [86]. In his work, using very similar geometric perturbation methods, he constructs a large class of heteroclinic orbits from the saddle Q to (for example) the sink P. These orbits have complex patterns of center-edge jumps, finite in number. While only transient behavior, the length of the transients is arbitrarily long. In any case, this set of heteroclinic orbits certainly demonstrates very complicated dynamics which depends sensitively upon initial conditions. Moreover, these orbits are associated with a "mixing and entangling" of the unstable manifolds. And, as the second measurement is not required to force the orbit to return to the saddle Q, these heteroclinic orbits exist for a full open set of external parameter values, without any codim 1 restriction.
4.4. Very recent work Our proof [ 131 ] of persistence of homoclinic orbits for dissipative, driven perturbations of NLS is very geometric. While beautiful, this geometric framework can be cumbersome and somewhat tedious. We continue to develop methods which rely upon geometric intuition, but which implement the actual calculations "more mechanistically"- within an integral equation framework, together with natural pde estimates. In [ 148] we prove the persistence of an orbit homoclinic to an isolated unstable fixed point for a nonlinear Klein-Gordon equation (a simpler situation than the orbit homoclinic to a circle of fixed points treated in [ 131 ]) by replacing Melnikov methods with a Lyapunov-Schmidt framework for the pde, together with pde scattering theory. Normal forms, while beautiful when they work as in the NLS case, often depend upon conditions which are extremely difficult to verify- as, for example, in the case of the persistence of an orbit homoclinic to a periodic solution in the sine-Gordon setting. Recently Shatah and Zeng [ 168] have used integral equation estimates to replace the normal form argument. They also have extended our NLS results [ 131 ] to include unbounded dissipative perturbations such as diffusion. They accomplished this extension by replacing our "fiber" representation of the stable manifold with long-time, integral equation estimates. Such improvements in the methods, while technical, are essential for the development of general procedures to establish qualitative results, valid globally in time, for p d e ' s - such as the persistence of homoclinic orbits for pde's. Another set of related questions concerns persistent tori, and persistence of associated periodic and quasi-periodic solutions, for Hamiltonian perturbations of integrable systems. This well studied "KAM" behavior in finite dimensional dynamical systems has recently been extended to infinite dimensional pde settings. While we will not describe these extensions in this review, we do note several representative references: [ 114,115,40,20-22,25, 13,17,48,185,54]. An important example is the persistence of the sine-Gordon "breather". The sine-Gordon equation on the whole line ( - c ~ < x < +cx~) has exact solutions which are periodic in
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time, exponentially localized in space, and which can be viewed as nonlinear bound states which consist of two solitons. The question of the survival of such temporally periodic solutions to small Hamiltonian perturbations is known as the "persistence of the breather". For perturbations which do not depend upon derivatives of the field, the breather does not persist [166,167,14]. Rather it decays extremely slowly, generating radiation as it decays at a rate which is exponentially small in the perturbation parameter. The work of Segur and Kruskal [ 166,167] uses formal "asymptotics beyond all orders" [ 165] to capture this decay rate, while that of [ 14] proves that the breather does not persist with mathematical arguments which begin from formulas of "soliton perturbation theory" [ 146]. This persistence problem provides one example of the important interactions between solitary waves and radiation in nonintegrable situations. (See [173,174,30] for others.)
5. Spatiotemporal chaos 5.1. Intuition In Section 4, we have discussed the existence and nature of temporal chaos which consists of spatially coherent localized waves which dance chaotically in time. As Figure 4 clearly indicates, these waves are very regular in space. Their time series at location x, {q (x, t) Yt }, appears to be statistically well correlated to the time series at location y =/:x, {q(y, t) Vt}. On the other hand, waves of dispersive turbulence should behave chaotically in both space and time. At least the time series {q(x, t) Vt} and {q(y, t) u should become statistically independent as the distance from x to y increases. Intuitively, this independence might be achieved by increasing the size e of the spatial domain. The numerical data shown in Section 4 was for small spatial domains, with only one instability and only one solitary wave under even, periodic boundary conditions. Recall that the number of instabilities, and hence the number of solitary waves present in the spatial domain, scales linearly with e. Moreover, with an increasing number of linearly unstable modes, there is, correspondingly, an increasingly large number of distinct classes of spatial excitations in the form many types of quasi-solitons- standing waves, waves traveling to the left and right, bound states which are quasiperiodic in time, etc. Therefore, increasing ~ will place more, and more complex, solitary wave structures into the spatial domain, and should decorrelate in space. Moreover, relaxing even symmetry enlarges the number of spatial locations at which these solitary waves can reside (from a discrete set to the continuum). (See [ 1] for fascinating effects which result from relaxing even symmetry.) Indeed, this decorrelation is seen in our NLS numerical experiments [35,34], provided the constraint of even symmetry is removed. And, similar phenomena occur much more widely than just for near-integrable waves. For example, a similar scenario occurs in studies of purely spatially chaotic, stationary waves [7,80], for which the temporal access of all of these stationary states remains to be fully addressed [41 ]. Specifically, we will describe spatiotemporal chaos for the driven, damped NLS equation,
iqt + qxx + 2lql2q = -iotq +/-'e i(~215
(5.1)
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Fig. 8. Evolution of system (5.1) with c~= 0.004, F = 0.144, 09- 1. The initial condition q = c + eexp(i27rx/e), c = 0.8, e = 2 • 10-5 . Plotted here is ]q(x, t)l. Left panel: Temporal chaos in the presence of one linearly unstable mode, e = 6.4; Right panel: Spatiotemporal chaos in the presence of two linearly unstable modes, ~ = 9.6.
with periodic boundary conditions,
q(x + g,) = q(x),
(5.2)
where ~ is the system length, and co and y are the driving frequency and phase, respectively. The damping coefficient ot and the driving strength F will be small. A natural question: Given a temporally chaotic solution of the perturbed NLS equation, how large a spatial domain, or how many instabilities, is required for effective decorrelation in space? An example with only one instability is shown in Figure 8 (left panel), while one with two instabilities in Figure 8 (right panel). Clearly, the two figures display drastically different spatial patterns. Before investigating such questions further, we need first to formulate a precise definition of the concept of spatiotemporal chaos.
5.2. A definition of spatiotemporal chaos There have been many definitions proposed to capture the essence of spatiotemporal chaos [41 ]. We prefer a "working definition" which includes two points: (i) A temporally chaotic wave q(x, t), (ii) for which the time series {q(x, t) gt} and {q(y, t) Yt} become statistically independent as the distance from x to y increases. For a definition, we must make precise the meanings of "temporal chaos" and "statistical independence". For temporal chaos we will accept any common definition, such as a bounded attractor with positive Lyapunov exponents. Statistical independence is often estimated through the decay of the two-point correlation function:
C(x - y) -~ lim T--,~T-To
[(q(x, t ) -
(q))(~(y, t) - ( ~ ) ) ] dt,
(5.3)
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where (.) denotes the temporal mean, and where we have assumed translational invariance of the system. However, the vanishing of the two-point correlation function is only a necessary condition for statistical independence; thus, we prefer to base the definition upon
mutual information. For two stochastic variables U and V, with probability density functions p(u) and p(v), respectively, and with joint probability density function p(u, v), the mutual information between these two variables U and V is defined as [ 16] 2-(U, V) = f du dv p(u, v) log
p(u, v) p(u)p(v)
(5.4)
In this application of spatiotemporal chaos, the probability distributions will be generated by the chaotic time series:
px(q)" {q(x, t) Vt} py(q): {q(y, t) Vt} Px,,.(q, r): {[q(x, t), r(y, t)] Yt}, where r(y, t) = q(y, t). Intuitively, px(q) dq is the fraction of time that q(x, .) ~ (q, q + dq), etc. Thus, we define the mutual information between points x and y by
Z(x, y) --
f
dq dr px.~.(q, r) log
px.~.(q,r) px(q)pr(r)
(5.5)
In terms of this mutual information between spatial points, we arrive at our WORKING DEFINITION. A wave q (x, t) is spatiotemporally chaotic if (1) q(x, t) is a temporally chaotic orbit (for example, as characterized by positive Lyapunov exponents); (2) its mutual information between two spatial points, Z(x, y), decays exponentially for large separatons, i.e., as Ix - Yl >> 0. It is well known that temporal chaos signifies a loss of information in time. (This loss of temporal information can be quantified by a positive Kolmogorov-Sinai entropy, which in turn can be estimated by the sum of positive Lyapunov exponents.) It is our view that a key feature of spatial chaos is a similar loss of information, but over space. Mutual information provides a natural measure. First, mutual information is closely related to entropy [ 16]: 2-(x, y) = R ( x ) + ~ ( y ) - R ( x , y),
where H(.) denotes the entropy. Here, H(x) is the entropy of q (x, t) at the space point x, and H(x, y) is the total entropy of q (x, t) and q (y, t) between the two spatial points. This relation shows that the mutual information Z(q(x, t), q(y, t)) measures the shared information between the two spatial points (x, y).
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Mutual information more faithfully captures the notion of statistical independence than does the two-point correlation, since the vanishing of mutual information is a necessary and sufficient condition for statistical independence. (The factorization of the complete infinite hierarchy of correlation functions to all orders is required for statistical independence - not just the factorization of two point correlations.) In addition, unlike the correlation functions, mutual information is invariant under invertible coordinate transformations. Thus, it provides an intrinsic description of the information propagated under the chaotic dynamics. Unfortunately, the numerical computation of mutual information is far more involved than that of correlation functions. However, we stress the conceptual advantage of mutual information over correlation functions since it renders a unified picture of chaos in time and space - spatiotemporal chaos giving rise to a loss of information in both time and space.
5.3. Information propagation in linear stochastic dynamics Before turning to the discussion of mutual information for chaotic NLS waves, we develop some intuition about the behavior of mutual information for several distinct classes of linear waves. In particular, we describe examples which illustrate very distinct behavior for the propagation of information in space for diffusive, wave, and dispersive systems. We need the mutual information between two random variables (X, Y), each individually normal with means m x, my and variances a 2 and %,, 2 respectively, and whose joint probability density is Gaussian 1 2zrcrxcrv(1 - p2)1/2
p ( x , y) -
{ • exp
1
mx)(y
2(1 - p2)
Ox
Ox
fly
Here p is the correlation coefficient, i.e., p = Cov(x, y)/(Crx~v). In this case, definition (5.4) becomes
Zf (X, Y) --
_
F
d x d y p ( x , y) oo
__1 ln(1-p
2
{
- - x ln(1 - p
/02. /92
m
l,
2
o. x
(5.6)
).
First, consider diffusion dynamics: ut - Duxx = f ( t ) g ( x ) ,
[(xmx2 1
- o o < x < +oo,
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where f (t) is a Gaussian white noise with zero mean and a-correlation: ( f (t)) = O,
(s(,)/(,'))
=
where (..-) denotes an average over noise. As any linear transformation of a Gaussian process remains a Gaussian process, the solution t
u(x, t) --
L
[
1
x2 ]
~/4zc D(t - t') exp - 4 D ( t
-t')
f (t') dt'
is a Gaussian process, whose correlation can be easily written as
(u(x, , ) u ( x ' , , ) ) -
-4rc----D
Ei( 't -t
'
where Ei is the exponential-integral function Ei(z) -- -
f
OG,
s - l e -s ds
for z < 0,
and where X
@ X t2
~-\ g-6 )" Therefore, according to Equation (5.6), the mutual information between u(x, t) and u(x', t) in the case of diffusion is
z(x,x')- - ~ 1l n ( 1--
Ei2(-~/t) IEi(-~l/t)Ei(-~2/t)l
) '
in which ~1 = x 2 / ( 2 D ) and ~2 = x ' 2 / ( 2 D ) . For a fixed time t and a fixed x' 5~ 0, we have C (x', t)
(5.7)
x
for large x, i.e., x >> ( 2 D t ) 1/2 and Ixl >> [x'l. Here C ( x ' , t) is a positive constant depending on x' and t. Equation (5.7) shows that, in the case o f diffusion, the mutual information decays with a p o w e r law in space. Now we contrast this result with that for wave dynamics: ut - ux = f (t)6(x),
- e ~ < x < +e~,
where f (t) is again Gaussian white noise. Since the solution is u(x, t) -- f (t + x),
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for x and t within the "light cone" such that - t ~< x ~< 0, we have
-x'). Thus, in the wave case, the mutual information between u(x, t) and u(x', t) for x and t in the light cone is
z(x,x,)_{o,~ , x#x', x = x'.
(5.8)
(Note that the correlation coefficient p = 1 for x = x' by definition.) Therefore, in the wave case, spatially distinct points do not share any information when driven by Gaussian white noise which is 6 correlated in time. If the noise is Gaussian but with a finite correlation function in time,
It-r t'l) '
(f(t)f(t')}-~,exp
(5.9)
then the mutual information between u(x, t) and u(x', t) becomes
Z(x,x')---~ln
1
[l_exp(_21x-x'l
r
)]"
(5.10)
At large distances, Ix - x'l >> r / 2 , the mutual information decays exponentially in space, i.e.,
Z ( x , x ' ) ~,exp - 2 I x -r x ' l ) "
(5.11)
Note that the equal-time two-point correlation function for the wave is
C ( x , x ' ) - (u(x, t)u(x', t)} ~ e x p ( - I x - x'l
(5.12)
Therefore, in this case, the lengthscale of the spatial decay of the correlation function is two times that of the mutual information. Numerical simulations have observed values of this ratio which are close to 2, even for nonlinear systems [ 192,159]. For linear Schr6dinger dispersive waves, i.e.,
iut + Uxx = f (t)6(x),
- o o < x < +c~,
where f ( t ) is real Gaussian white noise, it can also be shown that the mutual information between u(x, t) and u(x', t) is
Z(x,x,)_ { o,oc, xx#x' -- x". That is, the information is not shared between any spatially distinct points.
(5.13)
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These discussions show clearly that the propagation of information in space depends distinctly on the class of linear pde. Moreover, for linear Gaussian processes, these calculations illustrate the sufficiency of two-point correlations to compute mutual informations. In contrast, for the information propagation in the situation of spatially extended deterministic nonlinear dynamics, mutual information, in general, requires a full knowledge of joint probability distributions - and not just the two-point correlation functions [ 183].
5.4. Numerical measurements of spatiotemporal chaos for NLS waves Now we return to chaotic NLS waves (5.1), and use mutual information to establish the existence of spatiotemporal chaos. First, we calculate numerically the spatial correlation function C(x) (Equation (5.3)). Figure 9 (left panel) shows the dependence of the correlation function C (x) on the system length. For g -- 6.4, which corresponds to the one linearly unstable mode, clearly, the whole system is correlated. This is intuitively consistent with what one would conclude by observing the spatial structures of Figure 8, since, for the most times of the evolution, there is only one quasi-soliton in space. When the system size is increased so that there are higher numbers of linearly unstable modes present, Figure 9 (left panel) shows that their correlation functions rapidly vanish. Therefore, the system becomes increasingly decorrelated, indicating an onset of spatiotemporal chaos. As shown in the inset of Figure 9 (left panel), the correlation at the half system length as a function of displays a clear transition around the threshold above which the second linearly unstable mode enters (see Equation (3.12)). According to our definition of spatiotemporal chaos, we use mutual information to further corroborate the preceding results. Figure 9 (right panel) summarizes the mutual information as a function of the distance x between any two points in space for both one and two linearly unstable modes, which corresponds to the cases in the left and right panel of Figure 8, respectively. For one linearly unstable mode the mutual information remains nonzero
lO
\
1.4
~
1.2 1.0 S"
~
,.
o.o
/ ~ _ a - _
10~ , ',, ',,
~
0.8
L:9.6 10"
'
•
5
~:
0.6
\
,
0.4 0.2
',, \ / !
"
~~\"~'X~L ....
'\\
lO-~
00
\. L=9.6
0.0 -0.2
"
0.5
1.0 x
1.5
2.0
L=6.4
""'--...~ ......
0.0
011
0.2
0.3 x/L
0.4
0.5
0.0.000
O.10
0.20
0.30
0.40
0.50
x/L
Fig. 9. Numerical measurements. Left panel: Dependence of the correlation C(x) on the system size L. Inset: Transition of C(L/2) around L t h - - 2rr/c (dashed line) (cf. Equation (3.12)). Right panel: Mutual information Z(x). Fine line: one linearly unstable mode; Dotted line: two linearly unstable modes, as also shown in the inset on the linear-log scale (the straight line is a fit to an exponential form).
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across the system, signifying no sufficient loss of information over the whole system, while it vanishes rapidly for the case of two linearly unstable mode. It can be further determined that this decay is exponential as shown in the inset of Figure 9 (right panel); that is,
(x)
2-(x) --+ exp -~-
for large x
(5.14)
with a decay length ~ ~ 0.30. As solitons are phase-locked to the external driver, we anticipate that the driving frequency co controls this decay length, i.e., the soliton's frequency determines its spatial width, the coherence length in space. These results establish that spatiotemporal chaos exists for NLS waves, with the transition from temporal chaos to spatiotemporal chaos occurring at the system size at which a second instability arises. Only two instabilities seem to be required. Spatiotemporal chaos in such small systems is somewhat unexpected, as the prevalent belief in the physical literature requires very large systems with many unstable modes [41,89,82,57,58,170,137,200, 172,38,112,91,176,39]. (See, however, the recent work [80].) Although only two instabilities are required for spatiotemporal chaos, our most recent results indicate that many (i.e., spatial period very large) are required to ensure the coarsegrained field behaves statistically (and universally) as a Gaussian process.
6. Descriptions of the chaotic state Given a chaotic state, one seeks ways to describe and to understand it. For temporal chaos, dynamical systems theory has provided a framework and some concepts, including: "strange" attractors, horseshoes and symbol dynamics, Lyapunov exponents, different dimensions, and universal routes to chaos - as well as more statistical descriptions of the attractor, including invariant measures and entropy. The application of some of these concepts to temporally chaotic dispersive waves is described in the survey [ 149]. For spatiotemporal chaos much less is known, and we believe that statistical representation will be essential for its description. For waves which occur in nature, such as waves on the surface of the ocean, statistical behavior and properties in the mean become far more important than individual trajectories. Wave spectra are observable, and modelling these with deterministic initial-boundary problems seems unnatural and would be irrelevant. Spatiotemporal chaotic waves call for statistical descriptions. In this section, we briefly describe three such statistical theories- (i) invariant measures of equilibrium statistical mechanics, (ii) weak-turbulence theories, and (iii) effective stochastic dynamics.
6.1. Equilibrium statistical mechanics Nonlinear dispersive waves are frequently related to conservative mechanical systems. The Toda lattice and the sine-Gordon equation (as a continuum limit of coupled pendula) provide two examples. Equilibrium statistical mechanics is the traditional description of conservative mechanical systems with a large number of degrees of freedom; hence, it provides a natural starting point for a statistical description of conservative waves.
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We view the 1-D NLS equation under periodic boundary conditions,
iqt -- qxx - 2g(qO)q, q(x, t) = q(x + g., t), as a Hamiltonian system
3H
iqt= 6q' with Hamiltonian
H ( q ) = fo e [Iqx[ 2 -k- glql 4] dx. Statistical mechanics of NLS is the study of the Gibbs measure based upon this Hamiltonian. This measure, on the space of continuous functions, can be written formally in terms of the Hamiltonian H as
1
exp{-/~H[u(.), v(.)]}Du(.)Dv(.),
(6.1)
where q(x) = u(x) + iv(x), and where the normalization constant (partition function) Z is defined as
Z -- f c
(0.~)
exp{-~H[u(.), v(-)] } Du(.)Dr(.).
Here C(0, ~) denotes the space of continuous, periodic functions in which [u(x), v(x)] reside, and the positive parameter fl denotes inverse temperature. For those not familiar with function space integrals, reference [ 102] provides an intuitive introduction which emphasizes a view of the integral over functions as a "sum over paths", and which makes concrete the notation Du(.), etc. In the defocusing (g > 0) case, it is relatively easy to give precise meaning to these formal expressions by writing
1
exp{-/~H[u(.), v(.)] } Du(.)Dr(.) --
-lexp -fig Z
{ fo
_-- _1 exp - f i g Z
[u 2 +
v2]2dx exp - f l
[u 2 + v 2 2 dx
j
[u 2 + v2]dx
Dwu(.) Dwv(.),
where Dwu(.) Dwv(.) denotes unnormalized Wiener measure:
{Jo
Du, u(.) Du, v(.) =exp - f l
}
[u 2 + V2x]dx Du(.)Dv(.).
Du(.)Dv(.) (6.2)
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With this observation, Wiener measure can be used to give a precise mathematical definition of the Gibbs measure for the defocusing case [19,23,141-143]. Wiener measure is supported on functions which are continuous, but no-where differentiable. As such, these functions are "very rough"; for example, the energy space H 1 has Wiener measure zero. For the Gibbs measure to be invariant, the NLS equation must be well-posed for such rough data. Clearly, energy arguments will not work for such rough data. Resolving these existence issues requires delicate and interesting mathematical arguments [ 19,23,141-143], which establish that the Gibbs measure exists and is invariant for the defocusing NLS case. The focusing (g < 0) case is more subtle, as the formal expressions show. (Note, for g < 0 the integrand (6.2) is unbounded.) In one-dimension, control can be achieved by constraining with the L 2 norm (which is also invariant). The goal of an equilibrium statistical mechanics of waves is to use these invariant measures to extract statistical properties of typical wave configurations. Rose, Leibowitz, and Speer introduced these NLS measures, studied them both numerically and in "mean field", and posed some fascinating problems [127,128]. In particular, they conjectured a phase transition in the focusing case which involves solitons vs radiation- at high temperature (small/3), the typical configuration would consist in radiation, while at low temperature, it would consist in a gas of solitons. While recent evidence seems against this conjecture, the extraction of qualitative information about the statistical ensemble of waves from the Gibbs measure remains open mathematically. (There is a related calculation for the discrete Toda lattice which estimates the expected number of solitons as a function of the temperature, and which agrees well with numerical observations [155,154,138,33,179].) There are many fascinating issues, including: the thermodynamic (~ ~ cx~) limit, together with the possibility of the coupling constant g --+ c~; the extraction of spectra and other mean properties of the waves from the measure; the use of the measure or its moments to produce effective integration schemes, constrained by partial data [37]; "fluctuationdissipation theorems" for ensembles of waves; macroscopic transport [73]; the application of these ideas to vortex filaments of fluid mechanics [ 134] (which can be described by NLS and its perturbations [ 103,104,107,105,106]). Although these issues are fascinating mathematically, the fact that the measure is concentrated on rough functions remains troublesome physically. Typically, dissipation dominates at small scales - exactly where this rough spatial behavior appears. And this description of waves as a conservative Hamiltonian system neglects dissipation. Descriptions which focus upon steady fluxes of excitations between the different spatial scales, rather than upon equilibrium behavior, may be more relevant for ensembles of nonlinear waves. One such description is "weak turbulence theory".
6.2. W e a k - t u r b u l e n c e t h e o r i e s In order to understand dynamics of spatially extended, nonlinear wave systems, an important issue one must first address is the identification of fundamental excitations. In an appropriate coordinate system the fundamental excitations often acquire a very simple representation, such as a soliton in the nonlinear spectral representation, which is far simpler
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and more compact than its Fourier (plane wave) representation. Conceptually, these natural representations often allow us to capture the main dynamics of the system. The small residual interactions amongst the fundamental excitations can be then treated perturbatively. The theoretical power, as we demonstrated in preceding chapters, of the spectral representation toward understanding temporal chaos precisely lies in the fact that solitonic excitations and their interactions are the most important features in this dynamics. In this section we present another important theoretical formalism for nonlinear phenome n a - namely, weak-turbulence theories. The dynamical emphasis of this formalism is resonant wave-wave interactions. One origin of this formalism was a description of nonlinear phenomena in plasmas [ 198,197,196], such as the processes of modulational instabilities, decay instabilities, and wave couplings. It turns out that a Hamiltonian formalism, together with normal form transformations, provides a natural language for weak-turbulence, in which dissipative effects can be taken into account as small corrections. The waves described by weak-turbulence must be of small amplitude; and the weak-turbulence formalism fails to capture strongly nonlinear effects such as wave collapse and self-focusing. This is to be expected since these nonlinear phenomena involve a different kind of coherent degrees of freedom than simple resonant wave interactions. Weak-turbulence theories provide a statistical description for the kinetic evolution of correlation functions which describe wave spectra. In the derivation of these kinetic equations, a random phase approximation (i.e., a Gaussian assumption), as well as some technical assumptions, are invoked for the interacting waves, resulting in a certain closure scheme for the weak-turbulence description of the dynamics. These are strong assumptions which are difficult to verify, and often are not valid. Therefore, the applicability of weak-turbulence closures should be carefully examined. We will describe an explicit toy model which was introduced [ 136] to illustrate the hazards of a blind application of the weak-turbulence formalism. We will also describe a heuristic closure scheme [ 136] which provides an accurate representation of a wave spectrum observed numerically for this model problem. 6.2.1. Formalism. If there is only one type of wave dispersion co(k) present in a nonlinear medium, one can describe the waves in the absence of dissipation by the complex amplitude ak satisfying the Hamiltonian system
.Oak Ot
1
~
=
6H ~Dk
.
(6.3)
We consider Hamiltonians of the form H -- H0 + Hint,
(6.4)
where
- f co(k)akDkdk
(6.5)
is the Hamiltonian of the linearized problem, and Hint is the perturbation describing the interaction amongst those degrees of freedom represented by ak. Generally, Hint can be
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expressed in terms of power series in ak, and ilk, such as
Hint - -
f
(Pkk,k2{tkak, ak2 + /3kk,k2akik,~k2)6(k -- kl - k 2 ) d k d k l dk2
+ f(Qkk,k2akak, ak2 + Okk,k2akak, ik2)6(k + kl + k 2 ) d k d k l dk2 +
f
Rkk~kzk3akak~ak2ak36(k + kl -- k2 - k 3 ) d k d k l dkzdk3.
(6.6)
The dispersion co(k) determines the nature of wave interaction and its resulting turbulence properties. For example, if the following condition holds, co(k) = co(kl) + co(k2), (6.7) k = kl + k 2 , for some k, the wave interaction leads to the resonant interaction of the waves a k l a n d ak2 into ak~+k2- This situation is called three-wave resonance. If (i) Equation (6.7) does not have solutions, and if (ii) the following condition holds instead, co(kl) + co(k2) = co(k3) + co(k4), (6.8) kl + k 2 = k3 q-k4, then the four-wave resonance is responsible for the main energy transfer between weak dispersive waves. In this instance, it can be easily shown that a normal form transformation will place the Hamiltonian (6.4) in the form
.-f
f co(k )ak{tk dk + ] Skk~k2k3akak~ak2ak36(k + kl - k2 - k 3 ) d k d k l dk2 dk3. ,1
(6.9)
This is the general Hamiltonian system with four-wave resonances. Clearly, the "particle" number
N-fnkdk-fn~do~,
(6.10)
is conserved, where nk = [ak [2 and no, = nk dk/dco. In addition, the kinetic energy
E- f o n dk- f onodo is an important quantity.
(6.11)
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6.2.2. Direct and inverse cascades. These two conserved quantities under the four-wave resonance have direct implication on the flux of energy and wave number in the co space, when the system is forced at some wave numbers and damped at others. This can be easily seen from a global balance of "particles" and energy. Consider an idealized situation in which N particles are being created per unit time at frequency co, and N_ and N+ particles are being removed at frequencies co_ and co+. In a steady state, conservation of particles and energy leads to N--
N_ + N+,
coN -- co_N_ + co+N+.
Solving for N_ and N+, we have N_ --
N (co+ - c o ) co+
N+ =
N(co - co_) co+
,
(6.12)
.
(6.13)
-- co_
-- co_
Since N_, N+ > 0, co has to lie between co_ and co+. Without loss of generality, we choose co_ < co < co+. As neither N_, N+ nor co_ N_, co+ N+ vanish, there are fluxes of particles and energy in both directions from co. If co_ is near zero, there will be almost no energy removal at the low frequencies, and the energy will flow upward from co to co+, resulting in an upward (direct) cascade of energy from the low frequencies to the high ones. If co+ is very large, Equation (6.13) shows that the number of particles removed at co+ will be very small, and the particles have to flow from co to co_, creating a downward (inverse) cascade of particle numbers. As a consequence, if the dissipation takes place only at frequencies near zero and at very high values, there is an "inertial" range in which the energy flows upward from its source to the sink at the high frequencies, whilst the particles flow downward from their source to the sink at the low frequencies. As we will see below, these cascades provide a physical basis for understanding (nonequilibrium) steady state solutions in weak-turbulence theories. 6.2.3. A simple model. To further illustrate detailed aspects (such as the closure issues and wave spectra) of weak-turbulence via four-wave resonances, we describe a model system introduced by Majda et al. [ 136]. The governing equation of the system is
io,
+ 10xl ,,.(110xl ,,4 l'lo,
(6.14)
The equation has two parameters ot > 0. For fl -- 0, a standard cubic nonlinearity results. The parameter/3 is introduced to control the nonlinearity. The parameter ot controls the dispersion relation co(k) - [ k [ ~, which, for o~ < 1, leads to resonance quartets in this one-dimensional model.
(6.15)
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The essence of weak turbulence theory is a statistical description of weakly nonlinear dispersive waves in terms of a closed, kinetic equation for certain two-point spectral functions. Starting with the equation of motion for system (6.14) in the Fourier space, and introducing Gaussian randomness through the initial conditions, one has
f
2im(akl ak2hk3~k)
nt(k,t) =
ikll~/4lk2l~/4lk3l~/4lkl~/4
6(kl + k2 - k3 - k)dkl dk2 dk3 (6.16)
for the two-point function
n(k, t) -(ak(t){tk(t)).
(6.17)
The evolution of the four-point functions depends on six-point functions. Under a Gaussian random phase approximation, and the assumption that 0
ot (ak,ak2{tk3{tk) ~ O, one obtains Im(aklak2~k3~k) ~ --27r •
n2r/3nk
-~- n l n 3 n k
-- nln2nk
-- nln2n3
[kl [fi/4lk2[fi/4[k3]fl/4lk[ ~/4
6 (O91 -+- 092 - - 093 - - O ) ) ,
(6.18)
where n2 = n(k2, t), etc. Using this closure condition, one can close Equation (6.16) to arrive at
f hk -- 4re
nln2n3nk ( 1 k 1 [klk2k3k[~/2 -~ n3 •
6(Wl
-~- 092 - - 093 - -
1 1) n2 nl co)6(kl + k 2 - k3 - k) dkl dk2 dk3.
(6.19)
Equation (6.19) is the kinetic equation for n(k, t). 6.2.4. Zakharov's solutions. time-independent solutions n(co) = c ,
For an angular averaged kinetic Equation (6.19), the trivial
(6.20)
and C
n(co) ------, O9
(6.21)
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correspond to equipartition of particle number and energy, respectively. Using a conformal mapping, Zakharov showed that the angular averaged kinetic equations often possess additional (Kolmogorov) power law solutions of the form [ 195] c
nK(0)) = ~ ,
(6.22)
(2)9/
for y ~ 0, or 1, which describes the spectra for the cascades in nonequilibrium situations. These solutions are intimately related to fluxes of particles and energy in 0) or k space as we discussed previously [195]. For system (6.14), it can be shown that for ot = 1/2
n x (0)) = c 0)4/3fi-5/3,
i.e.,
n/((k) = clkl 2/3fi-5/6
for the inverse cascade, and nK(0))
=C0)
4/3/~-2,
i.e.,
nK(k) = clkl 2/3r
for the direct cascade. 6.2.5. Numerical results for the model. In [ 136], numerical experiments were carried out for the direct cascade of energy from long waves to short waves. For ot = 1/2 and/3 = 1, the Kolmogorov spectrum from the weak-turbulence theory is n/~ ~ Ik1-1/3. However, numerically this spectrum was not observed. Instead, the numerical measurements yielded a much steeper spectrum n "-~ Ik[ -3/4 over large inertial ranges. Moreover, in contrast to the weak-turbulence prediction of the existence of a spectral bifurcation at a critical 13, the experiment displayed no spectral bifurcations. Careful postprocessing of the numerical simulations shows clearly that the Gaussian approximation is satisfied. Therefore, one would expect that the weak-turbulence theory should work. It appears that the failure of the weak-turbulence theory prediction for this one dimensional model can be traced to the breakdown of the closure condition (6.18). Using the insight derived from the numerical results, a new closure condition was proposed [136]:
Im(ak, ak2ctk3gtk4) ~ C
(nln2n3n4) 1/2
(6.23)
0)1 -Jr- 0)2 - - 0)3 - - 0)4
for the evolution of the two-point function n(k). The scaling of the Kolmogorov spectrum under this new closure is found to be in excellent agreement with numerical scalings for the model. In general, to find a good closure scheme is a difficult problem. As is demonstrated by our example, although weak-turbulence theories provide a systematic approach to the closure problem, the validity of the closure thus obtained still needs to be carefully tested for the applicability of the weak-turbulence theories. Finally it is worth noting that the weak-turbulence theory for system (6.14) is insensitive to the sign of the nonlinear term. Recalling that modulational instabilities and solitons crucially depend on this sign, one appreciates that weak turbulence, when valid, must be restricted to nonlinear waves of very small amplitudes.
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6.3. Effective stochastic dynamics In weak-turbulence theories, one uses two-point correlations (6.17) to characterize the spectra of wave-wave interactions over many scales. However, one may also be interested in a "macroscopic" description of the longest waves in the spatiotemporal chaotic system. The long waves in the deterministic chaotic system will be effectively stochastic. One anticipates that their dynamical evolution will be described by a stochastic equation in which the chaotic waves on intermediate spatial scales will act as both a "source of a random stirring force" on the longest waves, and a "sink for the dissipation" of the longest waves. That is, long-wave instabilities create chaotic shorter waves, which, in turn, act as an "active heat bath" which causes the random forcing and dissipation of the longest waves. In contrast to weak-turbulence theory (which can be viewed as a stochastic description of the active heat bath), "effective stochastic dynamics" depends critically on properties of the nonlinearity because it demands the presence of long-wave instabilities. For example focusing, rather than defocusing, nonlinearity is required. Recently, this issue of "non-equilibrium fluctuation-dissipation theorems" has received renewed interest in statistical physics, particularly in the connection between the hydrodynamic limit of the Kuramoto-Sivashinsky equation and the Burgers-KPZ universality class [200,193,172,38,32,100,93]. The formalism used to describe the coarse-grained effective stochastic dynamics is a natural extension to a dissipative system of the Zwanzig-Mori projection formalism for a Hamiltonian system in thermal equilibrium [182]. When applied to the Kuramoto-Sivashinsky model in the spatiotemporal chaotic regime, a noisy Burgers equation results as the effective long-range, large-time dynamics [200,193,172,38,32]. There are two questions: (i) Does an effective stochastic dynamics exist which provides a macroscopic description of long waves in a chaotic deterministic system? (ii) Can a closure theory be developed which derives the effective stochastic equations from the original deterministic system? Most of the work in the literature assumes an affirmative answer to the first question, and develops formal closure schemes to address the second. Often, these heuristic arguments are based on ideas from renormalization group methodology [76,193,194], and are very difficult to convert into precise asymptotic analyses. In this article, we address the first question with numerical experiments designed to validate some necessary conditions for the existence of an effective stochastic dynamics. In [35,34] we extend the methods of reference [200] to perturbed NLS equation (5.1), focusing upon which aspects of chaoticity are necessary for the validity of its effective stochastic dynamics. Specifically, is spatiotemporal chaos necessary or is temporal chaos sufficient for an effective stochastic dynamics? Surprisingly, we find that numerical tests of necessary conditions for an effective stochastic dynamics for the perturbed NLS equation require only temporal chaos, in contrast to the Kuramoto-Sivashinsky equation for which spatiotemporal chaos is believed to play a crucial role for the validity of the effective stochastic dynamics [200,193,172,38,32]. But effective stochastic dynamics fails for quasi-periodic behavior. -
647
The nonlinear Schr6dinger equation
The representation of perturbed NLS equation (5. l) in the Fourier space is 2 iak = (k 2 - i o t ) a k - --~ ~-~aqapDp+q_l( 4-
eFe/(~~
(6.24)
q,P
where
ak-
fo g
q ( x ) e ikx dx
with k = 27rm/g, m being an integer. The effective dynamics is concerned with the dynamics for ak in the long wavelength limit. In other words, the aim is to construct an effective dynamics for the macroscopic observable 1
(1(x) -- -~ Z
ake-ikx'
]k[
= (k
- yE
+ q,P
2 Z'
Fk (t) = --i (ol -- Otk )ak -- --~
aqap{tq+p-k
'
0+ (6.25)
where y~< denotes summation over all [q[, [p], ]q + p - k] < A and y~', a summation in which at least one of wavenumbers ]q[, [p], [q + p - k[ is larger than A. In this setting, Equation (6.25) can be viewed as the effective stochastic dynamics, provided we regard Fk (t) as a stochastic force and &k as an correction to dissipation and/or dispersion (e.g., a k-dependent Re 8k will represent an effective k-dependent damping while Im 8k an effective dispersion). If Fk(t) truly acts as an "external" stochastic force, it cannot depend on the solution q(s) in the past; i.e., for s < t. This "causality condition"
(Fk(t)fik(t -- s))t - - 0
for s > 0,
(6.26)
where (...)t is the time average over t, determines an expression for the effective dissipation parameter:
2i y ~ , (aq(t)ap(t)Dq+p_k(t)Dk(t 6~k -- ot -- --~ (ak(t)Dk(t -- s))t
- s))t
(6.27)
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Relation (6.27) implies explicit s-dependence, which we denote as otk (s). The existence of an effective stochastic dynamics requires s-independence- at least over a coarse-grained time scale. The numerical computation of the full dynamics (5.1) shows that Equation (6.27) is indeed s-independent for the perturbed NLS equation, even in the regime of only temporal chaos (see Figure 10 (left panel)). No spatiotemporal chaos is required. In view of the the usual belief that an effective stochastic dynamics requires spatiotemporal chaos, this result is rather surprising. Of course, as is expected, this s-independence also applies in the regime of spatiotemporal chaos. However, an effective stochastic description is not valid for quasiperiodic dynamics, since quasiperiodic temporal behavior introduces a long memory time. And indeed, this was borne out in the numerical simulations [34]. In the presence of temporal chaos alone, the numerical construction shows that the effective Re ~k gives rise to a renormalization of dissipation for the longest waves (k -- 0, 1). In general, Im ~k has the form of/3o + r2 k2 -Jr-fl4k 4, leading to an additional modification of the Schr6dinger dispersion co - k 2 to 09 ~ (1 -Jr- f l l ) k 2 -Jr- r2
k4,
(6.28)
with fll and r 2 determined from a numerical construction. In addition to this test that otk(s) is independent of s, another necessary criterion for stochastic dynamics is that the effective stochastic force should have no long-time correlation. This is satisfied in the perturbed NLS case: it has been (numerically) shown that (Fk(t)Pk(t + s))t decays rapidly, and can be regarded as no correlation over the coarsegrained time-scale (see Figure 10 (right panel)). The construction procedure only demands the causality (6.26), which only involves correlations for the same k. For k r k', ( F k ( t ) h k , ( t - s))t is left unconstrained by the determination of the effective dissipation. However, the numerical results also show that, for the Fk(t) constructed above, (Fk(t)Dk,(t -- s))t ~ 0 also holds for k ~ k'. The fulfillment of causality in this general form is indicative of a deep self-consistency, and goes toward the validation of the inter-
0.5
~,,
i
<
Rea
:
~ line
-2
0
10
o
~
20
30
-0.5
0
10
20
30
40
50
60
Fig. 10. Numerical validation of effective stochastic dynamics: the s-independence of otk (s) (left panel), and rapid decay of the force-force correlation (right panel) (note that ( F (t) P (t + s))t / (I F (t)12)t = 1 for s = 0). Time unit is normalized by 2n/o9.
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649
pretation of Fk(t) as an external stochastic forcing. Of course, whether a Gaussian white noise can be used to replace Fk(t) in the effective dynamics requires a further statistical characterization of higher orders of correlations in the stochastic forcing. Finally, we emphasize that a significant separation of scales lies in the heart of the validity of effective stochastic dynamics. And we note in passing that the numerical simulations show that effective stochastic dynamics also works well with even symmetry imposed. For example, for one linearly unstable mode, with or without even symmetry, the effective otk are the same within numerical errors. In summary, these numerical studies have clarified the nature of the chaoticity which is required for effective stochastic dynamics, i.e., temporal chaos seems sufficient for the validity of the effective equation for perturbed NLS, while the absence of chaos invalidates effective stochastic dynamics.
6.4. Nonlinear localization In this survey, we have described deterministic chaos, together with some discussion (in the section on weak-turbulence theory) of the stochastic behavior induced by random initial conditions. However, stochastic waves can also be generated by a random environment. In contrast to the decoherence effects caused by spatiotemporal chaos, the spatial disorder of a random environment can cause the waves to localize. This phenomena is particularly striking for linear waves - where it can convert conductors into insulators [2], and prevent sound from propagating [169]! For example, consider the very idealized model for an electron propagating in a m e t a l the one dimensional linear Schr6dinger equation of quantum mechanics: iqt = - q x x + g V ( x ) q ,
-~<x<+~.
Here V (x) is a random potential which models impurities in the metal. This problem can be completely understood by analyzing the time-independent spectral problem - q x x + g V (x)q = Eq. Following the original work of Anderson [2], it is now understood (with complete mathematical rigor) that the spectrum of this one-dimensional problem consists of only point spectrum with no continuous spectrum. (See, for example, [ 177].) Any amount of randomness converts a problem with only continuous spectrum (g = 0) into one with dense point spectrum (g --/=0)! All eigenfunctions in the random system are exponentially localized in space, since they are associated with the discrete eigenvalues in the point spectrum. As such, the extended generalized eigenfunctions of the continuous spectrum in the deterministic g = 0 case (which are associated with conduction) are all destroyed by the randomness, and replaced by localized eigenfunctions which do not conduct. Extended waves are localized by the random environment. While this problem is completely understood for linear waves, it is essentially open in the presence of nonlinearity. Focusing nonlinearity causes waves to localize into solitary
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waves, while defocusing causes nonlinear spreading. Numerical experiments show that the competition between these deterministic nonlinear processes and the linear localization caused by spatial disorder produces some interesting p h e n o m e n a - phenomena which, as yet, is not well understood analytically. For the nonlinear Schr6dinger equation,
iqt = - q x x + V(x)q + 2glql2q,
(6.29)
where V (x) is a random potential, some theoretical results [55,108] related to localization have been obtained using a time-harmonic ansatz,
q(x, t) = e x p ( - i k 2 t ) u , k2u = -Uxx + V(x)u -k- 2glulZu, to study the resulting time-independent nonlinear Schr6dinger equation. However, in the presence of nonlinearity, this nonlinear eigenvalue problem approach inherited from the linear theory may not be sufficient. These time-independent solutions may be dynamically unstable, and hence irrelevant for the description of long-time behavior. This is indeed the case [28]: For both the focusing and defocusing nonlinearities, the time harmonic solutions of the random NLS equation are often unstable. Furthermore, it has been demonstrated numerically [28] that the disordered NLS equation exhibits rather different dynamics, depending on whether the nonlinearity is focusing or defocusing. For the focusing case, the final attractor of the dynamics is a state which consists of interacting, highly localized solitary waves, with widths far narrower than the localization lengths of the corresponding linear system. For the defocusing case, in contrast, the system settles down to a nearly monochromatic state with a spatial profile which can be approximately described by 2glq(x)l
2 ~ k2 -
V(x).
This profile is slaved to the random potential and its form indicates a lack of localization. Finally we mention that similar issues arise in the discrete NLS equation
i ~ l , , - - J ( q , , + l + q n - 1 ) - coq,, + Vnqn + 2glqnl2qn,
(6.30)
in the presence of disorder V,,, where J and co are constants. Similar localization phenomena have been observed. In the discrete case, the localized states are intimately related to discrete breathers, which are ubiquitous, robust nonlinear excitations in discrete nonlinear systems [6]. For the focusing case, in the weak nonlinearity limit, the localization is still "Anderson-like". With increasing of nonlinearity, the excitations become highly localized and are controlled by the nucleation of discrete breathers. This scenario suggests the existence of a phase diagram in disorder-nonlinearity space describing a crossover between a disorder controlled attractor and a nonlinearity controlled attractor. Again, as in the continuous case, it has been shown that the effect of nonlinearity on localization depends sensitively on the class of nonlinearities: the nonlinearity enhances the localization in the focusing case, whilst suppressing the localization in the defocusing case.
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651
7. Asymptotic long-time behavior of NLS waves In final two sections we return to the completely integrable NLS equation, in order to exhibit the level of precision in asymptotic description of nonlinear waves which can be extracted from the inverse spectral representation. In this section we will describe the Riemann-Hilbert formulation of the inverse spectral problem for the defocusing NLS equation. Then we will use this formulation to establish long time asymptotics for the defocusing NLS equation. The results described here are due to Deift et al. [45]. A more detailed description of the analysis contained in [45] is presented in [53].
7.1. Statement of the Riemann-Hilbert problem Before defining the Riemann-Hilbert problem, we begin with an auxiliary matrix-valued function: Given a function r(Z) E S(]R), the Schwarz space of C ~ functions which decay faster than any power as IZI --+ c~, we build a matrix vx,~ ()~) via
Ux.t()0 = e -i(2tZ2-xZ)~ (
l -r(Z)
r(Z) ) e i(2tx2-xZ)cr3 1 - Ir()~)l 2 J
(7.1)
The goal of the Riemann-Hilbert problem is to determine the unique 2 x 2 matrix valued function m()~, x, t) satisfying (7.2)-(7.4) below: m is analytic in C \ R,
(7.2)
with continuous boundary values for Z 6 JR, m+(Z, x, t) = lim~omO~-+-ie, x, t), satisfying m+ (7,, x, t) = m_(Z, x, t)Vx,t(Z).
(7.3)
Finally, m possesses the following asymptotics for x and t fixed: m ~ I
as Z ~ oc.
(7.4)
The fact of the matter is that if r()0 is the reflection coefficient associated with qo(x) as defined in (3.8), then the solution q (x, t) to the defocusing NLS equation is obtained from the matrix m via
q ( x , t ) - " 2[Lz~lim Z ( I - - m ( Z , x , t ) ) ] l
2.
(7.5)
That is, if m possesses the asymptotic description m = I + m l/Z as Z --+ oc, then q (x, t) = --2(ml)12. REMARK. In this section we have replaced t in the defocusing NLS equation with - t , and the equation becomes
iqt + qxx - 21q leq -- O.
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D. Cai et al.
The above Riemann-Hilbert problem (7.2)-(7.4) is one formulation of the integral equations of inverse scattering theory, as mentioned in Section 3. Indeed, if we set !/J (X, x, t) = _ m ( X , x , t ) e i)~x~r3,
(7.6)
then it turns out that for X 6 C \ R, !/J is a fundamental matrix solution of differential Equation (3.3), normalized by the following two conditions: tile -ix)~cr3 ~
I,
tl-/e -ix)~r3
as x --+ + e c ,
(7.7)
remains bounded as x --+ - o c .
(7.8)
The jump relation (7.3) expresses the fact that while q/(X, x, t) satisfying (7.7)-(7.8) is defined a priori for Im X # 0 only, it turns out that q/(X, x, t) has continuous boundary values for ~. 6 R, ~ + ( X , x, t) = lim q/(X 4- ie, x, t).
e$0
Since q/+ and q~_ represent two fundamental matrix solutions of (3.3), they must be related, i.e., qJ+ (X, x, t) = ~ (X, x, t)v(X, t), for some jump matrix v which is independent of x. REMARK. If one starts with this jump relation, and then uses (7.6), one arrives at a jump relation for m, which appears a bit different than (7.3), because the time-dependence is not explicit. However, one can compute the evolution of the matrix v(X, t) explicitly (see, for example, [53]), and (7.3) can be derived in this fashion. REMARK. The connection between the Riemann-Hilbert problem (7.2)-(7.4) and a set of integral equations for inverse scattering theory is classical: it turns out that existence and uniqueness of the solution of a Riemann-Hilbert problem is equivalent to invertibility of an associated integral operator. This is explained in many papers; we refer the reader to [53], where the connection is made particularly clearly.
7.2. Long-time behavior In this subsection we will explain the Riemann-Hilbert approach to the problem of computing the long-time asymptotics of the solution to the defocusing NLS equation. The idea is to describe the solution to the Riemann-Hilbert problem (7.2)-(7.4) for t --+ co, and then use the reconstruction formula (7.5) to compute asymptotics for q (x, t). To avoid technical issues, we will assume for the remainder of this section that the reflection coefficient r 0 0 can be continued analytically to a strip containing the real axis. This is satisfied, for example, if qo(x) is analytic, with sufficient decay as x --+ cxz.
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653
From a calculational point of view, the basic idea behind the method is that if we have a Riemann-Hilbert problem which is simple:
n(Z) analytic in C \ Z, n+(Z)
-
-
n _ ( I + err(A)),
n(Z)--+I,
Zes
asZ--+oo,
where Z is some oriented contour, and err(Z) is uniformly small on Z' [0(71 ) in the L l (Z,) A L or (Z) norm, for example], then the solution to this Riemann-Hilbert problem can be obtained by solving an associated integral equation. It turns out that because the term err is uniformly small, this integral equation can be solved by Neumann series. Unraveling this connection between the Riemann-Hilbert problem and the integral equation, one arrives at an asymptotic expansion for the solution of the Riemann-Hilbert problem l In the present setting, rather than arriving at a "simple" Riemann-Hilbert (in powers of 7)" problem as described above, we will arrive at a simplified model Riemann-Hilbert problem which can be solved exactly ([45], see also [53]). Now we will explain how one arrives at a "simple" Riemann-Hilbert problem. The fundamental observation is that we have the factorization of the matrix Vx,t (Z)
( , Vx,t(Z) --
0)(, r e2,,0)
_r(Z)eZitO
1
0
1
'
itO = itO(Z, x, t) -- i t ( Z Z 2 - t -1 x Z ) -- it(ZZ 2 - 4ZoZ),
(7.9) (7.10)
x
Zo =
--.
(7.11)
4t
Now although Vx,t(X) possesses rapidly oscillating terms (as t --+ oo), we observe that for Z < Z0, the second factor on the right hand side in (7.9) can be analytically continued above the real axis, and R e { - 2 i t O } < 0, i.e., the rapidly oscillating term becomes exponentially decaying! Furthermore, the first factor on the right hand side of (7.9) can be analytically continued below the axis, where the oscillatory term again becomes exponentially decaying. (These properties of i tO can be seen by noting that if Z = u + i v, then Re i tO -- - 4 t v ( u - Zo).) We now indicate how one splits a part of the real axis into two contours, deformed above and below the real axis, in order to exploit the exponential decay indicated above. We begin by re-writing the jump relation (7.3), using the factorization (7.9), m+ - m _
1 - r ( Z ) e 2itO
Now using the analyticity discussed above, we may write this equation (for Re )v < Z0) as follows,
(m(o1
- - r - - ~ e -2itO
,
) ) (+
(m
1
r ,e2i,o ol ) )
D. Cai et al.
654 f
TL1-- I)2
7Zl--TI~
7Zl---Trz
(
Ao -r(,~)e 2i~~
?zl z T/~
____~____---Fig. 11. The definition of the matrix n I .
and so if we define n 1 via
n l -- m ( 10 -r()Oe-2it~
n l -- m
(
1
_r(~.)e2it 0
0) 1
for ~ above the axis, Re )~ < ~0,
for ~ below the axis, Re )~ < ~0,
then n l possess no jump across (-cx~,)~0). Since r and T can only be continued to a strip containing the real axis, we cannot make this definition globally, and so we define n l as shown in Figure 11. Now n l is analytic for ~ 6 C \ r l , where r l is shown in Figure 12. Observe that the jump across the real axis for ~ < ~0 has been removed by this factorization. Putting this all together, we have transformed the first Riemann-Hilbert problem for m, into a new Riemann-Hilbert problem, for n 1: n l
is analytic in C \ S l ,
(nl)+()~
x ~ t)--(nl)
--
()~ ~ x ~ t ) v (Xl,~ ( ) ~ )
(7.12) for )~ 6 Z1 (V(xl] is defined in Figure 12), (7.13)
nl -+ I as ~--~ cx~,
(7.14)
which is equivalent to the original problem: if we have a solution to the new problem, then by using Figure 11, we have a solution to the original problem. REMARK. The contour Zl is oriented as shown in Figure 12, and we use the convention that the plus side of an oriented contour lies to the left as one traverse the contour. The + ( - ) subscript in (7.13) denotes the boundary value taken from the + ( - ) side of the contour Z l . The second, and more fundamental, thing to observe is that now, for Re)~ < )~0, the off-diagonal entries in the jump matrices v ~1) are exponentially decaying, and so the jump matrices are exponentially close to I. Unfortunately, for )~ to the right of)~0, this procedure does not work immediately. Indeed, for Re ~ > )~0, and ~ below the real axis, e 2itO is exponentially growing, and for Re)~ > ~0,
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655
r(A)e -2it~ )
1 0
1 A0
(
1
0
Fig. 12. The new contour E 1, and jump matrices V (1)
and k above the real axis, e -2itO is also exponentially growing. However, we can switch the order of the factors that appear through a lower/upper factorization:
Vx,t(k) --
l-lr(X)12
0
1-1r(X)
1
0
r(~) e2itO 1 "
1 --Jr(X)] 2
--j_l,.(x)12
(7.15) Now for )~ to the right of )~0, the first term on the right hand side of (7.15) can be deformed below the real axis, and the off-diagonal entry becomes exponentially decaying, while the third term can be deformed above the real axis, and again the off-diagonal entry becomes exponentially decaying. (It turns out that since the reflection coefficient is analytic in a m
strip containing R, the quantity r(k) -- r()0, )~ 6 R, possesses an analytic continuation to a strip, as does 1 - Ir()012.) So now we define n2 using Figure 13.
n2 z 7tl /
1
~2
0
( 1-F;.Dt)I ~(~) = e2it~ 2A~ r
~
~
r
Z
~
Fig. 13. The definition of n2.
l
(
1 0
1_1~(),)12e 1
1
)
D. Cai et al.
656
1
r(A)e-2it~ )
0
1
(
1
~(~) e2~to -- l _ l r ( A ) l 2
0) 1
0 ) 0
(
1
1)
(
0
-rlA)e2it~
1
l_ir(X)12
0
1
Fig. 14. The contour z~72, and the jump matrices
1-
~(~)1 ~
(2)
Vx. t .
We thus arrive at the following Riemann-Hilbert problem for n2: n2
is analytic in C \
~v'2,
(2) (n2)+ ()~, x, t) -- (n2)- ()~, x, t)Vx, t ()~),
n2--+I
)~ E r2,
as )~ --+ cx~,
where the new contour r 2 is shown in Figure 14, along with the jump matrices Vx(2]. There remains a diagonal jump matrix, for )~ 6 0~0, cxz), which, it seems, cannot be deformed away. Even if we attempt to deform this term off of the axis, there is no hope to gain exponential decay, because this jump matrix has no t-dependence. This piece of the puzzle is handled by first solving the following scalar Riemann-Hilbert problem: find 6 analytic in C \ R such that
~+(z) 6--+ I,
{ ~6_(k)(1 -Ir(X)12), _ (z),
)~ > )~0, k <~0,
as)~--+o~.
It turns out that this problem can be solved by formula:
600--exp
[lf~l~ ~
0
s-)~
lds '
Now, using this, we define
n3()0--n2()0
0
0)
6 -1
"
(7.16)
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O) 1
~ ~ ) 1 2
(1
--'r(A)t52 e 2it0
o)
o
~(x) ~5_2e_2~to ) 1
1
Fig. 15. The contour ~3 and the jump matrices
(3)
Vx, t 9
One may verify directly that the matrix valued function n 3 solves the Riemann-Hilbert problem n3 is analytic in C \ Z3,
(n3)+ ()~, x, t) -- (n3)_(k, x, t) Ux,t(3)()~), n3--+I as )~ ---->00,
)~ 6 Z3,
Now we have finally arrived at a Riemann-Hilbert problem which is in the fortunate situation that away from one point, )~0, the jump matrices are uniformly close to I. The last step to arrive at a Riemann-Hilbert problem which is "simple" in the manner described above, is to isolate the local nature of the Riemann-Hilbert problem for n 3. To do this, we introduce the scaling transformation, ~'(zk) ~ (8t)1/2()~ -- kO), which is a map from C to C, sending Z3 to Z4, shown in Figure 16. This figure also shows the new jump matrices, v(3) (s
Remark 1. We have chosen the contour E3 so that for a small neighborhood of )~0, the contour consists of 4 straight lines emanating from )~0, forming angles of 7r/4 radians with the real axis. Because of this, on a large neighborhood of ~ = 0 (of size O(t 1/2)), the new contour Z4 consists of 4 straight lines forming angles of Jr/4 radians with the real axis.
D. Cai et al.
658
0
1
0
1
"()'(r 1-1r(A(r
282e 2it~
1
i_ i,,(,x(~))l 2 1
Fig. 16. The contour E 4, and jump matrices v (3) (~(~')).
Remark 2. For ~" on the contour E4, we have the following representation for 6(;L(~'))e itO"
(~()~(~)) e i t O -
~iv ~(8t)iv/2 e i ~2/4 e-2it Zo2e~ (Zo+C / (8t)l/2) ~iv ~ e i ~2/4 e-2it z2 e~ (Zo)e~ (Zo+C / (8t) 1/2)--K (kO) (8t)iv/2
=
6(o)6(~)
(7.17)
where
x(~.)--
1 2yri
1 log(1 v - 2---~-
fOG logO~ -
s) d l o g ( 1 - I r ( s ) 1 2 ) ,
0
-Ir(~o)l
2)
8(0) = eK(~~ -2it;~2
(8t)iv/2
(7.18)
'
8(1) _ ~ivei~ 2/4eX()~o+~/(St)
1/2)-x (k0).
We now m a k e one further transformation, by defining n4 (~')"
n4(~') --
0
(6(0)) -1
n3 (Z(~'))
0
6 (0)
(7.19)
(recall from the definition (7.18) that 6 (o) is a constant). The matrix n4 then solves the
The nonlinear Schr6dinger equation
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0)
1 0
(
~-I~(~(r 1
0
1
i_1~(~(r
1
~
1 )
((5())-2 1
Fig. 17. The contour 274, and the jump matrices v(4)(ff).
following Riemann-Hilbert problem: n4 is analytic in C \ Z4, .(4) ( n 4 ) + (~', x , t) - - ( n 4 ) - ( ~ ' , x , t)Vx, t (~),
n4--+I
as )~ --+ oc),
where Z74 and l) (4) a r e shown in Figure 17. Notice that v (4) is obtained from v(S)()~(~')) by replacing 6e it~ by 6 (l). The last step is to solve this final Riemann-Hilbert problem for n4. Observe that r()~(~')) - r()~o + (8t),/2). It turns out that one can show that the solution to this problem is well approximated by the solution to the problem obtained by replacing r()~(~')) with r()~o), and 6(1)0~(~')) with 6 (l) -- ~ive-i(2/4. This model problem can be solved through the use of parabolic cylinder functions. Thus the final problem for n4 can be solved asymptotically. We will omit these details, and refer the reader to [53]. We now assume that we have obtained n4 by this procedure. In summary, by explicit, invertible transformations, m --+ n l --+ n2 ~ n3 --+ n4, we have arrived at a simple problem whose solution can be approximated easily. We can unravel all of these transformations, and hence we have obtained an approximation for the solution of the original Riemann-Hilbert problem (7.2)-(7.4). For example, if we take 2~ to lie in the region of the upper half-plane which is above the contour 273, then we find
m()~) --
0
0 t n4 (~"()~)) (,,0,0
3(01
0
(3(~ -1
0
0)
~()~)
"
In this region of the complex plane, one can compute asymptotics for )~ --+ cx~, and from those asymptotics, read off the asymptotics for q (x, t), using (7.5). We refer the reader to [53] for the details, and here only state the final result.
D. Cai et al.
660 Let n 4( l ) _
r
lim ~(n4(~)-
I).
If q (x, t) is the solution of the defocusing NLS equation, then there is a constant M so that as t --+ e~,
q(x t)=_(2t)_l/2(6~o))_2(n~41) ) ,
12-~-0
(logt) -7
'
x
for[~ol-- ~-~ ~<M.
(7.20)
In [53], the explicit asymptotic description of n4 is carried out. If we use this, and formula (7.18), then we have the following result: THEOREM 7.1. If q(x, t) is the solution of the defocusing NLS equation, then there is a constant M so that as t --+ ~ , i i •2•) I•2ei••4el• 12 q (x, t) -- - (2t)- 1/2 (8t)iv ei l4-7-e -2K(~~ r(~o)F(-iv) X
for I~.ol- ~- ~ M.
••2
log t ) +o
-7-
,
(7.21)
Such uniform long-time asymptotics is unprecedented in the theory of nonlinear waves, and can only be obtained because of the deep connection between the linear spectral theory and the complete integrability of NLS. Similar results have been obtained for other soliton equations (see, for example, [45]). We close this section by reiterating that a key step in the argument is understanding how to handle rapidly oscillating terms in Riemann-Hilbert problems.
8. Semi-classical behavior Consider the NLS equation in the form
ieqt -- eZqxx - 2g(q{l)q, q(x,O)--
Ain(x)expli
Sin(X)]
(8.1)
where 0 < e << 1. The limiting behavior of q (x, t; e) for fixed x and t, as e --+ 0, is known as the "semi-classical" or "vanishing dispersion" limit. This limit is very natural in the linear (g = 0) case, where it describes the semi-classical reduction of nonrelativistic quantum mechanics. In this setting, the parameter e denotes Planck's constant h, and the limit describes the reduction of Schr6dinger quantum mechanics to Newtonian classical mechanics
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as h --+ 0. In the nonlinear cases (g ~ 0), the limit describes vanishing dispersion. In this limit, beautiful rapidly oscillating wavetrains form and propagate. The goal of the "small dispersion" problem is to characterize and describe these nonlinear wavetrains. REMARK. Physically, the nonlinear Schr6dinger equation provides an asymptotic description [164] of the slowly varying envelope of a rapidly oscillating nonlinear wavetrain, which is (i) strongly dispersive, (ii) nearly monochromatic, and (iii) weakly nonlinear (of small amplitude). As such, properties of NLS solutions such as "blow-up in finite time" and "the development of rapid oscillations" tend to violate the assumptions of the NLS representation- assumptions such as slowly varying envelopes and small amplitude waves. Thus, the physical importance of these properties of NLS is not immediate; nevertheless, laser beams do filament and they can develop oscillations (which are associated to "optical shocks"). The validity of NLS in capturing such physical phenomena is a matter of scales which can be short on the envelop scale, while still long on the scale of the underlying wavetrain. In any case, often there is a correlation between beautiful NLS wavetrains and observable behavior in laserbeams [31,163,162,109,75]. -
8.1. Sample numerical simulations Figures 18, 19, and 20 illustrate the formation of rapid oscillations in the magnitude Iq(x, t; e)l for three cases: (i) linear (g = 0), (ii) defocusing nonlinearity (g > 0), and (iii) focusing nonlinearity (g < 0). The same initial data is used for each case:
q(x, O) -- Ain(x)exp
Ei
-Sin(X) 8
]
,
where A i , , ( x ) -- e x p ( - x 2) '
d Si,(x) = -tanh(x). dx
In the figures, e = 0.02. Notice that initially there are no oscillations in the data Iq (x, 0)l = Ain(x), but they form temporally in Iq(x, t; e)l. In the linear case, Figure 18 shows a severe focus of intensity, and the emergence of caustics which bound a region in space-time which supports rapid oscillations. These phenomena are well understood in this linear case, and can be easily calculated using elementary stationary phase evaluation of
q(x, t; e) --
(i /exp{i['x y'2 4fret
e
4t
--Si,l(y)]}Ai,,(y)dy.
The oscillations are the consequences of phase-interference of the quantum mechanical wave function q(x, t; ~). The mathematical theory of Lagrangian manifolds [4] was invented to describe such semi-classical phenomena in the presence of a potential.
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D. Cai et al.
o~
" S~
% !\
"4-.
J~L
Fig. 18. Semi-classical behavior: Linear case.
1
\.%
f~L
Fig. 19. Semi-classical behavior: Defocusing nonlinearity.
The nonlinear SchrOdinger equation
663
. 1
%.%
Fig. 20. Semi-classical behavior: Focusing nonlinearity.
The behavior for defocusing nonlinearity is m i l d e r - with no focus of intensity, but with two distinct regions of space-time which support rapid oscillations (Figure 19). In these oscillatory regions (for both the linear and defocusing cases), the convergence of Iq(x, t; e)l as e --+ 0 is only weak convergence. As e ~ 0, the wave-length of the oscillations goes to zero linearly with e, but their amplitude does not vanish. Rather, the amplitude itself converges to a nonvanishing limit, and the oscillations fill-in an "envelope" defined by the amplitude. This prevents strong convergence of Iq(x, t; e)l. The focusing case, Figure 20, exhibits the most severe behavior. (The intriguing second region of distinct oscillations was observed in [ 150].) To summarize, the oscillations may be ordered by their severity - from the least severe defocusing nonlinearity, through the intermediate linear case, to the most severe case of focusing nonlinearity.
8.2. Formal semi-classical asymptotics This ordering can be understood by the following formal asymptotic calculation [31], which applies before the onset of oscillations: We make the ansatz
q(x,t; E ) ~ A ( x , t ) e x p
E-iS ( x , t ) ] , 8
(8.2)
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D. Cai et al.
insert it into NLS equation, and balance powers of e to obtain Pt = 2 P pr + 4 g A A x , At = 2 P A x + P x A ,
where P = Sx. This is a first order system of pde's for A and P. It can be placed in Riemann invariant form: r,+
-
c +
,
(8.3)
where the Riemann invariants F + are defined by F + -- P -1- 2v/-gA,
(8.4)
and where the characteristic speeds are given by C + = 2[P -+- ~/-gA].
(8.5)
From these characteristic speeds (or "nonlinear group velocities"), one can understand the "ordering". In the defocusing case, the speeds are real and distinct, and the system of Equations (8.3) is "strictly hyperbolic" [122]. As such, the unknowns A and P are bounded; hence, the intensity A 2 cannot blow-up. (However, its derivatives can.) On the other hand, in the linear case, the two real speeds are identical and the system (8.3) is "degenerate hyperbolic". When the hyperbolic system is degenerate, it can have foci at which its solutions diverge. In fact, this linear NLS case is particularly simple: The equation Pt = 2 P pr can be easily solved by the method of characteristics [96], which can then be used to solve At = 2 P A x + Px A, whose amplitude A ( x , t) is seen explicitly to diverge at a focus of the characteristics. In the focusing case, the speeds are complex. The system is not hyperbolic, and the Cauchy problem is not well-posed. Instabilities are present which are related to the "modulational instability" as was described earlier. The situation is unclear, and quite unstable. Returning to the case of defocusing nonlinearity, the modulation equations (8.3) describe the propagation of the waveform (8.2), until it "breaks at a focus of the characteristics". Moreover, by replacing the waveform ansatz (8.2) with one based upon slowly varying elliptic functions [74], one can use the modulation theory of Whitham [189,190] to anticipate the evolution of those oscillations which form beyond "break-time". While this modulation theory provides beautiful representations of the oscillations, it is based upon the local ansatz of a modulating w a v e f o r m - an assumption which is not connected to, nor derived from, the nonlinear initial value problem. As such, modulation theory provides only a partial description- although a very detailed one.
8.3. The weak limit in the defocusing case In the defocusing integrable case, inverse spectral theory can be used to characterize completely the weak limit (8.1). This characterization connects the weak limit to the initial data;
The nonlinear SchrOdinger equation
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moreover, it characterizes the (phase) transition boundaries in space-time, across which the nature of the oscillations changes (see Figure 19). Lax and Levermore [ 123,125,124] first used inverse spectral theory to describe vanishing dispersion nonlinear wave problems in the setting of the Korteweg-de Vries (KdV) equation. That initial work, together with subsequent studies, is summarized in the survey [ 126]. The heart of the matter is a closed formula for the solution of the KdV equation obtained by neglecting the reflection coefficient (the formula involves only the solitons). A remarkable calculation then shows that, asymptotically as the dispersion parameter tends to zero, this formula is governed by an associated maximization problem in function space, in which x and t appear as external parameters. These methods were adapted in [94,95] to study the semi-classical limit of the defocusing cubic NLS equation, where the steps in, and organization of, the proof of the Lax-Levermore construction is clarified significantly. Today, the modern approach to these semi-classical limits combines the methods of Lax-Levermore with those of Riemann-Hilbert problems with rapidly oscillating kernels. (Here, the oscillations arise as the coefficient of dispersion vanishes, rather than as t --+ ~ . ) Recently, Deift et al. [51 ] have completed a Riemann-Hilbert analysis of the small dispersion limit of the KdV equation. They are able to obtain a very detailed asymptotic description of the solution. It is remarkable to note that the maximization problem identified by Lax and Levermore appears as an essential component for this Riemann-Hilbert analysis: the support of the maximizer determines the limiting subset of the real axis on which a model Riemann-Hilbert problem must be solved. Moreover, if this limiting set consists of finitely many (but more than one) intervals, then the asymptotic description of the solution involves an associated theta function, and one is able to connect the initial value problem with the higher genus modulation equations of [70]. So far such a Riemann-Hilbert analysis has not been carried out for the case of the semi-classical limit of the NLS equation, and there are some difficulties. One particularly interesting aspect is that in [51], the authors assume that there are no solitons present, and work exclusively with the reflection coefficient. The situation in which there are N solitons, and N ~ cx~ in the small dispersion limit (the easiest case for the Lax-Levermore approach), seems somewhat more difficult in the Riemann-Hilbert setting.
8.4. More on the modulation equations In spite of the success in the characterization of the semi-classical limit through inverse spectral theory, modulation theory still provides the quickest method to display the local space-time structure of the oscillations. While significant work [ 126,184,51 ] has been carried out toward extracting this local structure from the Lax-Levermore/Riemann-Hilbert framework, the modulation approach is still far more direct. Moreover, inverse spectral theory is restricted to the very special case of integrable nonlinear waves, and it would seem that the modulation approach will form the basis for studies of more general nonintegrable waves. In the completely integrable setting, a particularly beautiful representation of the modulation equations exists - an "invariant representation" in terms of meromorphic differentials. These are developed, and their consequences explored, in some detail in the KdV setting [70,144]. Similar results could certainly be developed for defocusing NLS [74].
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D. Cai et al.
One begins with an N-phase, quasi-periodic waveform and its associated ZakharovShabat spectral theory. Let {kl, k2 . . . . . k2N} denote the simple eigenvalues, and consider the hyperelliptic Riemann surface defined through these branch points:
The modulation of the waveform is described by letting the parameters be (slowly varying) functions of space-time, {kj(x, t), j = 1 . . . . . 2N}. Then, as shown in [74,95], the modulation equations take the compact form ats2(~)(k) = a~s2(t)(k),
(8.6)
where S2 (x) and t'2 (t) denote differentials of the form t"2(a)
p(ot) R
= --dk,
~-x,t,
(8.7)
and where P(~) denote polynomials which are uniquely defined in terms of the branch points {kj(x, t), j = 1 . . . . . 2N} through normalization conditions [74,95]. Thus, these differentials are uniquely specified by the branch points {kj(x, t), j = 1 . . . . . 2N}, and they depend upon (x, t) only through these branch points. Moreover, these differentials characterize physical features of the w a v e - such as its frequencies of oscillation and its nonlinear group velocities. Thus, the modulation equations (8.6) may be viewed as evolution equations for the branch points. As described in detail in [70,144] for the KdV case, Equation (8.6) is a particularly compact form of the modulation equations. All other forms may be extracted from it: (1) Expansion of the modulation equations (8.6) near k _~ kj produces the Riemann invariant form of the equations. The branch points are shown to be Riemann invariants, and explicit formulas for the characteristic speeds are deduced: C j = p(t) p(x)
),.=~j
(2) Expansion of Equations (8.6) as k ~ cx~ produces the "averaged conservation law" form of the modulation equations, as first deduced by Whitham [ 189]. (3) Integration of representation (8.6) around certain cycles on the Riemann surface produces a "canonical Hamiltonian-system form" of the modulation equations [70, 61]. 8.5. The f o c u s i n g case While the semi-classical limit of the defocusing integrable NLS equation is rather completely characterized through inverse spectral theory, the semi-classical limit for the focusing case remains o p e n - which many regard as the open problem in integrable nonlinear
The nonlinear Schr6dinger equation
667
wave theory [95,126]. In the focusing case, it is not even clear that the weak limit exists. Recently, there has been some progress: (1) By analysing the modified KdV hierarchy, Ercolani et al. [62] showed that the obstruction is not the nonself-adjointness of the Zakharov-Shabat operator. (2) Bronski [29] has computed numerically a fascinating "Y-configuration" in the Zakharov-Shabat spectrum in the semi-classical limit for one type of data. (3) Consequences of this Y-configuration in the spectrum have been observed recently in laboratory experiments in nonlinear optics [ 113]. (4) Miller and Kamvissis [150] have studied numerically the semiclassical limit for special analytic data. Their numerics indicates that the weak limit appears to exist and to be described, prebreaking, by the elliptic modulation theory. (5) Tian [180] has developed some numerical evidence toward the existence of a weak limit, as well as a particularly clean form of the log-determinant N-soliton formula in the focusing case. Given this recent progress on the semi-classical limit for focusing nonlinearity, we are optimistic that this central problem of integrable nonlinear wave theory will soon be solved. On the other hand, when the waves are not integrable, the semi-classical limit is completely open. The only known mathematical result is due to Grenier [81], who characterizes the prebreaking limit for defocusing nonlinearities. In this more general setting, the semi-classical limit and its generation of rapid oscillations is one example of the nonlinear transitions of excitations between spatial scales. As such, it is related to theories of wave turbulence (see Section 6.2). Recent promising mathematical approaches [ 116-119,18,24] to such transitions between scales combine pde and dynamical systems methods.
9. Conclusion
In this survey, we have used a class of nonlinear Schr6dinger equations to display typical qualitative properties of nonlinear dispersive waves, and to illustrate the interplay between the methods of partial differential equations and those of dynamical systems theory by which these properties can be understood mathematically. Specifically, for the study of global behavior for evolutionary pde's, we advocate implementing intuition from the theory of dynamical systems with methods natural for the pde. In addition, the central importance of scientific computation to the process is also emphasized throughout the survey, as is stochastic behavior. One very special NLS equation is the integrable case of cubic nonlinearity in one spatial dimension. As one of the soliton equations, it represents the most spectacular success of dynamical systems methods for pde's. The miraculous properties of the soliton, which were discovered numerically, have been understood through the realization that these soliton equations are completely integrable Hamiltonian systems in infinite dimensions. However, this understanding did not follow solely through intuition from dynamical systems theory. Rather it resulted from a totally new mathematical i d e a - the deep connection between certain special nonlinear wave equations and the spectral theory of linear differential operators. Moreover, for the rigorous calculation of asymptotic limits, the full exploitation of this deep connection requires the proper setting, skills, and methods from mathematical analysis - the Riemann-Hilbert formulation of inverse spectral theory.
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With the full power of the mathematical methods of the spectral and inverse spectral theory of linear differential operators, complete integrability (which seems so special from the viewpoint of nonlinear waves) has been shown to be quite universal throughout mathematical analyses where, in addition to the representation of integrable waves, it has been used: to solve classical problems regarding the asymptotic description of orthogonal polynomials [46]; to resolve open conjectures about the universality of random matrix theories [47]; to provide an understanding of sorting algorithms and of matrix factorizations [49] such as the "LU" and "singular-value" decompositions of numerical analysis; and, most recently, the solution of certain counting problems in number theory [8] which may indicate a relation of these integrable methods to the zeros of the Riemann ~" function. Again, we emphasize that this remarkable breadth of integrable techniques follows from both the new mathematical idea and its natural analytic framework. For nonlinear waves, integrable examples illustrate rich and fascinating global behavior; however, they do not indicate the generality of the phenomena. Once integrability is broken by perturbations of the equation, very little is known mathematically. In this survey we described some initial steps toward persistence, and toward the characterization and description of temporal and spatiotemporal chaos. However, most important non-integrable problems remain open, and we anticipate that the interplay between dynamical systems, pde, and stochastic analysis will play significant roles in their resolution.
Note added in proof Since this survey article was completed, several new results have appeared in the literature: (i) In the area of dispersive wave turbulence- D. Cai and D.W. McLaughlin, Chaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves, J. Math. Phys. 41 (2000), 4125-4153; D. Cai, A.J. Majda, D.W. McLaughlin and E.G. Tabak, Dispersive wave turbulence in one dimension, Physica D 152/153 (2001), 551-572. (ii) A complete description (pre and post breaking) of the semi-classical limit of the focusing NLS equation for real analytic data - S. Kamvissis, K.T.R. McLaughlin and E Miller, Semiclassical analysis of the focusing nonlinear Schrgdinger equation (submitted to Annals of Mathematics, Studies Series). (iii) A formal WKB asymptotic description of the Y-shaped configuration observed by Bronski has recently been presented by Miller- E Miller, Some remarks on a WKB method for the nonselfadjoint Zakharov-Shabat eigenvalue problem with analytic potentials and fast phase, Physica D 152/153 (2001), 145-162. In addition, a reference of particular historical importance for the use of RiemannHilbert formulations to provide connection formulas across caustics in weakly nonlinear wave systems - R. Haberman and R. Sun, Nonlinear cusped caustics for dispersive waves, Studies in Appl. Math. 72 (1985), 1-37.
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References [1] M.J. Ablowitz, B.M. Herbst and C.M. Schober, The nonlinear SchrOdinger equation: asymmetric perturbations, traveling waves and chaotic structures, Math. Comput. Simulation 43 (1997), 3-12. [2] P. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958), 1492. [3] V.I. Arnold, Instabilities ofsystems with several degrees offreedom, Soviet Math. Dokl. 5 (1964), 581-585. [4] V.I. Arnold, Mathematical Methods of Classical Physics, Springer, New York (1978). [5] V.I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, Benjamin, New York (1968); Translated from French by A. Avez. [6] S. Aubry, Breathers in nonlinear lattices: Existence, linear stability and quantization, Physica D 103 (1997), 201-250. [7] A.V. Babin and L.A. Bunimovich, Stable chaotic waves generated by hyperbolic pdes, Nonlinearity 9 (1996), 853. [8] J. Baik, E Deift and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), 1119-1178. [9] P.W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc. (1999). [10] R. Beals and R. Coifman, Scattering and inverse scattering for first-order operators, Comm. Pure Appl. Math. 37 (1984), 39-90. [ 11] R. Beals, E Deift and C. Tomei, Direct and Inverse Scattering on the Line, Amer. Math. Soc., Providence, RI (1988). [ 12] T.B. Benjamin and J.E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech. 27 (1967), 417-430. [13] R.E Bikbaev and S.B. Kuksin, A periodic boundary, value problem for the sine-Gordon equation, its small Hamiltonian perturbations, and KAM-deformations of finite-gap tori, Algebra Anal. 4 (1992), 42-78. [ 14] B. Birnir, H.E McKean and A. Weinstein, The rigidi~' ofsine-Gordon breathers, Comm. Pure Appl. Math. 47 (1994), 1043-1051. [15] A.R. Bishop, M.G. Forest, D.W. McLaughlin and E.A. Overman II, A quasiperiodic route to chaos in a near-integrable pde, Physica D 23 (1986), 293-328. [16] R.E. Blahut, Principles and Practice of Information Theory, Addison-Wesley, New York (1988). [17] A.I. Bobenko and S.B. Kuksin, Small-amplitude solutions of the sine-Gordon equation on an interval under Dirichlet or Neumann boundary conditions, J. Nonlinear Sci. 5 (1995), 207-232. [18] J. Bourgain, Approximation of solutions of the cubic nonlinear SchrOdinger equations by finite-dimensional equations and nonsqueezing properties, Internat. Math. Res. Notices (1994), 79-88. [19] J. Bourgain, Periodic nonlinear Schr6dinger equation and invariant measures, Comm. Math. Phys. 166 (1994), 1-26. [20] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geom. Funct. Anal. 5 (1995), 629-639. [21] J. Bourgain, Harmonic analysis and nonlinear partial differential equations, Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zfirich, 1994), Birkh~user, Basel (1995), 31-44. [22] J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear SchrOdinger and wave equations, Geom. Funct. Anal. 6 (1996), 201-230. [23] J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schr6dinger equation, Comm. Math. Phys. 176 (1996), 421-445. [24] J. Bourgain, On growth in time of Sobolev norms of smooth solutions of nonlinear Schr6dinger equations in R D, j. Anal. Math. 72 (1997), 299-310. [25] J. Bourgain, On Melnikov's persistency problem, Math. Res. Lett. 4 (1997), 445-458. [26] J. Bourgain, Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Notices 1998 (1998), 253-283. [27] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schr6dinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145-171. [28] J. Bronski, D. McLaughlin and M. Shelley, On the stability of time harmonic localized states in a disordered nonlinear medium, J. Statist. Phys. 88 (1997), 1077-1115.
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CHAPTER
13
Pattern Formation in Gradient Systems
Paul C. Fife Dept. of Mathematics, UniversiB' of Utah, Salt Lake Ci~, UT 84112, USA
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Models with c o m p l e x - v a l u e d sinusoidal minimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. T h e N e w t o n - K e l l e r criterion for linear stability
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2.2. Gradient flows: the n o n c o n s e r v e d case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Linear stability of solutions w h i c h are not m i n i m i z e r s . . . . . . . . . . . . . . . . . . . . . . . . . 3. N o n c o n s e r v e d gradient flows for real-valued functions
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679 681 681 682 689 690
3.1. D e p e n d e n c e of m i n i m i z e r s on p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
692
3.2. Bifurcating solutions
695
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3.3. Priority of small vs. large patterns
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697
3.4. D e p e n d e n c e of the solutions on k
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698
3.5. Further questions
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3.6. T h e S w i f t - H o h e n b e r g and related equations 3.7. A c t i v a t o r - i n h i b i t o r patterns in biology 3.8. Related models
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. S y s t e m s with an i m p o s e d conservation principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
699 700 703 705 706
4.1. Stable patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Metastable patterns: the C a h n - H i l l i a r d equation and another kind of competition . . . . . . . . . .
707 708
4.3. Spinodal d e c o m p o s i t i o n in higher d i m e n s i o n s
710
4.4. Interfaces
.............................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
712
4.5. Phase-field models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Patterns and phase evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
715 716
Acknowledgements
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
718
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
719
References
H A N D B O O K O F D Y N A M I C A L S Y S T E M S , VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 677
678
P. C. Fife
Abstract Stable and metastable patterned solutions of nonlinear evolution equations of gradient type are discussed. Examples include classes of higher order parabolic PDEs and integrodifferential equations. A governing theme is that patterns can arise as a result of a competition between opposing influences such as destabilizing and stabilizing mechanisms. The discussion is within the context of a general framework although in the case of conserved evolutions, most attention is given to the Cahn-Hilliard equation and metastable patterns. The focus is on rigorous results; however some important formal stability and modulational theories are also reviewed.
Pattern formation in gradient systems
679
1. Introduction
Spatial patterns observed in natural phenomena are often the product, in some perhaps loose sense, of competition between a destabilizing and a stabilizing mechanism. The main aim of this chapter is to explore the mathematical implications of this type of competition in the context of systems that have a Lyapunov functional. Within this restricted approach to patterns, we shall try to be general, in an attempt to clarify the essential mathematical ingredients in their formation. In most of the chapter our setting, in fact, will be evolution laws, for functions u (x, t), of the form ut = A u - p B u - f (u),
(1)
or this equation's conserved analog, where A and B are negative (not necessarily differential) self-adjoint operators on a space of functions of x, and f is a function such that u f (u) grows rapidly enough as lul ~ oo. The first two terms on the right of (1) represent the stabilizing and destabilizing influences, respectively, while the last term serves to limit the growth of patterned solutions. The number p will be our control parameter; stable patterns arise when it surpasses a critical value. There has been a vast number of mathematical investigations into patterned solutions of nonlinear partial differential equation models in the sciences; the reader is referred for example to [31 ] for an excellent comprehensive review. Our focus here is much more restrictive, in that we emphasize the role of mathematical rigor, and of course limit our attention to gradient systems. These restrictions exclude detailed treatment of a number of well-developed multiscale theories directed to the stability and other properties of patterns. They employ the tools of linear and weakly nonlinear stability analyses, amplitude equations, and phase evolution equations. We describe the ideas underlying such theories briefly in Sections 2.1 and 2.3, and especially 5. The prototypical applications of those methods have been to fluid dynamical instabilities (see, for example, the exposition in [38]), but the list extends far beyond that discipline. We are also excluding the extensive modeling, simulations, and asymptotic analyses that have been done for (nongradient) patterns in excitable media. One very good survey of this field is [64]. The study of patterns arising through bifurcation has been prevalent in the modeling literature since the seminal paper by Turing [82]. Again, we restrict our attention here to bifurcating solutions within our stated framework (Section 3.2). In our context, patterns will be defined as stable stationary spatially nonconstant solutions of evolution equations. The concept of stability can take various forms, all related to the expected behavior of other solutions of the evolution problem starting near the one in question. We shall speak of solutions being stable either in the sense of (1) their being minimizers of the Lyapunov functional, or (2) satisfying the formal requirements of some linear or weakly nonlinear stability theory. In the case of (1), a minimizer ~p has the property that no solution trajectory can leave it; i.e., there is no solution w ( t ) defined for all t, such that w(-cxz) --~p. In particular, there is no unstable manifold. This is because the Lyapunov functional would have to decrease on that trajectory, which would contradict ~p being a minimizer. In the case of linear stability criteria (2), one replaces the evolution equation by its linearization about the
680
P.C. Fife
given stationary solution 4~, and again asks whether there can exist any trajectory leaving 4~. The search is limited to solutions behaving exponentially in t, i.e., involving a factor e ~t. Stability occurs if all such coefficients cr satisfy ~o- < 0; if this is true except that some satisfy 3~o- -- 0, then further analysis, if possible, is called for. Very often, one or both of these notions of stability imply that for some Banach space X and any e > 0 there is a number 3(e) > 0 such that [lu(t) - 4~llx < e for t > 0 whenever u(t) is a solution with Ilu(0) - 4~llx < 3. See, for example, Comment 1 following Theorem 3. The simplest scenario in which patterns can be studied is that of equations which admit complex exponential solutions, and Section 2 is concerned solely with this case. A general linear stability criterion was developed by Newton and Keller for this situation; it is reviewed in Section 2.1. It applies to problems of greater generality than those within the stated theme of this chapter. Criterion (1) is the focus of the material in Section 2.2, which is based on [43]; in fact, we deal exclusively with conditions for the existence of nonconstant global minimizers for our general class of gradient flows, together with many properties of those (sinusoidal) minimizers. The problem is posed in the context of spatially periodic solutions with given wavelength k, although our intention is to discover properties of those solutions which change little with k when it is large. At the end we return to the linear stability criterion, which can be spelled out explicitly for our problems. We show that stability in this sense, as expected, encompasses a larger set of solutions than just global minimizers. We pass in Section 3 to consider problems which do not have complex exponential solutions. Less can be said in this case: since the solutions are not known explicitly, linear stability analyses would be far more difficult. However, necessary as well as sufficient conditions can be given for the existence of globally minimizing patterns. These two conditions just do not coincide, as they do in the cases considered in Section 2. The control parameter p is a measure of the relative strength of the destabilizing mechanism, compared with that of the stabilizing one. We are able to provide (as in Section 2) estimates for the amplitude of the patterns as a function of p (they generally grow unboundedly as p --+ cx~), but rather little about their dependence on k. At the end of this section, we indicate the relevance of our theory to the SwiftHohenberg equation and related equations, as well as to activator-inhibitor models, which are prevalent in biological modeling. Up to this point, the gradient flows will have been simple nonconserving ones with respect to the given Lyapunov (energy) functional. Section 4 is devoted to those for a conserved quantity. This class includes the Cahn-Hilliard equation. As far as the existence of minimizers is concerned, much of the previous theory still applies. But an important new phenomenon now arises: that of metastable patterns. When the energy does not admit interesting globally minimizing patterns, it still may be the case that the dynamics drives solutions to a neighborhood of patterned states where they remain temporarily, but for a long time. This is due to a different kind of competition, namely that between the system's desire to minimize the energy, which happens best at long wavelengths, and the kinetics of diffusion, which accomplishes patterning more quickly when the wavelength is small. This phenomenon leads to interesting mathematical issues. Some recent developments in this connection in higher dimensions, particularly due to Maier and Wanner with coworkers, are reviewed, together with "slow motion" results of Alikakos, Bates, Fusco, and others.
Pattern formation in gradient systems
681
This chapter does not attempt a complete review of the subject, nor does it attempt to assign priority to the ideas or results described.
2. Models with complex-valued sinusoidal minimizers There exist quite a number of nonlinear PDEs arising in mathematical physics that admit complex-valued stationary or traveling wave sinusoidal solutions. It is generally possible to pursue a linear stability analysis of such solutions. This has been done many times in the past for particular cases. The most complete results in this connection are probably those of Newton and Keller [69,70], who showed that the linear stability criterion can in most cases be reduced to an algebraic condition. We begin by quickly reviewing this in Section 2.1. In the following Section 2.2, we specialize to gradient flows for an energy functional admitting complex-valued functions, and ask for minimizers of the energy. They are guaranteed to be stable solutions of the corresponding evolution problem. We prove a threshold criterion for their existence, in terms of the "destabilization" parameter p, and also derive many properties of these solutions, such as their asymptotic structure (wavelength and amplitude) for large values of p. At the end (Section 2.3), we return to the linear stability criterion in this special context.
2.1. The Newton-Keller criterion f o r linear stability This criterion applies to a wide class of problems with sinusoidal traveling wave solutions. The basic focus of these authors was on systems of differential equations of the form
f(io,,-iax, lul2)u-O,
(2)
for a matrix ~ of differential operators. There exists a solution of the form uo(x, t) -- a R e i(b'-c~
(3)
provided that the real parameters a, k, co satisfy detf'(co, k, a 2) = 0
(4)
and the unit vector R is a right nullvector of the matrix .T'(co, k, a2). A linear stability analysis can be performed for solutions u0. The analysis is for solutions defined on the entire real line. It is found to be convenient, and no restriction, to write the perturbed solutions in the form u(x, t; e) -- uo(x, t) + e ei(kx-wt)dp(x, t) + o(e).
(5)
This expression, when substituted into (2) and the higher order (in e) terms disregarded, yields a linear system for 4).
P. C. Fife
682
It turns out that this linear system has solutions of the form
q~(x, t) -- C1 e ilx+crt + C2 e -ilx+6t
(6)
for some constant vectors C1 and C2, if the numbers 1 (real) and ~r satisfy the solvability condition that the following matrix have zero determinant: ( a~(co -k- i cr, k + / , a 2) + a 2 [f'a 2 (co, k , a 2 ) R ]-1~
a2[~a2 (co' k' a2) R ]R
a2 [~a2 (co, k, a2)R]R
~(co - icr, k - l, a 2) -k-a2 [~a2 (co, k, a2)R-]R
)
(7) If there is such a pair (/, cr) with 9]cr > 0, then the basic wave u0 will be unstable according to this criterion. For stability, the criterion is that 9~cr ~< 0 in all cases, and also that every solution of the linear equation for 4~ be expressible as a combination of solutions of the form (6). Outside of this theory, there are important weakly nonlinear instabilities operating on long time and space scales; they will be reviewed in Section 5. The stated criterion was applied in [69,70] to various examples, including the nonlinear Schr6dinger, Klein-Gordon equations and coupled systems, the Zakharov system for Languir waves in plasmas, and the envelop equation for deep water waves. It yields the classical Eckhaus instabilities in the case of the simple Ginzburg-Landau equation u,-
Uxx +
u(1 -lu12).
The given method actually applies to a broader class of problems that those stated above. We shall see that the evolutions considered next fall into this broader category.
2.2. Gradient flows: the nonconserved case This section is based on [43]. The basic ingredients of the evolution equations considered in this section are linear self-adjoint operators A and B and a real function F(lul growing rapidly enough at cx~. We shall consider gradient flows
2)
Ou Ot
=
6E[u] 8u
(8)
for the energy functional
E [ u ] - - - ~ 1 (au, u ) + -~ p (Bu, u) + -~lf0Z F(lulZ)dx,
(9)
which is therefore a Lyapunov functional for the evolution. The flow takes the form (12) below. The roles of A and B are those of a stabilizing and destabilizing mechanism, respectively. The function F serves to limit the growth of any developing patterned mode, thus producing stable finite-amplitude states.
683
Pattern f o r m a t i o n in gradient systems
More specifically, the closed self-adjoint operators A and B are negative and densely defined on the L 2 space of complex-valued L-periodic functions of x. Space is onedimensional in this section, but in later ones we extend to higher dimensions and at the same time restrict u to be real. We use the scalar product
lfo
(u, v) - -~
u~dx.
Important examples are: (a) a - - - 0 4 , B - - 0 2. (b) A -- 0 2, B -- G 9 u - u, where G 9 u means convolution with a continuous function G in L I ( R ) , where G ~> 0 and f _ ~ G ( x ) d x -- 1. The wavelength )~ is a variable parameter, which can take on any positive value. Thus A and B implicitly depend on ~, being operators on a space which does. Generally, we think of )~ as being large; then the minimal wavelength of our solutions will typically be much smaller than ~. Further assumptions on A and B are: A (A1) Smooth symbols A ( k ) and B ( k ) , defined for all real k, independent of )~, exist such that when e ikx is )~-periodic, A'(k) - e -ikx A[eikx], and the same for B. These functions are real and even. (A2) A
lim A (k) = cx~, k ~ B (k)
A( k ) < O. lim sup B k~
(10) A
(A3) The nAullspaces of A and B are the set of constant functions. It follows that A ( k ) and B (k) are strictly negative for k > 0. We have this Fourier representation: if u - ~-~,,, Um e i 27rmx/~. then A u -- Z
umA(2rcm/)~) e i2rr'''x/;~,
m
and the domain of A is the set of functions u such that this series converges in L2. A similar statement holds for B. Note that (A2) (10) implies the inclusion of domains D ( A ) C D ( B ) . In the example (a) above, A'-- - k 4, B - - k 2. In (b), A'-- - k 2 and B " - G(k) - 1, where
~(k) - f _ ~ G(y) e -~k:' dy. Concerning the function F ( w ) in (9), we assume it is real and differentiable for w >/O, with a strict minimum of 0 at w - u02/> O, such that lim
F'(w)
-- ~.
(11)
ttl-----~ O ~
We note that F ( w ) -- (u 2 - w) 2 (u0 > 0) and F ( w ) -- ~ w + w 2, c~ >~ O, satisfy these assumptions. In the former case, when u is real F ( u 2) is a double-well function of u with equal depth wells at u - +u0" in the latter case u0 - 0 and it is a single-well function.
684
P.C. Fife
The L2-gradient flow for (9) is (12)
ut -- A u - p B u - f (u),
where f (u) - 2 u U ( l u l 2 ) . We assume this has a global solution u ( t ) E D ( A ) A Lvc, t >~ O, given any initial condition in that space. Our main goal will be to investigate the possibility of stable stationary nonconstant solutions of (12). In fact, we deal mainly with global minimizers of E. We shall call a minimizer ~b nontrivial if 4~ 7~ const. In view of (A3) and F ~> 0, a sufficient condition for this is that E[4~] < 0, since constants have E ~> 0. 2.2.1. The m i n i m i z e r s a n d their properties. A
M(p,k)
We define
A
(13)
= A(k) - pB(k),
an increasing function of p, and p * ( X ) - inf{p" M ( p , km) > 2 F ' ( u 2) ---- t' for some m},
(14)
where 2zrm
km = ~ ,
m-1,2
X
.....
(15) A
We have 0 < p* 0 0 < c~, for if p* (X) - 0, then A (km) - 0 for some m, contradicting (A3). If u0 > 0, then y = 0 in (14). Note the following alternate characterization of p* in that case: A
A(km)
P* 0 0 -- inf ~ A
when u0 > 0.
(16)
m O(km) A
A
A
In fact, M (p, kin) -- - B (km) [p - A (km) / B (km) ], and this is positive for some value of m exactly when p is greater than p* as defined by (16). Below, we speak of functions a e ikx . The solution set of (12) and the set of minimizers of E are invariant under rotation, i.e., multiplication by a constant e i~ . Therefore we may, and shall, always assume that a is real and nonnegative. In this sense, our minimizers are really equivalence classes. The following theorem provides existence and nonexistence results for nontrivial minimizers, as well as several properties enjoyed by them. See the explanatory comments after the statement of the theorem. THEOREM 1. (a) For each X > O, p > p* (X), there exists a nontrivial global m i n i m i z e r o f E o f the f o r m ae ikx f o r s o m e a -- a ( p , X) > uo, k -- k ( p , X) > O. I f uo > O, this is also true f o r p -- p*" then a ( p * ( X ) , X) -- u0.
Pattern formation in gradient systems
685
(b) For p < p*(k), the only global minimizers are constants with ]u] = u0. (c) The functions a and k satisfy lim a ( p , X ) = o o , p--~ cx:~
lim
a(p,L)=uo.
(17)
pSp*(k)
(d) If A
A
inf B (k) < B (k)
f o r each k
(~8)
(the former could be - o o ) , then limp~o~ k(p, ~.) = oo. (e) Suppose F' (w) < cw r f o r some r > O, c > O, and all large enough w. Then f o r some C>0, a(p, Jk) >~ Cp 1/2r
(19)
f o r sufficiently large p. (f) Let p ~ ( ~ O) -- the critical (least) value o f p beyond which M ( p , k) > Of o r some k (unrestricted by (15)), and f o r p > p~, let fc(p) > 0 be the least positive value of k at which M (p, k) is maximized (neither Po nor k depend on X). For p > Po, 0 < k (p) = lim inf k (p, X) ~< lim sup k (p, X) < oo )~---->o c
(20)
k----> o e
and u0 ~< l i m i n f a ( p , Jk) ~< l i m s u p a ( p , Jk) < oe. )~--+ o o
(21)
)~--+ o o
Commentary. (1) Items (a) and (b) are threshold results, showing that nontrivial global minimizers exist for p > p*(~.) (>~ p*(X) in the case u0 > 0) and not for p < P*0~). Moreover, these minimizers are given explicitly. In Lemma 2 below, it will be shown that in typical cases, these are the only global minimizers, although other stable stationary solutions may exist, as shown in Section 2.3. One can prove that these minimizers satisfy a somewhat stronger stability statement in d2
the case ~ F (u 2) > 0 for u -- a. Namely, if 4~ is a minimizer of this type and e is any positive number, then for some 6(e) > 0, any solution u(t) with Ilu(0) - 4~11 < 6 satisfies Ilu(t) - ell < e for all t > 0. Here the norm is in L 2. We omit the details. (2) Items (c), (d), and (e) give information about the amplitude and minimal period of the global minimizers which will have been constructed. The amplitude a grows without bound as p -+ e~, and under a reasonable additional assumption (18) the wavenumber k does as well. Thus the minimal wavelength shrinks. The relation (19) gives a more precise lower bound for the rate of growth of the amplitude. For the case F ( w ) -- (w - u2) 2, we have a >~ O(p 1/2) as p -+ c~. It should be noted that this lower bound depends only on the function F. It was shown in [65] that such a bound may sometimes be improved by taking into consideration the operators A and B as well. In fact in the case of the fourth
686
P.C. Fife
order problem given in Section 3.6.2 below and this same convex function F, those authors established that the amplitude grows at least at the rate O(p) as p --+ c~. As p approaches its threshold value p* from above, the amplitude approaches the position u0 of F's well. The behavior of the wavelength for p near p* can also easily be obtained. (3) Item (f) explores the effect on the global minimizer when the size of the basic period interval becomes very large. We conclude that there is no important effect. The wavenumber k and amplitude a are bounded above and below independently of )~. An examination of the proof will show, in fact, that in typical cases they approach finite limits as ~ ~ c~. This means that the structure of stationary patterns is little affected by the domain size ~, when the latter is large. (4) It will be shown in Lemma 1 below that when u0 > 0 the family of exponential minimizers which exist for p ~> p* can be continued as stationary solutions of (12) of exponential type for an interval of values of p < p* as well; however the extended solutions are not global minimizers. The question of their stability is considered in Section 2.3; see also the comment following Lemma 1 below. 2.2.2. Proofs LEMMA 1. Let Um,a(X) = a e ikmx, a > 0 real km given by (15). There exist functions a*(p,)~) and pl(~) ~< p*()~) such that f o r some m, Um,a is a stationary solution of (12) if P > Pl and a = a*. Also a* > uo if p > p* ()~), a* (p*,)~) = uo, and a* (cx~,)~) = cx~.
PROOF. Substituting this function into (12), we obtain the following necessary and sufficient condition for it to be a solution: either a = 0 or M(p, km)-2F'(a2)--O.
(22)
Let Fc(w), w > u 2, be the greatest monotone increasing function with Fc(w) <~ F ( w ) for every w in that range. Its graph is the convex hull of that of F. We have that F~(w) is nondecreasing, and by (11), Fc~(cx~) = cx~. The range of 2F'(w) contains [y, cx~) (see (14)), the same as that of 2F~(w). Thus there exists a solution such that F ' ( a 2) -- F~(a2),
(23)
provided that M ( p , km) >/y for some m, which is true if and only if p ~> p* ()~). Moreover, the quantity 2 F ' (a 2) takes on all negative values in some interval [-c~, 0] as a 2 ranges from 0 to u 2. Let Pl ()~) be such that for each p E (pl, p*), M ( p , km) E (-or, O) for some m(p). There exist exponential solutions for this range of p as well. (Some of them satisfy a linear stability criterion; but generally their minimal wavelength is )~ itself, so they are almost constant when )~ is large.) This proves the existence part of the lemma. The stated properties of a* follow from (22). D
Pattern formation in gradient systems
687
LEMMA 2. When p >~p*(~.), there exists a finite integer m - - m * ( p , ~.) which maximizes M(p, km). It can be chosen so that limp,z,, m*(p, k) -- m0 < oo. lf p >~ p*()v) and a satisfies (22) with m = m* and (23), then Um*.a is a global minimizer of E. PROOF. It follows from (A2) (10) that for any fixed p, M(p, k) < 0 for k large enough. Therefore when maximizing M (p, kin) over m, we need consider only a finite number of values of m, namely those for which M > 0. This establishes the first two statements of the lemma. We shall show that Um*,a is a global minimizer of E when a ~> u0. In the following, we drop the subscripts on u, let v be any function in D(A) A Lee, and calculate E[u + v]. We find that
{A(u + v), u + v}-- (Au, u) + 2~(Au, v) + (Av, v),
(24)
and the same for B. We have chosen the amplitude a so that (23) holds, which means that
F(a 2 -+- Wl) > / F ( a 2) -+- w l F ' ( a 2) for all
1,01.
Choosing
1/31 - -
2~(u~) + Ivl 2, we note that a 2 -Jr-wl - lu
-+-
13]2. Hence
f0 v F(lu + vl 2) dx
fo
/> k F ( a 2) + 2 F ' ( a 2)
3~(uf~)dx + F ' ( a 2)
fo
Ivl 2 dx.
(25)
Combining these expressions, we obtain
E[u + v] >~ E[u] - 2.qt<[Au - pBu - 2F'(a2)u], v ) + E[v],
(26)
where ~ Ivl 2dx, E [ v l - - E o [ v ] + ~ 1 F , ( a 2 ) fa0 1
p
E 0 [ v ] - - ~ (av, v) + -~ (By, v).
(27)
The expression Au - pBu - 2F'(a2)u in the middle term of (26) vanishes because u is a solution of (12). We conclude that u is a global minimizer of E if E[v] >>,0 for all v. We may express Ely] -- - 89(Lv, v), where L is the linear self-adjoint operator given by
Lv--Av-pBv-2F'(a2)v. Therefore E will be positive, as desired, if all eigenvalues # of L are nonpositive. The eigenvalue equation is
Av-pBv-2F'(a2)v--#v.
(28)
688
P. C. Fife
This may be solved by Fourier expansion over the period interval (0, X). It therefore suffices to look for solutions of the form v - e ikmx, km given by (15). We obtain the set of eigenvalues # m = m (,o, k m ) - 2
F' (a 2)
m
--
1
2,
(29)
We subtract (22) with k - km* from this to obtain #m
-
-
-
M (p, km ) - M (p , km* ).
(30)
Therefore since m = m* was chosen to maximize M ( p , km), we obtain from (30) that all eigenvalues satisfy #~<0, and hence E[u + v] >~ E[u] for all admissible v. Thus Ua,m* is a global minimizer. It will be the only one if E[u + v] > E[u] for v 7~ 0. This can be verified in case m* is unique and F ( w ) is strictly convex. [3 REMARK. Values of m which do not maximize M ( p , km) generate stationary solutions which may be stable but not global minimizers. This issue is taken up in Section 2.3. LEMMA 3. / f p ~< p*, then E[u] >~0
(31)
f o r any u in the domain o f E. Moreover if equality holds in (31), then necessarily [ul = u0. I f equality holds and p < p*, then u - const. PROOF. By (9), (27), E[u] >~ E0[u]. The form E0 is positive if all the eigenvalues of - A + p B are nonnegative, i.e., all the eigenvalues of A - p B are ~< 0. These latter eigenvalues can be found by using exponential functions, and are given by (29) with the last term set equal to zero, i.e., #m = M ( p , km). But when p ~< p*, this is nonpositive for all m. This completes the proof of the first statement. If equality holds in (31), the last term in (9) vanishes, so that F(lu] 2) = 0, hence lul 2 - u 2. If moreover p < p*, then also (Au, u) = 0, so that u = const, by (A3). D PROOF OF THEOREM 1. Parts (a) and (c) follow directly from Lemmas 1 and 2, and part (b) from L e m m a 3. Consider now part (d). Recall that our construction in L e m m a 2 has m - m*, so that M ( p , km*) is maximal. If the assertion in (d) were not true, then there would exist a sequence Pn --+ ec and an integer rh with m*(pn,X) = rn for all n. Since m* maximizes M, we have that for everAy m > rhAandevery n, M (Pn, km) <~ M (Pn, k,~). Thus from (13), setting AA = A(kcn) - A(km), A B -- B(k,~) - B(km), we get
Pattern formation in gradient systems
689 A
However, by the condition (18), we can always choose a number m so that AB > 0. Letting n --+ ec gives us then a contradiction, which proves item (d). Part (e): From (22) and (23) and the hypothesis, we have
M (p, kin*) < ca 2r. Now let m be any fixed number > 0. We have M ( p , kin*) >/M(p, km) =-- C! -Jr C2p, where the Ci depend on km and C2 = 89 > 0. Thus a 2r > C'l + C~p. This yields (19). Consider now part (f). Let I (p) be the bounded closed interval on the k-axis where
m ( p , k ) >~0. For all )~, the maximization of M (p, kin) over m is the maximization over a discrete set of values of k in I (p), and that in the maximization of M ( p , k) is taken over all of I (p). Therefore the lim sup and liminf appearing in (20) lie in I, and in fact liminfz~oc kin. = ~: > 0. This proves (20). Finally, (21) follows from this and (22). This completes the proof of the theorem. D 2.3. Linear stability of solutions which are not minimizers We now return to the linear stability analysis of solutions uo - a e ikx of (12). The procedure in [69] applies. It is with regard to solutions on the entire real line, rather than on a finite period interval, as in our context. Recall (5), (6) for the perturbation 05. In our case, the equation for 05 takes the form
r
A (d# eikX ) e -ikx _ p B (r eikX ) e-ikx _ 24)F'(a 2) - 4a23tc/) F" (a2).
(32)
Seeking solutions of this equation in the form (6) with scalar Ci, we find the following pair of equations for C l and C2"
Cl[o- - M ( p , k + g) + 2F' + 2 a 2 F ''] + 4 a 2 F " C 2 --0, C2[o- - M ( p , k - ~) 4- 2F' -Jr 2 a 2 F ''] -+-4a2F"C1 --0,
(33)
where the functions F', F " are evaluated at a 2. The determinant condition for the existence of nontrivial solutions leads to the following second order equation for o-, where we set M+ (k, g.) = m ( p , k) - m ( p , k -t- g.) and 27 (k, e) - M+ (k, e) + M_ (k, g)" o-2 + o-(4a2F '' + 27) + M + M _ + 2 a 2 F " Z - - 0 .
(34)
Here we have used (22). The larger of the real parts of the two roots of this equation is ~< 0 for all g iff
4a2F"(a2)+r>~O
and
-2a2F"Z<~M+M_
forallg.
(35)
For stability, (35) must hold. This is our linear stability criterion. It is seen that stability cannot occur if F " (a 2) < 0, because 22 --+ 0 as g --+ 0, and the first inequality would be
690
P.C. Fife
violated for small ~. Similarly, it can be seen that if F " (a 2) = 0, stability can only happen if the given value of k maximizes M (p, k). An important case is when the index k and the function M ( p , k) are such that r > 0 for all ~ # 0 (it is always the case that r > 0 for large ~). This is true if the concave hull of the graph of the function M (p, k) touches the graph itself at the chosen point k and the concavity is strict there. Then from (35) we obtain a sufficient condition for stability, in terms the following function R ( p , k) (which might be 4-c,z), defined simply in terms of the function M and number k: - ( M ( p , k) - M ( p , k + g ) ) ( M ( p , k) - M ( p , k - g~)) R (p, k) -- sup e:/:o ( m ( p , k) - m ( p , k 4- ~)) 4- ( m ( p , k) - m ( p , k - g,))
(36)
Then we have the following linear stability criterion in the case Z'(k, () > 0 f o r all ~#0: R (p, k) ~< 2a 2 F" (a2).
(37)
When it is not true that r > 0 for all ~ # 0, then we revert to the above more general criterion (35). The criterion is easily modified to handle problems in a finite period interval A, as we were doing. Then the numbers ~ are restricted to be multiples of 27r/A. All the above continues to hold. In the case of the global minimizers we have been considering, k has been taken to maximize the function M ( p , k) under the restriction that k be of the form (15), and we also have F" (a 2) >/0. Therefore the quotient in (36) is negative and the condition (37) clearly holds, as expected. But (37) also indicates that other solutions, not global minimizers, may be stable as well. In fact when F " > 0, values of k which are near but not at the maximizer for M satisfy the condition.
3. Nonconserved gradient flows for real-valued functions The foregoing theory falls short in relevance to many applied problems, because it allows complex-valued minimizers; moreover the class of nonlinearities F is restricted to functions of lu] 2 alone and the theory is one-dimensional. We may extend the theory to partially remove these difficulties while losing some completeness of our conclusions, by (1) restricting the class of admissible functions competing for the minimization to realvalued functions, (2) allowing more general nonlinearities, and (3) extending the theory to higher space dimensions (although if A and B are isotropic, our criteria for the existence of patterns differ little from those in the 1-D case). In the N-dimensional context, the operators A and B act on A-periodic functions u (x), where now x - - ( X l , x 2 . . . . . xu) and A - (Al, A2 . . . . . A N ) . The assumptions (A1) to (A3) of Section 2.2 are still assumed, with the obvious notational changes: k = (kl, k2 . . . . . kN), e ikx means e ik'x, and the limit in (10) is taken as Ikl ~ ~ . Let Dr be the set of real-valued functions in D(A).
Pattern formation in gradient systems
691
We restrict u to be real-valued, and consider now energy functionals of the form 1 P (Bu u)+ E [u ] -- - -~ ( a u , u ) + -~ ,
1 L H(u)dx
(38)
-~1
where the C 1 real function H has a minimum of 0 at some value u -- u0 (if it attains this minimum at more than one point, let u0 be the maximal one). The integral in (38) is the integral over one period cell A, and IA[ is the measure of that cell. We also assume that H grows superquadratically as lul ~ ec: lim
[u I--> cx~
H(u)
= cx~.
U2
(39)
The evolution equation (8) now takes the form (40)
ut -- A u - p B u - H ' ( u ) ,
and the minimizers ~p of E in Dr n L ~ are stable k-periodic solutions of (41)
A O - pBcp - H ' (4)) = O.
In the real valued case we do not know the minimizers explicitly, and this diminishes the completeness of our results. For one thing, the linear stability analysis of Section 2.3 is not applicable to patterns which are not sinusoidal, as they will be in this section; we therefore restrict attention completely to global minimizers of the energy. A second point is that the existence of minimizers is not always clear. We shall assume that for each p, k there is ~pP 6 7P~ (A) n L oc such that min E[u]uEZ),.(A)NL~
E[qSP].
(42)
In general condition (42) might be difficult to verify without making additional assumptions on the operators A, B. On the other hand in special cases proving (42) usually involves checking that E is coercive and weakly lower semicontinuous. The condition (39) will be relevant to establishing coercivity. We illustrate this by considering case (b) near the beginning of Section 2.2:
Remark.
A u - Uxx,
Bu = G,
u - u,
H ( u ) -- F ( u 2 ) ,
resulting in the equation (85) below. Here G 9 u represents convolution of u with a nonnegative kernel with unit integral. The corresponding energy is
l ffo
lfokl~ lu ,2[ d x
E [ u ] -- -s
l
+ ~
F(lul
k 4
G(y)lu(x
+ y)-
u(x)[ 2 d x d y
.
dx
(43)
P.C. Fife
692
We set XD to be the characteristic function of the set D. For each K > 0 there exist constants C l (K, p), C2 (p) such that
4
G(y)lu(x + y) - u(y)l 2 dx dy
ffo
~< CI (p, K) q-
C2(p)
fo
U2(X)X{u2>K} dx.
(44)
On the other hand
f0 F(u 2)/>f0
(45)
F(u2)X{u2>K}.
Using the fact that F ' ( w ) ~ ec as w --+ ec we conclude that for sufficiently large K we have [F(u 2) - uZ]g{uZ>K} > 0 and thus combining (44) and (45) we see that there exists a constant C3 (K, p) such that
1Efo 1 lu 2I o -2
E[u] >~ --s
C3(K, P)
]
(46)
and thus E is coercive in H 1. From the embedding C a (0, X) ~ H l (0, X), ot E [0, 1) we conclude that E is weakly 1 lower semicontinuous with respect to Hper(0, )~) norm, where H~er(0, X) denotes the space of ,k-periodic H 1 functions. From (46) and the weak lower semicontinuity we can verify (42) by a fairly standard argument. We observe that the minimizers are in fact smooth.
3.1. Dependence of minimizers on p Associated with the function H , we define two other functions H* and H0. In accordance with our periodicity constraint, we consider (as in (15)) wavenumber vectors km _
(kt?l, ,km~ ~2~,.
.
mN .,k u )
(47)
with
mi ki
27cmi --
Xi
(48) '
not all of the mi vanishing. Let ]m] be the number of integers i ~< N with mi ~ O. When k is of this form, it is clear that the integral
l IAI
(49) l
Pattern formation in gradient systems
693
depends only on a, b, and ]m I, i.e., how many indexes m i vanish. This is because when mi 7~ O, the integrand is nontrivially periodic in xi of period h i / m i . For example if all the mi except one vanish (]m[-- 1), we have that the integral on the left of (49) is equal to 1
fo' H(acos(2yrml/)~lxl) + b)dxi = fo H(acos(Zrrx) + b)dx, and if [ml = 2, it is
fo'f'
H (a cos(2yrXl) COS(2yrx2) + b) dxl dx2.
In any case, the function H* is even in a and has a minimum of 0, attained at (a, b) =
(0, u0). We now set
Ho(a, [m[)= minH*(a,b, Im[),
(50)
b
with the minimum attained at a value b = b*(a, ]ml). Let
M*([m]) = inf 4H0(a, Im[) a>0
a2
'
(51)
which is either attained at a positive value a* of a (finite because of the superquadratic growth of H ) or approached as a ---> 0 (in which case we set a* = 0). It is important to emphasize that M* depends only on the nonlinearity H and ]m l. For example, in the case H(u) = (1 - U2) 2 and N = 1, it turns out [43] that M* = 0.899. Essentially this same calculation, leading to the same sufficient condition for the existence of stable patterns, was done in the context of the equation (75) below by Mizel, Peletier, and Troy [65]. Recalling the definition (47) of k m, we define r
-- inf{p: M ( p , k m) > M * ( [ m ] ) f o r some m}
(52)
and p~) ( L ) = inf{p: M ( p , k m) > 0 for some m}.
(53)
Note that if H(u) -- F(u 2) and u0 > 0, the number p~ coincides with p* given by (14). THEOREM 2. There exists a number pc(X) 6 [p(~(k), fi(k)] such that: (a) for each p < Pc, there exists no nontrivial global minimizer of the functional E (38)
in the class of real-valued functions; (b) for each p > Pc,, there exists such a nontrivial real global minimizer ckp of E in
Dr N L ~ with E[4) p] < 0.
(54)
694
P. C. Fife
Let E l ( p ) - - l ( a * ) 2 [ M ( p , km) - M*(Iml)], where m is chosen and fixed so that M ( p , k m) > 0 for some p = Po. Since M is an increasing function of p, this will be true for all p > Po as well. For a > 0, let
max(uH'(u)-2H(u)).
P ( a ) -- lul~<~
It can be shown that (39) implies lim,,--->oc P ( a ) = ec; in fact if u H ' ( u ) - 2 H ( u ) were bounded for large u by some number K, then integrating the inequality u H f (u) - 2 H (u) < K would imply that H ( u ) / u 2 is bounded. THEOREM 3.
For p > Pc, let qbp be a minimizer o f E, and a ( p ) = max 14>p (x)l. Then
P ( a ( p ) ) >~--2El(p).
(55)
Since - E l increases linearly with p and P ( a ) increases to ec as a --+ ~x~, we see that (55) provides a lower bound on the amplitude which grows toward eo as p ~ eo. 1. Assume H ( u ) = clu[ r + O([u[ r - i ) as u --+ ex~, r > 2, and that the corresponding differentiated relation H ' ( u ) = rculu] r-2 + O(]u] r-2) holds. Then
COROLLARY
a ( p ) ~ Cfi 1/r
(56)
f o r large p, where C depends only on the function H. COROLLARY 2.
In the case H ( u ) = (1
-
U2) 2, we have, f o r p > Pc,
maxl~bP(x) [ ~> 1.
(57)
PROOF OF THEOREM 2. Any global minimizer ~bp with E[~ p] < 0 must be nontrivial, because constants have E ~> 0. To emphasize dependence on p, we write E[u] = Ep[u]. Let Pc - - i n f { p" E p has a minimizer 4>p with E p [~bp] < 0 }. If Pc < ~ , the assertion (b) holds by virtue of this definition and the fact that Ep[cb] is a decreasing function of p. Now suppose that for some number P0 < Pc, Epo has a nontrivial global minimizer ~bp0. Then Epo[dpPo] = 0, since the minimum of Ep for every p is always nonpositive. Since 4~p0 is nontrivial, we have from Assumption (A3) that (Bob po, 4~p0) < 0. Hence Ep[cb po ] is strictly decreasing in p, so that E p[cbpo] < 0 for p0 < p < pc. The global minimizers for these values of p must be nontrivial, contradicting the definition of Pc. Thus part (a) follows. 9 M ( p , k m ) < 0 for all m, so We show that Pc lies in the indicated interval. If p < Po, that the operator A - p B is negative definite, and since H ( u ) > 0, it follows from (38) that E[u] >~ 0 for all u, hence by our definition of Pc, P <~ Pc, and we conclude that Pc >~ Po"
Pattern formation in gradient systems
695
To show that pc ~ P, we choose a = a* (51) and b = b*(a*) (50) to obtain from (38), (51 ) that E a*Hcos(k7/x)+b*
]
---~
l(a*)2[M(p , km) - M* ([rnl)]
<0
(58)
i
for some m for p > ft. Hence the minimizer CP for such p has negative energy and must be nontrivial, so that p >~ Pc. [-] PROOF O F THEOREM 3. Since CP is a minimizer, it satisfies (41). Take the scalar product of (41) with CP:
1 L ,PH'(,P)dx
((A - pB)c~ p , 4)p) - -~[
=0,
(59)
so that
1 L (ckPH'(cp p) -
2 E [ r p] + ~-~
Note from (58) that El (p) = E[a* E[ck p ] <~ El (p), hence from (60),
2El (p) + ~
2H(r
Hi COS (kl"/x)
1L (4)p H'(r
-0.
+ b*]. Since CP is a minimizer, we have
- 2H(c/)P))dx >~ O.
Since P(a(p)) ~ - ~1 f a (r H ' (OP) - 2H(cpP))dx, we obtain (55) This completes the proof. PROOF
(60)
(61)
D
OF COROLLARIES. In the case of Corollary 1, we have P(o-) -
max (rclu[ r - 2cIu] ~ + O ( l u [ " - ' ) )
I.(~
(r - 2)~o -r + o ( o r - ~ )
(o- ~
~).
We also have E I (p) >~ cl p - r for some positive constants Cj. Thus (56) follows easily. We omit the proof of Corollary 2. 5
3.2. Bifurcating solutions Continuing with the energy functional given by (38), we assume the dimension is one. Suppose u0 = 1. Again, we consider real-valued steady state solutions r of (40), periodic in x with period k. We set f ( u ) -- H'(u). There is of course the trivial solution 05(x) = 1, which is stable for p = 0. As p increases, however, there will be a point where it loses its
P.C. Fife
696
stability. In fact, an easy linear stability analysis shows that it becomes unstable as soon as M ( p , km), for some m, surpasses f ' ( 1 ) (km is given by (15)). Then generally other solutions will arise, some of them being stable. In this section we establish the existence of an infinite number of bifurcating solution branches, p being the bifurcation parameter. For the purpose of this section we shall make the following additional assumption on the operator A: o~
1
(A4)
- Z
"A(km) < 00.
m=l
It is easy to check that (A4) holds for the operator A = A. THEOREM 4. Let Pm()O = inf{p: M ( p , km) > f'(1)}. For each positive integer m, Pm is a bifurcation point of real-valued steady states from the trivial solution dp -- 1. PROOF OF THEOREM 4. Write 4) = 1 + 7t. Let Z = L2 nspan{cos(kjx), j ~ 0}. Note that 7t 6 Z implies gr(x) = 7r()~ - x). Furthermore we define X = Dr(A) n Z n C~ )~). Now X equipped with the norm 117~112 - IIa~ll 2 + IIB~II 2 + II~ll 2 + I1~112c o is a Banach space. Set
f ( ~ , p) = (T - pB)gff - h(~p)
(62)
where TTt = A~p - f'(1)~p, h(Tt) = f ( 1 -4- ~ ) - f'(1)Tt. If ~ e X has the Fourier expansion
7r -- Z
aj cos(kjx)
(63)
then
ATt -- y ~ a j A ( k j ) c o s ( k j x ) ~ Z" and a similar formula holds for B. For 7r E X we also have that h(gt) 6 C~ ~,) hence h(~p) E Z. Using this and the smoothness of f it follows that ,T 6 C2(X; Z). Moreover T and B are bounded linear operators from X into Z. It can be checked easily that h(0) = Dh(O) - - 0 (using the Frechet derivative D). We shall use the well known theorem of Crandall and Rabinowitz [28] to show the existence of solutions of ~(7r, p) = 0
(64)
bifurcating from simple eigenvalues of the pair (T, B) at 7r = 0, p Recall that p e II~ is a simple eigenvalue of the pair (T, B) if dimA/'(T
-
pB)
[B./V'(T - p B ) ]
=
1 =
codim~(T
-
@ 72~,(T - p B ) -- Z ,
pB);
=
Pm.
(65) (66)
Pattern formation in gradient systems
697
where A/', 7-4.denote the null space and the range of T - p B, respectively [25]. Let d / E A/'(T - p B ) with Fourier expansion as in (63). We then have E
aj[A(kj) - f'(1) - pB(kj)] cos(kjx) = 0
Z
a j [ A ( k j ) - f ' ( 1 ) - p B ( k j ) ] ( c o s ( k j x ) , c o s ( k m x ) ) = O.
hence
It follows that d/ E A/'(T - p B ) if and only if, for some m, ~ E span{cos(kmx)} and p = Pro. Thus dimA/'(T - p B) = 1. We fix n >~ 1 and p = p,,. We will show that codimT-C(T - p B ) -- 1. Let
7, -- Z
bm cos(kmx).
Ill ~ 1 l
Since A is closed, a function ~ E Z defined by rl - 2.., am cos(kmx),
b177 a117 m
A
A
A(km) - f ' ( 1 ) - p B ( k m )
!1l ~z 1l
solves the equation (T - pB)rl -- d/. From (A2) we have for all sufficiently large m laml~<2
A]bm[ -- A (kin)
~<2
AII~PI[ - A (kin)
hence using (A4) we conclude that r/ E D ( A ) n C ~ Thus we have r/ 6 X and s p a n { c o s ( k m x ) , m :/: n} E T4.(T - p B ) . On the other hand if ~ E T~(T - p B ) then for some 77 E X we have 7t -- (T - pB)rl E s p a n { c o s ( k m x ) , m :/: n}. It follows that span{cos(kmx), m ~ n} -- T4.(T - p B ) . A This establishes (65). Since B[cos(kmx)] = B ( k m ) c o s ( k m x ) , (66) holds as well. The assertion of the theorem then follows by [28,29]. D
3.3. Priority o f small vs. large patterns We now have two criteria for the appearance of patterned solutions, when the nonlinearity is bistable with stable zero at (say) u0 = 1: the one arising from Theorem 4, namely M (p, km) > f ' (1), and that arising from Theorem 2, namely M (p, kin) > M* (in the case N = 1). The second is only a sufficient condition, and the first provides bifurcating solutions whose stability would have to be checked. As the control parameter p increases in magnitude, it will be interesting to determine which of these two criteria is first satisfied. This depends simply on the relative magnitudes of M* and f ' ( 1 ) , which in turn depend only on H. Thus this priority will be independent of A or B.
698
P. C. Fife
In the case H ( u ) = (1 - u2) 2, it was indicated following (51) that M* --0.899 . . . . whereas f ' ( 1 ) = 8. Therefore in this case, patterns with amplitude around 1 appear much earlier than the solutions bifurcating from 1. The latter will in many cases probably be local but not global minimizers of the energy. It is also interesting to note that the criterion involving f~(1) can be obtained formally by constructing a number M~ analogous to (51), but by holding b - 1 throughout.
3.4. D e p e n d e n c e o f the solutions on )~ We now fix p > pc and ask how the minimizers depend on )~. Again for simplicity, we restrict to the dimension N = 1, although analogous results hold in general. We suppress the dependence on p and write the minimizer 4} as ~bz. We now emphasize the )~-dependence of the energy functional E by writing it as E z. Finally, we write E0~) -- EZ[4}z]. Since the estimate (55) is independent of)~, we know that the amplitudes of our minimizing patterns remain bounded away from 0 as )~ --+ ~ . The following result on the minimal energy holds. THEOREM 5. For every positive n u m b e r )~o, the sequence E (v)~0) is nonincreasing in the integer v and approaches a finite negative limit as v --+ cx~.
PROOF. Any )~0-periodic function u is also a v)~0-periodic one; moreover, its v)~0-energy is identical to its )~0-energy. This is because (a) -sl f o H (u) d x is clearly the same, and (b) if we expand u in Fourier series oo U -- Z
Un e i2nrcx/)~~
l,I--n ~ Un,
-(30
we see, defining E~[u] -- 1
Eo~
Z
((A - p B ) u , u), that
[Unl2M p'
/7~--00
where Um - _ Un for m that
vn
E(v)~o) <~ EVZ~
2nzr]_
1 ~
2M(2mrr)
--~0J
2m=
--(N:)
[~m[
P'T~-0
'
(67)
; blm -- 0 otherwise. The right side of (67) is ~0 ~-'~z~[u]. It follows
01 -- EZ~
E()~0),
which proves the monotonicity of E (v)~0). To show that E(v)~o) approaches a limit, it suffices to show that EZ[u] is bounded below, independently of ;~ and u, for fixed p. Let F(u 2) be a nonnegative smooth convex function of u 2 satisfying F(u 2) ~< H (u) and (11). Let E" be the associated energy (9). Thus E'[u] ~< E[u] for all u. In Theorem 1 we found the minimizers of E', among complexvalued periodic functions, explicitly. They are exponential functions. Their energies are
Pattern formation in gradient systems
699
verified to be bounded independently of )~. The minimal energies among real-valued functions are no less than they are among complex-valued ones. Therefore the energies E[u] are likewise bounded below. This completes the proof. Consider now the minimizer 4~0 for any period interval ,k0. Since it is also a solution of (41) with period v)~0 for any positive integer v, it is a stationary point for E vz0 on that larger interval, but is no longer necessarily a minimizer. If ,k0 is large, however, 4~z0 is at worst only weakly unstable with respect to the interval v)~0, in the sense that solutions of the evolution problem starting near 4~0 move away from 4~0, if at all, only slowly. Specifically, we have the following result about the L 2 norm of the velocity ut: D THEOREM 6. Consider the evolution (40), where u is required to be a v)~o-periodic function of x for each time t, and to satisfy the initial condition u(x, O) = uo(x). Let 6()~) be any function of ~ approaching 0 as X --+ ~ . There exists a function 61 ()~) independent of v with l i m z ~ 61 (X) = 0 such that if EVZ~ ~< EVZ~ 0] + 6()~0) = E()~0) + 6(~0), then
fo ~ I[u,(., t)II 2 at
~ ~, (~o).
(68)
PROOF. We calculate
dtd E[u(., t ) ] - - ( A u - p B u - H ' ( u ) , u t ) --
- Ilu,(, t) II2,
(69)
hence
dt -- E'~Z~
- lim E~Z~
., t)] ~< E(),,o) + 6()~o) - E(vXo).
t-----~ o o
The conclusion (68) follows by the previous Theorem 5 and our assumption on 6.
D
3.5. Further questions The most important continuing questions about the minima of E (38) have to do with their dependence on )~. It may be expected that in many cases the global minimizers, or translates of them, will approach some periodic function as )~ --+ cx~ uniformly on bounded intervals. If that is true, then the pattern's properties will be more or less insensitive to the size of the domain in x-space. Simple as this concept may be, apparently the only case which has been successfully studied from this point of view is that of the functional (74) below and its generalizations to some other second order functionals. The results of Leizarowitz, Marcus, Mizel, and Zaslavski will be reviewed below in Section 3.6.3. Our Theorem 3 establishes a lower bound on the amplitude of ~b~ which is independent of ,k; but it is quite possible that no sequence of them, as )~ ~ oo, will approach a periodic stationary solution.
700
P. C. Fife
3.6. The S w i f t - H o h e n b e r g a n d related equations 3.6.1. M o t i v a t i o n f r o m f l u i d convection. simplest form is
The Swift-Hohenberg equation, which in its
u, -- - ( V 2 + 1)2u + otu - u 3,
(70)
together with its generalizations, have been, and continue to be, extremely popular as models for various kinds of patterns in nature. Many references can be found in [31 ] and [27]. Especially interesting are its more recent uses in modeling localized patterns, such as "oscillons" seen in vibrating granular materials [77,78,30] (see [76] for a review of various mechanisms capable of producing localized patterns). This equation is sometimes generalized to have complex coefficients; then traveling wave solutions may exist. The equation is no longer gradient in that case. For examples pertinent to lazers, see [52,53], and to patterns of swimming microorganisms [63]. In their original paper [81 ], Swift and Hohenberg studied B6nard convection with random thermal fluctuations by means of a weakly nonlinear analysis of the Boussinesq equations. Because of the inherent symmetry of the model, they predicted a first-order transition from disordered to ordered state past a critical value of the Rayleigh number, although they comment that this first order transition will be unobservable in their setup because of the smallness of the fluctuations. The weakly nonlinear analysis leads to a fourth order parabolic differential equation with an added cubically nonlinear integral term with complicated kernel. Making the approximation that the kernel is constant reduces the integral term to an ordinary cubic nonlinearity. The evolution equation which they obtained is of the form Ow Ot
8F --
c~--+
8w
r/,
(71)
with free energy given by
F [ w ] --
f
+
+ q2)2)w(x)
+ yto4].
(72)
Here the solution w represents the amplitude of the most unstable mode (a linear combination of fluid velocity and temperature), q is a critical wavenumber, and R, or, 13 are physical parameters. In fact, R is a scaled Rayleigh number. The last term ~ is a random forcing term due to thermal fluctuations. The gradient flow is R 1 Ut -- --(V 4 -Jr-2q2V 2 -+- q4)w -~- -- W - - - y w 3 + o
---- - V 4 w - 2q2V2w - lZW - K w 3 + rl,
(73)
Pattern formation in gradient systems
701
where K = V / ( 4 f l ) and q2-
R/fl
~/4~ In typical cases, # > 0 so that the function analogous to our f (u) in (1) is monostable (at the origin) rather than bistable. Equation (70) is a scaled version of this. Returning to (70), we see that it is form (40) with A = - V 4, B = V 2, p = 1, and H ( u ) a quartic nonlinearity. The control parameter is c~ rather than p. Our gradient flow does not have the fluctuation term ~ (73). In [81], attention is drawn to a previous model of Brazovskii [13] which has many of the same characteristics. Brazovskii's model is less grounded in specific differential equations; it is a general model for the transition between a disordered (as for a liquid) to an ordered (as in certain crystals) phase of a material in the presence of fluctuations. Again, a first order transition is found, occurring at a critical value of the relevant parameter which is determined by the criterion that the energy difference between the two phases changes sign. A comparison of the two energies is in effect the main tool which we have used in our analysis of the appearance of stable patterns. 3.6.2. Motivation from materials science. A fourth order equation similar to (70) was also proposed as a model for nonuniformities in a nonlinearly elastic rod by Coleman, as described in [54]. Consider such a rod of unstressed length ~., and subject it to tension, sufficient to make its stretched length equal to g. Let X (x), x 6 [0, ~.], be the deformation, so that X (0) = 0, X (~.) = g. Let u ( x ) = X ' (x) be the strain. It is postulated that the equilibrium value of u (x) minimizes the energy functional
E[u; )~1 --
-~ u-~x - -~ pu x + H ( u )
dx
(74)
under the length constraint fo~ u (x) d x - ~. (Here I have translated their notation into that of the present paper.) It is expected that when p > Pc > 0 for some critical value Pc, the indefinite nature of the energy will result in the minimizing strain being oscillatory. 3.6.3. Previous and current (1998) results. In the paper [54], a detailed mathematical theory is given for the generalization of the problem of minimizing (74) to one obtained when )~ ~ ec, so that the rod occupies the half-line [0, ec) (analogous results hold for ( - e c , ec)). Moreover, the integrand in (74) is generalized to other functions, still involving first and second order derivatives. An extensive mathematical treatment of the analogous problem with a mass constraint was given in [26,58-61 ]. Such variational problems on an infinite interval are sometimes called "infinite horizon variational problems". Marcus and Zaslavski ([87] and later papers) have further developed the theory of second order infinite horizon problems. Considerable discussion is given in [54] to the question of how to formulate a reasonable variational problem on the infinite interval, so that the energy is finite for bounded (not
702
P. C. F i f e
necessarily periodic) functions u. Moreover, the minimizer should automatically minimize the energy "locally" in some sense. Finally, the requirement of fixed length clearly is not appropriate; the authors instead impose a given tension. In [54], the existence of minimizers in their sense was proved, and also the existence of periodic minimizers. The method of proof relies on a local minimization problem making sense: the local problem is to minimize the integral analogous to (74) under the constraint that boundary values for u and Ux are prescribed at the two ends of the interval. This extension to an infinite interval is designed to exhibit a possibly patterned stable configuration for the strain which is independent of any boundary effects coming from the ends of the rod. In the generality of our problem (40), the approach in [54] is not likely to be possible, because it is not appropriate or possible to prescribe boundary conditions at the ends of a finite interval. As I mentioned before, an analysis of the corresponding problem on an infinite domain with mass constraint was initiated by Coleman et al. [26] and continued by Marcus and Zaslavskii [58-61 ]. For both the unconstrained and constrained problem, it was proved in [59] that any sequence of minimizers for the corresponding finite domain problem, in which the length of the domain tends to infinity, has a subsequence converging to a minimizer of the infinite domain problem. Rates of convergence are found and much more. Many of the results concerning the constrained problem were extended by Marcus and Zaslavski for a much more general family of second order integrands, in [61 ]. In [60], the authors deal with unconstrained problems and discuss the shape of minimizers on (0, c~) and in particular the shape of periodic minimizers. All this is done for a wide class of integrands, not just the basic model. In addition they show that generically, in a very precise way, there is uniqueness (up to translation) of the periodic minimizer. This means that by arbitrarily small perturbations of a given integrand one obtains integrands with this uniqueness property. In a paper under preparation, these same authors obtain the precise behavior of optimal solutions for the unconstrained problem on the half line and show that at infinity each of them converges to a periodic solution. The existence of periodic stationary solutions of the Swift-Hohenberg equation (70) was given in [27] for small positive values of c~ - 1. Motivated by [54], extensive further investigations into solutions of Ut = - - t t x x x x
-- PUxx
-~- tt - - tt 3
(75)
were made by Mizel et al. in [65]. The same concept of variational problem for functions on the whole line was used as in [54]. A specific estimate for a critical value p = Pl was given, beyond which nontrivial minimizers exist. It coincides with the number/5 (52) calculated for this example. Symmetry properties of the periodic minimizers were proved; for example, they must be even with respect to maxima. Upper and lower bounds for the minimal energy E (p) were given in terms of p; in particular it was found that E(p) approaches - c ~ like the 4th power of p. Finally, the authors constructed global branches of stationary periodic solutions of (75), the parameter again being p.
Pattern formation in gradient systems
703
3.7. Activator-inhibitor patterns in biology Activator-inhibitor models are usually pairs of nonlinear reaction-diffusion equations whose nonlinearities have certain monotonicity properties, suggesting that one of the two functions in the solution p a i r - the activator u - serves to increase the rate of both reactions, and the o t h e r - the inhibitor v - serves to inhibit them. With assumptions on the relative magnitudes of the two diffusivities, these systems are often used to model patterns (not necessarily stationary ones) in biology. See, for example, [47,62] among many references. A rough intuitive description of the basis of these models is this. Consider a solution of the evolution system which is constant in both space and time. If this uniform distribution is perturbed near a single location by the introduction, say, of an extra amount of u, this additional activator will cause both u and v to be produced at that location in greater amounts. The inhibitor v, however, diffuses away rapidly, spreading its effect over a larger territory. Its inhibiting effect is thereby diminished at the original point of production. As a result, the initial surplus of u can continue to increase locally to form a spike-like inhomogeneity, or pattern. Nonlinear effects prevent it from becoming too large, and a stable spatial pattern results. In short, unequal diffusivities may cause the uniform distribution to be unstable, with typical instabilities growing to form spatially patterned states. The competition between the activator and inhibitor influences is not the same as the competition between the stabilizing influence of the operator A and the destabilizing influence o f - p B that we examined previously (12), (40). However, we shall draw a connection now between the two settings. Consider the system ut-
D , V2u -? f (u, v),
v1 -- Dv
V 2v
+ g(u, v),
(76)
in which the D's are constants and the derivatives of the reaction terms f and g satisfy, for some M > 0, f . > 0, fv < O,
for iu[ < M and all v,
g. > 0 gv < 0
for all u, v.
(77)
Thus an increase in the amount of u serves to enhance both reactions when u is not too large, namely OuR < M. Commonly the behavior of at least the function f is different for larger values of ]u i, it being such as to effectively limit the growth of u. This results in a limitation on the amplitudes of the patterns. Apparently the first stability analysis of large amplitude patterns for systems in this class was given in [39]. It was for functions f and g of a special type. The following system of FitzHugh-Nagumo type is an example of an activator-inhibitor system (space, time, and the variable v have been scaled to produce a system with few parameters): ut - D V Z u - f (u) - p ( v - u), 6 l ) t - - V 2 1 ) -Jr- u -
1).
(78)
704
P C. Fife
Here the function f (u) is decreasing in a finite interval u 6 I, and an increasing function outside of I. In fact, l i m l u l ~ f ' ( u ) / l u l = ec. If D is an O(1) quantity, the parameter 1/e is a measure of the relative magnitude of the diffusivity of v, as well as of the strength of the v-reaction. The parameter p represents a measure of the strength of v's inhibition and u's activation of the first reaction. 3.7.1. Fast kinetics. Consider, for simplicity, bounded solutions of (78) in the entire space R 3, in the limit e --+ 0. The second equation can be solved for v in terms of u: v
-
-
( - V 2 + 1) -1 u - G . u ,
(79)
where G is Green's function for the operator (--V 2 -~- 1). With this, (78) becomes ut = D V 2 u - p ( G * u - u) - f (u),
(80)
which has the form of (40) with A u = DV2u,
Bu = G 9u - u
(81)
and H ( u ) -- fo ~ f ( s ) d s . The function H has the properties required in Section 3, namely superquadratic growth in lul. If f ( u ) = 2 u F ' ( u 2) for a function F ( w ) with superlinear growth, it fits into the framework of Section 2.2. Models of this sort were presented in [72,80,48]. Models bearing some similarity were studied in [71,55,12]. We consider one space dimension, as entirely analogous results hold for higher ones. In 1D, the convolution involves an integral over the whole line with the Green's function kernel G ( y ) -- 89e -lyl, and the function u is periodic with period ~. It is easily checked that G 9 u is then also periodic with the same period. The operator B in fact satisfies all the requirements of our theory with G ( k ) given below. We have A
AA(k)
--
- O k
2 ,
M ( p , k) - - D k 2 +
_k 2 B'(k) -- G(k) - 1 - 1 + k 2 ~ 0, pk-~2
(82) (83)
l + k 2" The function M attains positive values only for p > D, which is the critical number identified as l i m z ~ p*()~) (14), or the number p~ in Theorem l(f) in the case f ( u ) 2u F'(u2), u0 > 0. All the conclusions of Theorems 2-6 hold (and Theorem 1 as well, if f ( u ) = -2uF'(]ul2)).
P a t t e r n f o r m a t i o n in g r a d i e n t s y s t e m s
3.8.
Related
705
models
The Green's function G ( x ) used in the previous Section 3.7.1 is just one example of a class of influence functions G in (80) for which our conclusions are valid. We simply require
G(x)
~ O,
F
G(x)
1.
dx-
(84)
OC~
Then B u - - G 9 u - u is a negative self-adjoint operator. If A u - - Uxx, then these two operators satisfy our assumptions throughout, and our conclusions hold regarding patterns for the equation u t - - U.rx -
p(G
* u -
u) -
(85)
f (u).
A complementary model is obtained when p < 0. Although such an equation is still a gradient flow, when f is bistable it has traveling wave solutions defined on the whole line. In [8], the existence, uniqueness, and some stability results were obtained for such waves, with and without the second derivative term on the right of (85) (but with p < 0). This paper was nominally for the equation without the x-derivative, but the results clearly hold with it present. If it is absent, an interesting feature of the traveling waves is that their profile may be discontinuous when the velocity is zero. Finally, we consider another sort of evolution, when A and B are both convolution operators. Let G + (y) be two functions, each satisfying (84). We consider the evolution equation ut
If we set
-- G +
, u - u - p
G -- G + -
ut -- G 9 u -
pG-,
lu -
(G- *
u-
u)
-
f(u).
(86)
then this equation takes the form f(u),
where I - - f-~vc G ( y ) d y . We therefore have an integrodifferential equation similar to that in [8,23,24,45], but with a kernel which can change sign. Identifying A u - - G + 9 u - u , B u - - G - 9 u - u , we obtain (1). Also AA(k)
-- G+ (k) - 1,
B'(k) - G - (k) - 1.
(87)
Although assumptions (A1) and (A3) are satisfied, (A2) is not; in fact A ( k ) / B ( k ) --+ 1. The following theorem applies to a general situation suggested by this example. We consider two cases (one may apply at one value of p > p*()~), and the other at a different value of p): (i) M ( p , k i n ) has a maximum with respect to m at a finite value m * ( p ) . (ii) M ( p , k ) approaches its supremum with respect to k at k = oc.
706
P. C. Fife
THEOREM 7. Assume that (A1) and (A3) hold, and A
lim k~
A(k)
= ~.
(88)
B (k)
If p is such that case (i) holds, then the applicable conclusions of Theorem 1 hold for that p. On the other hand if case (ii) holds, then there is no global minimizer of exponential form for that value of p. However there is a minimizing sequence of exact stationary solutions of (12) of exponential form, along which the wavenumbers approach cx~. The energy levels of these solutions approach a finite limit, as do their amplitudes. The proof is along the lines of the foregoing, and will be omitted. If M(p, k) approaches its supremum only at k = cx~, there exist infinite sequences {km} (see (15)) and am, defined for sufficiently large m, which satisfy (22). They generate exponential solutions whose amplitudes approach a solution of (22) with k = o
4. Systems with an imposed conservation principle In place of (8), we now consider the evolution law (89)
0u : V2 S E [ u ]
Ot
6u
'
again for the time evolution of ~.-periodic functions of x E ~N. This equation is still relaxational with respect to E, because along solution paths u (., t) the following holds, where the scalar products and norms are for L2 (A).
Although there are many others, this is the most frequently encountered type of relaxation law with the property that the average value
t7 -
IAI
u(x, t) dx
(90)
is independent of t. It represents a gradient flow, not in L 2 (as (8)), but with respect to another scalar product (., .). One considers periodic functions w(x) with 6) = 0, solves vZqb -- W for a A-periodic potential q~ with q5 - 0 . The mapping w --+ q~ is unique. The scalar product is then (wl, w2, )x - ( V ~ l , Vc~b2)L2. This is in fact can be taken as the scalar product for the dual space ( ~ l ) , , where /_~1 is the subspace of H l consisting of functions with zero average.
Pattern formation in gradient systems
707
For our familiar energy (38), we have the evolution ut -- - V 2 [ A u - p B u - f (u)].
(91)
Again, we consider solutions which are periodic in space with period cell A. Because of the generality of (91), we avoid giving any specific conditions for existence of solutions, and assume that A, B, H are such that the initial value problem is globally well posed, possibly in a weak sense, in some subspace Y of LZ(A) (i.e., periodic with period cell A) containing the domain of E (38).
4.1. Stable patterns In view of the constancy of/7 (90), it is appropriate to look for a global minimizer of E (9) or (38) under the restriction that its average is a prescribed number/7 = b (say). These solutions satisfy Au - p B u - f (u) - cr = 0
(92)
for some constant ~r. If b = 0, the functional E is as in (9), and we allow complex-valued functions, then necessarily o- = 0 and the global minimizers studied in Section 2.2.1 are the same as those in the present context. On the other hand if we restrict to real-valued functions, then we have a result analogous to Theorem 2, under the additional assumption analogous to (42), but with/7 = b prescribed. For fixed b, set Mt, (im i)
_
i an f H * ( a ,ab 2, Iml)
(93)
and tSb -- inflP" M ( p , k m) > M ~ ( l m l ) f o r some m}.
(94)
THEOREM 8. There exists a number pc(b, X) ~ [p*(X), fib(X)] such that (a) f o r each p < Pc, there exists no nontrivial global minimizer o f the function E (38) in the class o f real-valued functions satisfying/7 = b; (b) f o r each p > p~:, there exists such a restricted nontrivial real global minimizer ckp o f E in Dr A L ~ with
E[~ '~ < 0 . The proof is similar to that of Theorem 2.
(95)
708
P. C. Fife
4.2. Metastable patterns: the Cahn-Hilliard equation and another kind of competition In the case of the Cahn-Hilliard and similar equations, it is appropriate to take Bu -- - u , so that the destabilizing influence is
- p B u -- pu
(96)
and M ( p , k) -- A(k) + p. In this case the maximum of M ( p , k) with respect to k is always attained, like A", at k - 0, and the globally stable states have maximal wavelength. This contrasts with the minimizers in Section 2.2, for which the wavelength was never maximal for large enough p. The difference is that this choice of B violates part of our assumption (A3). Nevertheless the most essential parts of the foregoing theory still apply. As an important example, take the quartic nonlinearity H(u) -- 88 4 + / 9 2 ) . Then we may write the last two terms of the energy (38) as
1
lfA
-~ p{Bu, u) + -~]
1)
lfA(1 lU4 H(u) dx -- IAI -- -210U2 -+- -4 + -~ 102 dx.
(97)
If Au -- V2u, we then obtain from (91) the Cahn-Hilliard equation Ut -- - - V 2 ( V 2 u + p u -- u3).
(98)
This equation was proposed in [17,19] as a model for some important features of solidifying molten binary alloys, such as spinodal decomposition. One interprets u as the relative concentration of one component of the alloy. For (98), global existence theories are available; for example this equation is included in a class treated by Henry [51 ]. Because the global minimizers constrained by/~ = b have maximal wavelength, even for large X, they do not always provide interesting patterns. Nor are these patterns the most relevant in typical applications. The reason is that a second kind of competition arises in the conserved case, which may result in temporary but long-lasting (metastable) patterned solutions. To examine this competition, we fix p = 1 in the above example (97) and (98), and in this section consider the problem in one space dimension, with X = 1. Then the minimization of the last two terms of E, i.e.,
1
-~ p<Bu, u} +
fo' H(u) dx - f(
l u 2 ) dx - l / ( u 2 H(u) - -~
- 1)2 dx,
is achieved when u is • To minimize 89 u) + fo H ( u ) d x , therefore, one must require the function u to take the value 1 on one set in [0, 1] and - 1 on the remainder of that domain. This is often called "phase separation". Since the total mass f u dx is prescribed, the measures of these sets are chosen to satisfy this constraint. However, we must also consider the additional term - 89(au, u ) - 89f~ u x2 d x in the energy (38). It should be as small as possible; this dictates that the transitions between
Pattern formation in gradient systems
709
-+-1 should be smooth rather than abrupt, and that there should be as few of them as possible. In other words, the desire to minimize the energy drives the configuration toward longer wavelengths (but they must all fit into the given period interval, which we are taking in this example to be (0,1)). This is the first of our new pair of competing influences. The second of these influences involves the kinetics of diffusion. It arises because to effect this separation with mass conserved, a redistribution of the particles whose concentration is u will be necessary. This must be accomplished by diffusion (diffusion is provided for by the fourth order term of (98) except when u is near +1; then the diffusion is governed by the other two terms on the right). Diffusion takes time, and the time required is longest when the sizes of the regions where u ~ 1 or u ~ - 1 are largest. The competition is therefore between (1) the fact that the energy is least when the length scale of the separation is largest, and (2) the fact that the time it takes to do that increases with this same length scale. This competition results in a compromise: there is a preferred range of characteristic lengths for the patterning mechanism which is neither too small nor too large. Here the terminology "preferred" takes the following dynamical meaning. Suppose that the initial state u(x, 0) is a small random perturbation of the constant solution u = / z and that that constant state is unstable. Then a whole range of patterned modes, whose amplitudes start small, will grow. Each mode has its own characteristic spacing. Those modes which have the fastest rates of growth will dictate the characteristic length scale of the emerging pattern. This is the length scale which results from the competition described above. The modes of growth from those almost constant initial data can be determined from a linearized stability analysis around the constant solution u = # (see, e.g., [49,7]). This linear analysis provides a dispersion relation between the growth rate of each mode and the wavenumber k characterizing that mode. The first rigorous analysis of this part of the separation process was given by Grant [49] for the 1D case. In this stage, we may take "separation" to be synonymous with "patterning". He proved roughly that if the initial state is a sufficiently small random perturbation of the constant state, then with high probability the evolution drives the system to a neighborhood of a stationary periodic exact solution with wavelength predicted by the linear dispersion relation. Such periodic solutions are unstable for )~ large enough, and the solution will not remain in that neighborhood forever. This temporary patterning process is the mathematical analog of spinodal decomposition [18], an event common in the processing of alloys. The subsequent evolution of the system away from the unstable separated solution takes place more slowly. It is also the mathematical analog, in this model, of an important physical process, namely the coarsening of alloys. An extensive study involving both the modeling and numerical simulation of the statistics of coarsening in 1D was undertaken by Eyre [36]. Modeling more complicated alloys leads to systems of Cahn-Hilliard equations, which have even more interesting separation dynamics. Among the notable mathematical studies of these models, I mention the work of Eyre in [37]. Next, we review some mathematical results on separation in higher dimensions.
710 4.3.
P.C. Fife
Spinodal decomposition in higher dimensions
The process by which patterns are formed spontaneously under Cahn-Hilliard dynamics is considerably more involved in dimensions greater than 1. One reason is that although in 1D the patterns which are observed have maxima and minima and zeros which are approximately evenly spaced (so that the spacing is approximately equal to a number which we may call the characteristic length of the pattern), no such property takes place in 2 or 3 (say) dimensions. The zero set of a solution does not consist of approximately evenly spaced lines, which might be the analog of the 1D situation. Instead, they typically are irregular and curved. Yet they do appear to the eye to possess some characteristic length. To put the phenomenon of spinodal decomposition on a more satisfactory mathematical footing in higher dimensions, therefore, one needs to provide a notion of characteristic length for a given function and somehow to quantify the assertion that most solutions starting near an unstable constant evolve temporarily to a configuration with an identifiable characteristic length. We shall consider the Cahn-Hilliard equation (98) with p = 1 and a small parameter e inserted as follows: Ut - - - - V 2 ( 8 2 g 2 U
--
f(u))
(99)
where f is the cubic function indicated in (98), or more generally a function with the same qualitative properties. Then the characteristic length referred to turns out to be O(e). The equation (99) corresponds to the Ginzburg-Landau energy
1 f A [ l-~e 2 E[u]- -~1
IVul 2 + H ( u ) ]
dx,
(100)
where f (u) -- H'(u). The papers by Maier-Paape and Wanner [56,57] serve to quantify the intuitive picture we have been describing and to prove some form of it. They consider solutions of the Cahn-Hilliard equation which have evolved from a small neighborhood of an unstable constant solution #, but which have not evolved too far from it. Thus these solutions start in a neighborhood N ~ of it in H 2, and are considered before they leave a larger neighborhood Ne. In 3-dimensional space, both of these neighborhoods have size of order e 3. In other dimensions, this exponent 3 here and below should be replaced by the dimension of the space. To develop a reasonable assertion, one first considers the eigenvalues of the CahnHilliard operator linearized about #, and collects in a group those which are close to the maximal one, which is positive. The number of these eigenvalues is strongly e-dependent and will be very large when e is small. They generate a strongly unstable finite dimensional invariant manifold Me for the flow, which is imbedded within another finite dimensional inertial manifold M~ with much larger dimension. The crucial statements are that solutions starting in N o are exponentially attracted to M~, and that most of them exit Ne being close to Me. The meaning of "close" and "most" is this: one can choose an arbitrarily small number 6e proportional to e 3 and a probability p arbitrarily close to 1 and conclude that
Pattern formation in gradient systems
711
if the diameter of N ~ is taken small enough relative to e 3 and initial conditions are taken with projections onto the tangent space of M~ in a random manner, then the probability of the orbit exiting Ne at a point closer than 3e to Me is at least p. The second conceptual result is that those orbits which are close to Me have a characteristic length which is O(e). This idea needs to be clarified. They propose tying it to an upper bound for the distance between adjacent portions of the nullset for the function u, which we can think of as being the solution of the CH equation at the point in time of its exit from Ne. This means that one looks at a connected set X2 in space bounded by a set where u vanishes. Thus u > 0 or u < 0 on X2. One takes a ball of radius r contained in s and asks how large r can be. An upper bound, given in terms of e, for such radii can be established for most of the solutions mentioned above. This upper bound is identified as the characteristic length of the pattern. First of all, this upper bound is proved for functions which are a linear combination of those eigenfunctions of the linearized CH operator corresponding to the chosen eigenvalues close to the maximal one. A technical restriction is placed on the center of the ball. Since at the exit time most of the solutions are close to the invariant manifold Me, which is in turn close to the span of these eigenfunctions, the property holds for them as well. In short, these authors have proved that with high probability, solutions exhibit a mathematical phenomenon like spinodal decomposition, a type of phase separation, by the time they leave a ball of radius O(e 3) in H 2 centered at/~, provided that they begin randomly in a smaller neighborhood with radius of the same order of magnitude. Numerical simulations (of which there are many; see for example [34]) show that separation with characteristic length persists outside of that small neighborhood, when in fact the amplitude of the separated solution is O(1). Extensive Monte-Carlo simulations in 1D were performed by Sander and Wanner [79] in order to gain insight into the question of how long the decomposition persists. They made a systematic study of the distribution of exit points on the boundaries of balls Ve of various radii centered at #. Their observations indeed confirm the persistence of the separation mechanism beyond the confines of the small neighborhood figuring in the proofs in [56, 57]. In fact they indicate that the solutions of the Cahn-Hilliard equation remain close to the solutions of the equation linearized about the constant u = # much longer than one would expect - far into the nonlinear regime. This is important since the solutions of the linearized problem are built up from the various linear modes growing according to their known growth rates, and it is easily seen that the fastest growing ones will eventually dominate. As mentioned above, this domination is another way of expressing the spinodal decomposition event. The unexpected validity of the linear approximation for solutions growing to larger amplitudes can be given a mathematical justification. The following argument is based on that given in [79]; this and more have been proved in a forthcoming publication. Consider (99) with f ( u ) = - u + u 3, and # - 0. We may decompose the operator on the right as --A(e2A + 1)u § A(u 3) -- LIU § L2u.
(101)
Let u be the solution with u ( x , O) = 6dp(x), where 6 << 1 and 4) is an eigenfunction of the linearized operator L l corresponding to one of the group of fastest growing modes.
712
P.C. Fife
For those modes, it can be calculated that the eigenvalues x of L I a r e O ( 8 - 2 ) ,
and that
I [ L l t t l l L 2 ~ c s - 2 l l U l l L 2.
During this time, it is expected that in most cases the L ~ norm of u will be bounded by unity; the largeness of L lu is due to its large derivatives. On the other hand, L 2 u c a n be estimated to be bounded in the form I[L2UIIL 2 ceZllull 2H 2 IlUllL2 as long as u remains close to the linear mode, due to the fact that 4~ is an eigenfunction of the Laplace operator with eigenvalue O(1). Therefore the magnitude of the nonlinear part IIL2 U IIL2 relative to that of the linear part IIL 1u II/~2 will be of the order of 84 Ilu 11229This allows Ilu II/-/: to grow quite large before the nonlinear terms become significant. The conclusion is that the nonlinearity is relatively insignificant during the time the solutions grow to moderate size. This heuristic result in 1D of Sander and Wanner has actually been proved and improved by them, and extended to dimensions 2 and 3 in a forthcoming paper. They have shown that similar estimates for the nonlinear versus linear part of the solution are satisfied not just for solutions starting as 64~(x), but in fact on the whole dominating subspace, i.e., for solutions u which start as superpositions of several dominating eigenfunctions, and in fact for those which start "close" to that subspace.
4.4. Interfaces The previous section was about patterning in solutions of the Cahn-Hilliard equation, in the sense of the spontaneous formation of nonconstant solutions whose spatial configurations are dominated by structures with characteristic lengths. The configurations are not stable stationary states, however. In time, numerical simulations indicate that other structures appear with separated length scales: spatial domains appear within which the solution u(x, t) of (99) has a degree of smoothness which is independent of e, and in each of which u assumes values near one of the stable zeros of the function f(u). These domains are separated by narrow internal layers of characteristic width O(e). Within these layers, the solution makes a transition between neighborhoods of the two stable zeros of f , namely the ones belonging to the two domains on either side of the layer. At this stage in the dynamics of (99), the characteristic length of the pattern has increased from O(e) to O(1). Thus coarsening has progressed considerably. In terms of the competition discussed in Section 4.2, this shows that in time the first influence, namely the fact that the energy E is smaller at larger characteristic lengths, wins out. The properties of these layers, especially their dynamics, have been studied to a considerable extent. We shall briefly review the results. First, we recount results for the nonconserved evolution analogous to (99), namely a form of the Ginzburg-Landau equation studied in a materials science context by Allen and Cahn: Ut = 82V2U --
f (u).
(102)
This is the nonconserved equation Ou/Ot = - 6 E / 6 u sharing the same Ginzburg-Landau energy (100).
Patternformation in gradient systems
713
4.4.1. Ginzburg-Landau-Allen-Cahn interfaces in 1D. The slow dynamics of layered solutions of (102) was discovered by J. Neu (unpublished), when the function f is of "balanced bistable" type. For simplicity, we take the stable zeros of f to be at u = • 1. Then "balanced bistable" in the present context will mean that f ' (-+-1) < 0, f has only one zero between the two stable ones, and f l I f (u)du = 0. Neu considered the problem
ut=e2Uxx-f(u),
x6(0,1),
ux=0,
x=0,1.
(103)
Throughout the following, we assume that 0 < e << 1. There are solutions such that u(x, t) = 0 for x = ~(t) for some time interval, u(x, t) is exponentially (in e) close to 1 for x > ~(t) + 8, and to - 1 for x < ~(t) - 6, for any ~ > 0 independent of e. Moreover, the following order of magnitude relation holds for the motion of the interface: ~'(t) ~ Ce(e -(l-~('))/C - e-(~('))/e).
(104)
Thus, the motion is exponentially slow and is directed toward the nearest boundary. Moreover, the velocity increases dramatically when the distance to that boundary is decreased by an O(1) amount. This result was generalized to the multilayer case and established rigorously by Carr and Pego [20] and Fusco and Hale [46]. The asymptotic analysis of this phenomenon was further developed in considerable detail by Reyna and Ward [83,75]. There are many stationary solutions with layered structure at equally spaced locations on the interval (0,1). During the period of time when the slow motion of the interfaces described above persists, the solution stays close to the unstable manifold of one of these layered states. This remains true until two layers are close enough to each other, or one of them is close enough to the boundary. At that time, the velocity of these layers will have reached the order of magnitude O(1), and the two layers quickly annihilate each other (or one is annihilated at the boundary). This event decreases the number of layers, and shifts the solution to a different unstable manifold. Key features in the rigorous justification of parts of this picture are the construction of good approximations to the unstable manifolds in terms of coordinates that correspond to physical space, and careful estimation of the exponentially small eigenvalues of the linearization of the differential operator about the layered solution. In the same spirit as the above, a problem in higher dimensions related to the perturbation of a layered equilibrium state by changes in the shape of the domain was considered by Alikakos et al. [6]. 4.4.2. Cahn-Hilliard slow motion in 1D. equation (99) in one space dimension: ut---(e2Uxx-f(u))x
x,
Similar results hold for the Cahn-Hilliard
Ux--Uxxx--0
forx--0,1.
(105)
Because this equation conserves "mass" fo u(x, t) dx, the dynamics of the layers are different, and in fact when there is only one of them, it does not move.
714
P.C. Fife
The extension of the results of Fusco-Hale-Carr-Pego to this equation in the case of two layers was done in the paper of Alikakos et al. [1 ]. Later, this was done for more layers by Bates and Xun [9,10]. Again, the careful construction of approximate unstable manifolds was a key feature of the proof. Some of the spectral estimates used the results of [7]. 4.4.3. Cahn-Hilliard slow motion in higher dimensions. Consider now solutions of (99) in higher dimensions. The analog of the layers discussed above are curves (in 2D) or surfaces which are either closed or intersect the boundary. Again, these curves separate domains in which u takes values near one of the two stable zeros of f . When they are very close to being circular, the motion is again exponentially slow. The study of solutions structured in this manner relies, as before, on careful estimates of the spectrum of the operator linearized about a layered function. This was initiated by Alikakos and Fusco [3], who did the analysis in 2D; it was followed by Chen [22] in higher dimensions. In [4], the spectra of circular or spherical solutions with a single layer (which they term a "bubble") was studied, and exponentially small eigenvalues revealed in any dimension. In [5], these authors established the exponentially slow motion of layered solutions which approximate a single bubble, in higher dimensions. (If more than one bubble are present with different diameters, the motion speeds up; the larger one will grow while the other decreases, at a rate faster than exponentially slow. Thus motion with more than one bubble is inherently unstable.) One of the new features of this paper is the development of a method for determining the equilibrium positions of the bubbles. In this paper, the authors use what they term the "manifold approach". Consider "bubble" solutions u ~ of fixed radius centered at the variable point ~. As ~ varies, this family of functions forms an approximate invariant manifold for the evolution (99). That equation says that motion at a point u on an invariant manifold is in the direction s -- -V2(e2V2u - f ( u ) ) , so that s is tangent to the manifold. On the other hand, the derivatives Ou~/O~i form a basis for the tangent space to the above-mentioned approximate invariant manifold at the point u ~ . We therefore have the approximate relation
3u ~ ~-~(lg~ ) -- Ci (~ ) O~i
for some velocity functions Ci (~), for which estimates can be obtained. This is a kind of global Lyapunov-Schmidt reduction to a finite-dimensional dynamical system. The equilibrium position of a bubble corresponds to the condition c(~) = 0. Despite the clear geometric picture which emerges, the proof of its validity is extremely involved. In [ 14], Bronsard and Kohn gave a simple energy argument establishing that a layer for the 1D Allen-Cahn equation takes superalgebraically long to move a distance of the order O(1). Their method was refined later by Grant [50] to give exponentially long time behavior, also for systems of equations. However, the Bronsard-Kohn method does not establish "exponentially slow motion until two layers get close", and persistence of shape until they get close. These results were strengthened in [2], in which the energy approach of [ 14] together with the spectral facts of [4] were used to establish in a relatively simple manner slow motion in higher dimensions as well until "two layers get close together, or
Pattern formation in gradient systems
715
the bubble gets close to the boundary". The difference between this result and that of [5] is that the vector field now is only estimated, not determined precisely enough so that equilibrium can be determined. 4.4.4. The role of exponential asymptotics. We have given a brief outline of the phenomenon of exponentially slow dynamics for the Allen-Cahn and Cahn-Hilliard equations, together with associated analytical results. The details of this dynamics can be discovered by formal asymptotics based on the smallness of s, but this procedure is not standard, in the sense that exponentially small quantities must be accounted for. Considerable effort has been expended to construct exponential asymptotic theories for "layered" and "spike" solutions in these and several other contexts, including viscous shocks, the viscous Cahn-Hilliard equation, and the Gierer-Meinhardt system. We refer the reader to the surveys found in [84-86].
4.5. Phase-field models Our discussion of pattern-forming conserved evolutions would not be complete without mentioning the increasingly popular diverse class of phase-field models. These models are used to investigate, both numerically and theoretically, the properties of materials which undergo phase transformations, especially the dynamical and structural properties of the interfaces between two phases. The phenomenology of these models parallels that of the Cahn-Hilliard equation. In fact the mathematical spinodal decomposition event occurs [41,42], and interfaces form and move. The spinodal decomposition phenomenon appears not to closely mirror any important laboratory observation. As for the interfaces, the analogy with the Cahn-Hilliard model breaks down here, because interfacial motion is not in general slow. Rather it is typically governed by some known free boundary problem like a modified Stefan problem. Examples of formal reductions to such problems can be found in [16,15,44,21 ]. The following is a representative but certainly not universal setting for these models. One begins with an order or phase parameter function 4~(x, t) which is meant to represent the phase of a material that can exist in two phases, except that it is a smooth function with range an interval on the real line. Pure phases are represented by extreme values of 4~, and other values represent the state of the material at locations where it may be in the process of changing phase. One also has an internal energy density function e and an entropy density, which is a given function s = s(e, cp, V4~) of e, 4~ and the spatial gradient of~. We consider constrained gradient flows for the total entropy
S[e, 4)l -- f s(e, 4), V4)) dx,
(106)
where the integral is over the spatial domain occupied by the material in question. The constraint is that the total internal energy
E(t) -be constant.
f
e(x,t)dx
(107)
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P C. Fife
The corresponding gradient flow for the functions e(x, t), cp(x, t) is: Oe Ot
6S = -VDV -6e '
a~
as
= K --. at 6~
(108)
Here D and K are positive functions. In (108), the second equation would be the usual gradient flow for the functional - S if the variable e were not present. The first equation would be a constrained gradient flow analogous to (89) if there were no other variable ~p. If both equations hold, then it is easy to check that ~ S[e(., t), ~(., t)] ~> 0, so that - S is a Lyapunov functional and global minimizers are stable. In the simplest cases, 6 S / 6 e = 1 / T where T is the temperature of the material, so that with proper choice of D, (108)1 can be made into the usual heat conduction equation, in which the flux of internal energy e is proportional V T. In that case, one should express e as a function e(T, ep, V4)) in a way that is thermodynamically compatible with the function s(e, c~, V4~); the system then becomes one for the evolution of T and 4~. Also in the simplest cases, 6 S / ~ is like the Ginzburg-Landau bistable operator on the right of (103) and ( 108)2 assumes the form O~b 0t :
K
-T--[82v2q~ -- f(~b, T)],
(109)
e << 1, where the function f , in its dependence on 4), is bistable and grows fast enough as ~p --+ oo, as we assumed of the function f in (12). I have sketched the derivation of typical simple phase-field equations on the basis of a principle that the total entropy in the system should increase. The postulate in doing this is that the evolution should be the corresponding gradient flow. However, a local entropy principle can also be used to produce the same result without this second postulate. See, for example, [21 ]. In its place, one postulates certain phenomenological relations designed so that the local entropy production is nonnegative.
5. Patterns and phase evolutions Patterns whose amplitude and wavelength vary slowly and on long spatial scales are seen repeatedly in experiments and numerical simulations. Multiscale perturbation methods, going back at least to the work of Eckhaus [33] and Benjamin and Feir [11], have been useful in deriving approximate evolution laws which govern these modulated solutions of nonlinear PDE's. For example, small amplitude patterns arising when the control parameter is just past its critical value have dynamics given by amplitude equations [68]. Amplitude equations are derived formally by a weakly nonlinear analysis of the equations with the ansatz that the solutions can be expressed as functions not only of the original space and time variables but also of long range ones. If e is the ratio of small to long space scales, the solution
Pattern formation in gradient systems
717
is then expanded formally in powers of e, up to a few orders. This expanded form of the desired solution is put into the original equation and each order considered separately to derive, with the help of solvability conditions belonging to singular operators, equations for the various terms in the expansion. What results is typically a lowest order equation for the evolution of the amplitude in the long space and time variables which is much simpler than the original PDE. This simplicity is of course paid for by the restrictive assumptions on the form of the solution. Nevertheless, instabilities in the original unmodulated regular pattern are found this way, and it is widely believed that these modulations are an accurate reflection of nature. Larger amplitude patterns in higher dimensions for values of the control parameter well past criticality are also capable of modulations on such long space-time scales. When these basic patterns are rolls, i.e., periodic in one spatial direction and constant in others (and in the case of traveling waves also periodic in time), one sees slow modulations in the wavelength, orientation and amplitude of the solutions, of which the first two are the most important. These modulations may result in rolls which are now only approximately straight, or whose wavelength is only approximately constant, in local regions of space. The mathematical formalism for this scenario was developed in perhaps the greatest generality by Cross and Newell [32]. Here we briefly describe their procedure. First, consider a pattern of unmodulated rolls. The state u of the system at a given point in space and time can be specified exactly in terms of the value of the phase 0 of the (spatial and/or temporal) oscillation, 0 living on the unit circle. With no loss of generality, we may take 0 to be O(x, t) = k . x - cot (mod27r) for some constant vector wavenumber k = V0 pointing in the direction orthogonal to the rolls. Since we shall deal only with derivatives of 0, we shall ignore the "mod2zr" provision. The solution is then given by a periodic function u - u (0). For simplicity we consider only the case co = 0. For modulations, the wave-form u(O) will vary slowly in space and time, and the wavenumber k will do so as well. The first natural ansatz in this case is therefore that the field function u = u(O, X, T), which locally is a function only of 0, is also a function of the long space and time variables X and T. As before, let e << 1 be the ratio of short to long space scales: X = ex; it may be taken to be arbitrary. The second ansatz is designed to express the modulation of k, while retaining its meaning as the spatial gradient of 0. In the unmodulated case with k = const., we have that cO(x) = e k . x = k . X =_ tO(X) is a function only of X. In the modulated case, one still assumes that the long-scale phase 69 defined as 69 = e0 is a function only of X and T (it may also depend on e). The slowly varying wavenumber is then given by k(X, T) = Vx6)(X, T).
When the assumed form u = u (0, X, T) is substituted into the basic equations, one identifies Vx0 = Vx69, so that V~u = k.uo + e V x u = V x O . u o + E V x u .
Similar expressions are used for time derivatives. Then one proceeds as before to expand u and 69 in low powers of s, substitute these expansions into the original PDEs rewritten for functions of the long-range as well as
718
P C. Fife
the original space/time variables, and examine each order (power of e) separately. The resulting equations are again analyzed with the use of solvability conditions. The Cross-Newell equation is an equation for the field tO (X, T), and is derived as indicated. It is usually written in the form (110)
o T - M~ (~)V./r - M2(k) ( ~ . V ) ~ -- O,
where k is the unit vector in the direction of k. Since as a second order equation for tO of the form
02to
Or -- Z Mi,j(VO) OXiOXj = 0 i,j
k -
Vxto,
this equation can be written
(111)
for some nonlinear functions mi,j depending on the original field equations and the original roll solutions. Under some special conditions, this equation takes on a divergence form, so that it is a gradient flow for an energy functional whose integrand is a nonlinear function of k. This condition makes the analysis easier. In typical cases, when the original periodic roll solution is stable according to the linear stability criterion, Equation (111) is parabolic, so that its initial value problem is well posed for at least a short time interval. There has been work directed to using the Cross-Newell equation to explain pattern defects such as disclinations (places where the vector k suddenly changes direction or is ill-defined, dislocations (where a roll terminates) or grain boundaries (boundaries between regions with different pattern orientations). It is surmised that near defects, the local wavenumber k is forced outside the domain where the corresponding roll would be stable. At these locations, (110) loses its well-posedness, and therefore its physical relevance, unless other interpretive considerations are added to it. However, a weak solution or a regularized solution of (110) can in many cases be defined. The singularities of weak solutions and analogous properties of regularized ones are then interpreted as defects, including domain boundaries between differently oriented regular patterns. See [73,67,66] for explorations of these issues. In particular, [66] contains a thorough analysis of the Cross-Newell equation, in part from a geometric point of view.
Acknowledgements I am grateful to N. Alikakos for supplying a description of results on slow motion for the Cahn-Hilliard equation, and to T. Wanner for help with Section 4.3. Most of Sections 2 and 3 report joint work with M. Kowalczyk [43]. I profited from information supplied by V. Mizel and M. Marcus. This work was supported in part by NSF Grant DMS-9703483. Part of it was also supported by the Center for Theoretical Sciences, National Tsing Hua University, Taiwan, during my stay there.
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CHAPTER
14
Blow-up in Nonlinear Heat Equations from the Dynamical Systems Point of View
Marek Fila Institute of Applied Mathematics, Comenius University, Mlynska dolina, 842 48 Bratislava, Slovakia
Hiroshi Matano Graduate School of Mathematical Sciences, Universi~ of Tokyo, 3-8-1 Komaba, Tokyo, 153 Japan
Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Finding blow-up profiles via center manifold theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Explanation of the underlying idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Other approaches to blow-up profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Another application of dynamical systems theory to a blow-up problem 2. Beyond blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Global L 1-solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Complete blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Complex-time-continuation beyond blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Connecting equilibria by blow-up solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Structure of equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Global solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Classical connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Uniqueness results for classical connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. L l-connection from 4)/,- to 4)0, k ~> 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Conjecture and open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References
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Blow-up in nonlinear heat equations
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O. Introduction In some classes of nonlinear heat equations, solutions may not exist globally for t/> 0 but may develop singularities in finite time. Such a phenomenon is called 'blow-up' and it has been a subject of intensive mathematical studies in connection with various fields of science such as plasma physics, combustion theory and population dynamics. Early studies of blow-up problems, including the pioneering work of Kaplan [50] and Fujita [24], were mainly concerned with the question as to under what conditions blow-up occurs. Since the middle of 1980s, people started to pay more attention to the structure of singularities that appear as solutions blow up (see [23,36,28,80] for earlier works in this direction). Over the last decade, dynamical systems theory has turned out to be exceedingly useful in the study of blow-up problems. In some cases it provided powerful analytical tools to reveal the fine structure of singularities or to understand some global features of blow-up solutions. In other cases, where the dynamical systems theory does not apply rigorously because of lack of some crucial estimates, its ideas nonetheless served as a reliable guiding principle in making right conjectures. In this article we will survey briefly some of the recent developments in this field. In Section 1 we will show how the center manifold theory sheds light on the rich structure of blow-up profiles. We will also review some other dynamical systems approaches to blow-up phenomena. In Section 2 we will discuss whether or not solutions have a continuation beyond the blow-up time. Interestingly, some solutions can be continued beyond the blow-up time in a suitable 'weak' sense. Section 2 will be a preliminary section for Section 3, where we study orbital connections between equilibria via blow-up solutions.
1. Finding blow-up profiles via center manifold theory 1.1. Explanation of the underlying idea In this section we will present three examples in which the Center Manifold Theorem provides an important guiding principle for finding out detailed information about the 'blowup profiles'. The first example comes from the paper [20] in which Filippas and Kohn use center manifold analysis in order to make a reliable guess of the second term of the asymptotic expansion of blow-up solutions. The second example is taken from [22] where Filippas and Merle use a formal center manifold approach to study stabilization properties of blow-up solutions. After obtaining a formal picture, they then make it a mathematically rigorous result with the help of other analytical tools. The third example comes from [6,7] where Bressan uses center manifold analysis as a guiding principle to find out asymptotics of blow-up solutions as well as the final profile of the solution at the blow-up time. We now describe how center manifolds appear in these three examples. First one rescales all variables (dependent and independent) so that the behavior of an unbounded solution
726
M. Fila and H. Matano
near the finite blow-up time corresponds to the asymptotic behavior (as the new timevariable tends to infinity) of a bounded solution of the rescaled equation. The rescaled solution converges to an equilibrium and this convergence behavior provides much information about the nature of blow-up of the original solution. In order to obtain a more detailed information about the convergence, one linearizes the rescaled equation around the limit equilibrium. It turns out that the linearized operator has zero as its eigenvalue, hence the center manifold theory becomes relevant. It seems intriguing that the linearized operators that appear in Examples 1 to 3, despite their apparent differences, all have the same set of eigenvalues with exactly the same multiplicities. In fact, this is not surprizing because those operators are obtained from a similar source using different scalings. Therefore, if a center manifold occurs in one of the problems it must occur in the others, too. We note that there are problems where a center manifold does not appear in the blow-up analysis (see, for instance, [8]). Let us remark that the center manifold theory does not yield results on 'blow-up profiles' without considerable extra effort. Other powerful tools such as matched asymptotic expansions, for instance, are needed to find the final profiles at the blow-up time.
1.2. E x a m p l e 1 Let us consider a positive solution of the semilinear heat equation ut=Au+u
p
i n n N x ( 0 , T), p >
1,
which blows up at the time T > 0 at the point a sup x E I~ N
u(x,t) < ~
6 ]1~N .
f o r t 9 (0, T),
This means that
l i m s u p ] I u ( . , t ) IIL~(RN .. =~ t -+ T
and there exists a sequence {(Xn, tn)} C x,,--+ a,
(1.1)
tn --+ T
and
I[{ N •
(0,
T ) such that
u(x,,,tn)--+ ~
asn--+oc.
If we rewrite Equation (1.1) in the similarity variables (cf. [36,29]) w ( y , s) := ( T - t ) J ( P - l ) u ( x , t),
y := (x - a ) / ~ / T - t, s := - ln(T - t),
then w solves 1 1 Ws -- - V ( p V w ) 4- w p - ~ w , p p-1
p ( y ) "--e -lyl2/4.
(1.2)
Studying the behavior of u near blow-up is equivalent to studying the behavior of w as s ~ oc. If (N - 2)p < N + 2 then it was shown in [36-38] that w ( y , s) --+ k := (p - 1) - l / ( p - l )
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Blow-up in nonlinear heat equations
uniformly on the sets lYl ~< C, C > 0. In order to learn more about the way w approaches k, let us formally linearize Equation ( 1.2) about k. If v (y, s) "-- w (y, s) - k, then 1
Vs -- - V ( p V v ) + v + f (v), P
f (v) "-- ~k v2 -+- g(v),
(1.3)
g(v) - - O ( v 3)
as v --+ 0
and v --+ 0 uniformly on compact sets in y. Let Lp2 denote the space of functions 4} such that
fiRN di)2 (y)p(y) dy < cx~. The linear operator s
1
"-- - V ( p V v ) + v P
(1.4)
defines a self-adjoint operator in L 2p with eigenvalues 1, l, 0, _ l , _ plicities are 1, N,m "-- 89
1, ... , whose multi+ 1) . . . . . respectively. Denote by {e+} u+l the eigenfunc-
tions of s that correspond to positive eigenvalues and similarly {e?}j%l
denotes
the eigen-
functions that correspond to the eigenvalue zero and {el }f~-l eigenfunctions corresponding to negative eigenvalues. (The eigenfunctions are in fact Hermite polynomials.) Let us decompose Lp2 as
Lp2
_
X +
G X0 OX-,
where X +, X ~ , X _ are the closed subspaces spanned b y / ,e j + , N] j+=Il ' {e? } m j - - l ' { e l } j _-- 1' respectively. The presence of a nontrivial null space for the operator s namely the space X ~ suggests the use of center manifold theory. A center manifold for the equilibrium v = 0, which we will denote by W c (0), is defined to be a locally invariant manifold whose tangent space at v = 0 is equal to X ~ Unfortunately, a standard center manifold theory (cf. [75], for example) does not apply. The nonlinear term does not have the required properties in the natural function spaces (cf. [20, Section 3] for a more detailed explanation). Nonetheless, the center manifold theory can work as a good guiding principle. To see this, let us expand v as N+ 1
v(y,s) -- Z
m
cx~
flJ(s)ef (Y) + E ~ J ( s ) e ? (y) + E F j ( s ) e - f (y)
j--1
= "v+ + vo + v - .
j=l
j=l
(1.5)
M. Fila and H. Matano
728
Here v +, v ~ v - are nothing else than the projection of v onto the subspaces X +, X ~ X - , respectively. As we are looking at solutions of (1.3) that tend to zero as s --+ ~:~, the initial data should lie on the center-stable manifold of v = 0. Now, according to the center manifold theory (pretending that it applies), almost all trajectories that converge to v = 0 are asymptotically tangential to W e (0), or, in other words,
to+l+l,
I-o(1
~
,ss--,
and the rate of the convergence is of the polynomial order in 1/s, while some exceptional trajectories (namely those lying on the stable manifold) converge to 0 exponentially fast and satisfy
Io+l+l
~
This formal observation suggests that the following estimate holds "generically":
as s --+ c~ j=l
j--1
(1.6)
j=l
and that there are some exceptional solutions that converge to 0 exponentially fast and in which Y~j~--1 y2 will be the dominant term. To see how the solutions converge to 0, we plug the form (1.6) into Equation (1.3), to obtain ordinary differential equations which govern the motion of the neutral modes c~j:
-
+ o
,
j = 1, 2 . . . . . m,
(1.7)
i=l
where Jrj0 denotes the orthogonal projection onto e j0 and v ~ is as in (1.5) If N = 1 then the eigenvalues of E are simple which makes the analysis easier. The normalized eigenfunction corresponding to the eigenvalue 0 is
e~ c(1
C:---
27/.4 '
and the solution of (1.7) where m = 1 can be written as
Otl = - ~
2"cps + ~
-s
"
Although the center manifold theory does not apply rigorously, the above formal observations suggested by the center manifold theory have been confirmed by Filippas and Kohn [20] by using analyses based on the special structure of Equation (1.1). Among other
Blow-up in nonlinear heat equations
729
things they have established in [20, Theorem A] that either v --+ 0 exponentially fast as s ~ ec or else for any e > 0 there is so such that N+ 1
cx~
!17 2
j=l
j=l
j=l
holds for s ~> so. They have also justified the "equation on the center manifold" (1.7) for solutions of (1.3) that do not tend to zero exponentially. As a consequence, they have been able to show that for any C > 0 there is so such that sup v(y, s) - ~ lyl
1- -~
--o
(1) -
,
s>~so,
(1.8)
s
provided v does not tend to zero exponentially fast. Restated in the original variables, (1.8) becomes
(T - t ) l / ( P - l ) u ( x , t) = k +
2pl ln(T -t)l
(1-
Ix - al e ) 2(T - t )
+ o(]ln(T - t ) ] - l )
as t ~ T, uniformly in parabolas Ix - al 2 ~< C ( T - t). A link between center manifold analysis and the geometry of the blow-up set was also established in [20] by showing that the center of scaling a is an isolated blow-up point provided N = 1 and v does not tend to zero exponentially. This result is weaker than a result from [ 12] which says that if N = 1 blow-up always occurs at isolated points. On the other hand, unlike in [ 12], the argument in [20] is not intrinsically one-dimensional. The refined asymptotics (1.8) was generalized to higher space dimension in [57] and [4] under the assumption of radial symmetry of u. The nonsymmetric case was considered in [21 ] where it was also established that (1.7) can be put in a remarkably simple form and then solved explicitly. Concerning the refined asymptotics of u, it was shown in [21 ] that
(T - t ) l / ( P - l ) u ( x , t )
k
-- k + 2 p l l n ( T _ t)l
(
trA0-
(x - a)T A~- t) - a ) )
+o(]ln(r- t)l-' ) as t --+ T, uniformly in parabolas ]x - a] 2 ~< C ( T - t) provided v does not decay exponentially fast. Here
AO-- Q
IN-j 0
0 )
Oj
Q-
for some j E {0, 1 . . . . . N - 1 }, where Q is an orthonormal matrix, IN-j is the (N - j ) x (N - j ) identity matrix and 0j is the j x j zero matrix. A similar result was proved independently in [77] where refined asymptotics is obtained also for the case when v decays exponentially fast.
M. Fila a n d H. M a t a n o
730
We remark that in the work of Herrero and Vel~izquez on blow-up profiles [40-47,7679], dynamical systems ideas also play an important role. We shall discuss some of their results in Section 1.5.
1.3. Example 2 In this section we discuss a result from [22] on blow-up of solutions of the equation
Ut -- AU 4-IUIP-IU
in ]1~N X (0, T), p > 1,
(1.9)
where U is a function from ]1~N X (0, T) to Ii~M. Introducing the similarity variables as before, we obtain a vector-valued analogue of (1.2) which now has a continuum of stationary points, namely an (M - 1)-dimensional sphere of radius k which will be denoted by S~4-1 . Since the scaling properties and the gradient structure are the same as in the scalar case, many arguments from [36,37] apply. In particular, it follows from [36,37] that if N = 1 or p < (3N + 8 ) / ( 3 N - 4) then inf
[
WoES~4-1 JRN
] W ( y , s ) - W o l 2 p(y)dy--+O
as s --+ cx~,
and
Iw(y, s ) - Wor ~ o
inf
a S S ---> OO,
WoESf f -I
uniformly for lY[ < C. The question addressed in [22] is: Does W(y, s) approach a single point on S~4-1 ? Assume for simplicity that M = 2 and introduce complex notation as in [22]. The following parametrization plays a crucial role in [22]:
W(y, s) - e i~
(V(y, s) + k),
where 69 (s) is obtained for each s by
f l w ( y s) ,
-
ei~ 2p -
min (-) E S l
here ]. I represents the complex modulus. If E is defined as in (1.4) and
0
s
0)
then V satisfies the equation
V~ = LV + N(V),
f
Iw(y,s)-e~C~
P~
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Blow-up in nonlinear heat equations
where 1 2 4- Or2 4- gi (V), -s 1 N(V) := (~kk v 2 4- ~-~v
-- O(vi 4- k) 4- g2(V) ) T
with Ig i (V )l ~ CIVI 3 for i - 1,2 and IVI small. The operator L is self-adjoint on (L2) 2 2l , ' " ., and the analysis of the neutral modes is a key step in
with eigenvalues 1, 89
[22] in order to prove that there is Woc E S~4-1 such that W ( y , s) --+ W~c
as s --+ oc,
uniformly for [Yl < C. More precisely, Filippas and Merle show in [22] that V decays exponentially or with the order s - l as s ~ cx). The smallness of V than implies the convergence of tO(s) as s --+ cxz, from which the convergence W --+ W~c follows. The assumption (3N - 4)p < 3N 4- 8 guarantees that W is bounded.
1.4. E x a m p l e 3 In this section we consider the problem ut = A u + e u, u =O,
x E S-2, t E (O, T ) , x E OY2, t E (O, T ) ,
u (., 0) = u0,
x E S2,
where s C ]t~ N is a bounded domain with smooth boundary. In the case s = R N, a study as described in Section 1.2 was performed in [4,57] for radial solutions. Here we focus our attention on the approach of [6,7] which leads to an elegant center manifold analysis in a different manner. Instead of the similarity variables, Bressan introduces the ignition variables
r = - ln(T - t),
x-b
r/= ~/-(T
V(~, r) = u ( x , t) 4- ln(T - t),
- t) l n ( T - t)
here b is a blow-up point and T < 1 is the blow-up time. This change of variables was first suggested in [ 15,28]. Performing the ignition change of variables we obtain
V~=-~
V(~,r) = -r V(r/, - l n T )
O . VV +e v-l+-
1(, r
- . VV + AV 2
ifb4-r/~/re -rEOf2,
-- u o ( b + r / ~ / - T l n T ) + l n T .
)
1
--" A ( V ) + - B V r '
M. Fila and H. Matano
732
Since V is expected to stabilize as r --+ c~ to the solution
of A (V ~ ) = 0, it is natural to linearize the equation
Vr = A(V) formally around V ~ to obtain
Wr = D A ( V ~ ) W , here DA(V ~) is the differential of A at V ~ :
DA(V~)W---rI
W
. VW + 2
1 + 1012/4 .
The eigenvalues of the operator DA(V ~) in C ~ ( R N) are 1, 89 . . . . with multiplicities 1, N, 1 N (N + 1) . . . . . and, as before, the presence of a nontrivial null space suggests the use of center manifold theory. However, the domain of V (., r) depends on r, and the first order approximation is not uniformly valid. Using some delicate a priori estimates and a topological argument, it is shown in [7] that there are solutions which behave as suggested by center manifold considerations. More precisely, if ~2 is convex and b E 12 then there are solutions which blow up at b at the time T and whose final profile satisfies lim (u(x, T ) + 2 In Ix - bl - In]In Ix -
x--+b
bll -
in 8) = 0.
This asymptotic behavior is proved to be stable with respect to small perturbations of the initial conditions.
1.5. Other approaches to blow-up profiles 1.5.1. Here we describe the answer to the question as to which blow-up profiles occur for positive solutions of Equation (1.1) with N = 1. It was shown in [40] that if u blows up at x = 0, t = T and w(y, s) is as in Section 1.2 then one of the following cases occurs: (i)
w(y, s) = k,
(ii) w ( y , s ) - k - ( 4 z r ) l / 4 k
,/~p
H2(Y)+o(~)
a s s - - + cx~,
s
(iii) w(y, s) = k - Ce(1-m/2)SHm(y) +o(e (1-m/2)s) as s ~ oo, where m/> 4 is an even integer and C > 0.
Blow-up in nonlinear heat equations
733
Here Hi is the i th normalized eigenfunction of E, and the convergence in (ii) and (iii) takes j,ot place in H} (R) as well as in Clo c (Ii~) for any j >~ 1 and c~ 6 (0, 1). Herrero and Vehizquez say: "Our approach is deeply influenced by dynamical systems theory . . . " (cf. [40, p. 134]) but their proof does not follow by standard applications of semilinear parabolic theory (as developed in [39], for instance). Its elements include elaborate estimates for the heat equation, Harnack and nondegeneracy estimates as well as a new change of variables. The existence of solutions behaving as in (ii) was conjectured in [48], where (ii) was formally derived for the case p = 3 by singular perturbation techniques. Further formal analysis can be found in [28,26]. In [26], the flatter behaviors (iii) were obtained formally by the method of matched asymptotic expansions. As we mentioned before, it was shown in [20] that either w(y, s) ~ k exponentially (as s ~ ~ ) or (ii) holds. In [40,41], Herrero and Velfizquez proved also that (ii) occurs if u0 has a single maximum and that there exists an initial value u0 such that the solution behaves as in (iii) with m = 4. It was shown later in [9] that all flatter behaviors in (iii) actually occur for some initial data. The main tool in [9] is the renormalization group. In [43], the genericity of the blow-up behavior given by (ii) was established. Analogous results hold also for the exponential nonlinearity (cf. [40,44,9]) which can be treated in the same way using the methods of matched asymptotic expansions and renormalization group. Generalizations to higher space dimension can be found in [76-79]. 1.5.2.
Sign-changing blow-up solutions of the equation
ut--mu+lulP-lu
inIR N x (0, T), p >
1, ( N - 2 ) p < N + 2 ,
were studied in [16]. Let S be the set of all blow-up points. It follows from [62,63] that under some restrictions on p there exists u* E C(oc (It~u \ S) such that
u(x, t) --+ u*(x)
in Clo c as t --+ T.
There are solutions for which u* (x) -- U! (x - a) + o(x - a)
as x ~ a, U l ( x ) " - -
Splloglxll)l/(P-1) ( P _ 1)2lxl 2
and other solutions such that for some integer n ~> 2 and Cn. p > 0 one has
u*(x)-C,,,pU,,(x -a)+o(x
-a)
a s x --+ a, U,,(x) "- Ix[ -2''~(p-l),
cf. [42,78,9,63]. The main aim in [16] is to show that if/~ blows up at (x, t) - (0, T) and (i) lift(x, t)llL~ <~ C ( T - t) -l/ 0, (ii) for some M, R > 0 we have
I~(x, t) I ~ g
for Ixl ~ R, t E [0, T),
734
M. Fila and H. Matano
(iii) limt__,~fi(x,t) -- fi*(x ) f o r x ~ O , LI*(x) m UI(X)-'~-O(X) a s x --+ 0, then there is a neighborhood V of tT0 in L ~ and 6 > 0 such that for any u0 6 ~; the solution u blows up at the time T -- T ( u o ) at a unique point a -- a(uo), u ( x , t) >>.0 for (x, t) [ - 6 , 6] • [T - 6, t) and limu(x,t)=u*(x) t--+ T
forxr
u* (x) -- U1 (x - a) + o ( x - a)
as x -->. a.
Moreover, (a(uo), T ( u o ) ) --+ (0, T) as u0 --+ rio. The proof uses a dynamical systems approach similar to what we described in Section 1.2. It consists in showing that a behavior analogous to (1.8) is stable under perturbation of initial data. 1.5.3. Recall from Section 1.2 that a sufficient condition which guarantees the convergence w ( y , s) --+ k as s --+ cxz is: (N - 2)p < N + 2. It is natural to ask whether or not such a condition is also necessary in some sense. In [46,47], there is given an example of a positive radial solution u of (1.1) with
N~>ll,
p>
N-
2x/N-
N-4-2x/N-
1
(1.10)
1
such that u blows up at t - T, x = 0 and lim sup(T - t ) l / ( P - l ) u ( t , 0) -- cx~. t/ZT
(1.11)
To indicate how (1.11) is obtained one first observes that 113"(y) :-- C p , N l y 1 - 2 ~ ( p - l )
p
p,X"
-
1 N - 2 - ~
p - 1
'
is a singular stationary solution of (1.2) if (N - 2)p > N and it is in Hlloc near y = 0 if (N - 2 ) p > N + 2. Then one shows that there exist solutions of (1.2) which converge to w* as s --+ c~ in an appropriate way. To do this one linearizes around w* by setting
q, (y, s) -
w ( y , s) - w* (y).
Then !/, satisfies (pointwise for y r 0) the equation __ 1 yVq/+ q/s -- Aq/ 2 /
+
+ N
--" a ~
+ f(ff,).
p
cP-I p,N
ly[ 2
tit
p-
1
p-1 Cp,u p-I -- Cp, N w * ( y ) - p ly12
Blow-up in nonlinear heat equations
735
To prove that ~ (s, y) --+ 0 as s ~ ec in a suitable way, a key point consists in showing that the linear operator A can be extended in a suitable self-adjoint manner. This can only be done if (1.10) holds. 1.5.4. Equation (1.1) with p ( N - 2) > N + 2 was studied in [60,61] where some results on the behavior of w ( y , s) as s ~ cx~ can be found. 1.5.5.
In [1 ], the following system in I~ N is considered ut -- A u + v p,
p ) 1,
(1.12)
ut -- A u -q- u q,
q >~ 1.
(1.13)
Positive solutions which blow up at x -- 0, t -- T are studied there. If one rescales variables as q+l
p+ I
u ( x , t) -- ( T - t) x -
y,/r
,'q ' ~ ( y , s ) ,
v ( x , t) -- ( T - t)
,'q-' O ( y , s ) ,
s -- - ln(T - t),
-t,
then one is lead to the system 1 4) s - -
A c/) -
p+l
-~ y . V c/) +
~ P
1
- q- -
'
(1.14)
q+l - - 0 -
1 Os -- A O - ~ y . V O +~b q
pq-
(1.15)
1
This system possesses the trivial solution 4~ -- ~ -----0 and the positive constant solution (4~, ~ ) = (F, y), where gp =
p+l pq
-
_ 1 I-',
I -'q
q+l pq-
1
y.
Under some assumptions on p, q it is shown in [ 1] that any nonnegative and bounded solution of (1.14) and (1.15) which is defined for all ( y , s ) E ~ U X ~ is either one of the constant solutions or satisfies otherwise
I1~(., s) -/~11 + I10(., s) - •
~
0
I1~(-, s)ll + II0(., s)ll --, 0
as s ~
-oo,
a s s -----~ o o ,
where II" II denotes the L 2p ( ] t ~ N ) _ n o r m . This Liouville-type theorem is then employed to prove the existence of solutions of (1.12) and (1.13) exhibiting a behavior near blow-up which is different from those that occur in the scalar case.
M. Fila and H. Matano
736
1.5.6. A classification of connections between equilibria of (1.2) was given in [64] for (N - 2)p < N + 2. Under this assumption, a bounded nonnegative solution of (1.2) defined for (y, s) E R N x R is one of the following: (i) w = 0 o r w = k ,
or
(ii) there exists so 6 IR such that w ( y , s) = 9(s - so), where q)(s) -- k(1 + eS) - 1 / ( p - ' ) . Note that q9 is the unique global solution (up to a translation) of (/9
p
~Os . . . . p-1
t- ~o
satisfying q) --+ k as s ~ - o o and 9 ~ 0 as s ~ cxz. This classification is used in [64] in order to obtain some optimal bounds for solutions of (1.1) at blow-up time. 1.5.7. A remarkable blow-up behavior was studied in [31,32]. There, Galaktionov and V~izquez considered the problem
ut -- Uxx + (1 + u) ln ~(1 + u),
xeR,
u(x, o) - uo(Ixl)/> o,
xEIR,
t>O,
where fl ~> 2 and u0 E L ~ (R) is nonconstant and nonincreasing in Ix l. This problem was introduced in [27] (see also the references of [73, Chapter IV]). By the results from [31,32], the blow-up behavior is governed by the first-order equation of Hamilton-Jacobi type:
Ut
I+U
+ (1 + U) lnf(1 + U).
To explain the appearance of this Hamilton-Jacobi equation one introduces the change of variable
v(x, t)"-- ln(1 + u(x, t)). Then v satisfies the equation
Vt --Vxx + Vx2 + v ~ 9 If one now rescales as O(~, r ) " - - (T
-
-
t)l/(~-l)v(~(T
Ixl
9-- ~ , (T-t)
m
-
-
t) m, t),
[3-2 m
.m
2(~ - 1)'
r "-- - ln(T - t),
Blow-up in nonlinear heat equations
737
then one arrives at the equation (9r -- .4((O) -+-e-r/(/~-l)(o~,
1
.,4(69) "-- (O~ - m~ (O~: - /3-----~
(O + (O~
Hence, the equation for 6) can be viewed as an exponentially small (as r --~ co) parabolic perturbation of the autonomous nonlinear Hamilton-Jacobi equation hT = Ah.
(1.16)
It is shown in [31,32] that there exists a solution S of the equation .A(S) = 0 such that S >~ 0, S ~ const and (O(~, r) --+ S(~)
as r --+ oo, uniformly in ~.
The proof is essentially based on a general result on w-limit sets of perturbed dynamical systems which was introduced in [30]. A certain reduced co-limit set of the HamiltonJacobi equation (1.16) is shown to be uniformly stable. This makes it then possible to pass to the limit as r --+ oc in the parabolic equation for 6) to get the convergence to the unique stationary solution of (1.16).
1.6. Another application of dynamical systems theory to a blow-up problem The study of critical exponents for parabolic problems was initiated by Fujita in his classical paper [24]. He studied Equation (1.1) and proved that every positive solution blows up in finite time if p < 1 + 2 / N while global positive solutions exist if p > 1 -4- 2/N. For surveys of a number of later extensions and generalizations we refer to [56,14]. Here we only discuss the results from [65,66] since dynamical systems theory plays a crucial role in [65]. Consider the problem
ut=ux~+lul p-iu, u (-, 0) = u0,
xeR, t>0, x 9 R,
where p > 1. For f E HJ (R), er(x) -- e x2/4, f ~_ 0, define z ( f ) "-- sup{j" there exist - oc < xi < x2 < ... < xj+l < oo with f ( x i ) f ( x i + j ) < 0 f o r i -
1,2 . . . . . j},
let
.-Is
<' (R).
pk:=l+
k+l
M. Fila and H. Matano
738
The main result from [65] reads as follows. If p ~< Pk then any solution with u0 6 ~'k blows up in finite time. If p > Pk then there exists a global solution with u0 6 I7k. To explain the conditions on p, rescale the problem as follows"
v ( y , s ) "--(t + 1 ) l / ( p - l ) u ( x , t ) ,
x "-- y x / t + 1, s "-- ln(t + 1).
Then v satisfies the equation y 1 Vs - Vyy -4- ~v,, 4- ~ w p-1
-4-Ivl p - l v.
"
Let L be defined by y 1 Lq9 "= ~o~,~,+ ~Oy + " p-1 Then the j t h eigenvalue
1.j=
)~j
i0.
of L in H2 is
1
j+l
p-1
2
and )~k < 0 if and only if p > Pk. The eigenfunction corresponding to ~,j is
qgj(y) = H j ( y ) e -y2/4,
Hj is a suitable modification of the j t h Hermite polynomial and z(q)k) = k. If p > pk then the stable manifold of the trivial solution contains a function belonging to r k , which proves the global existence part of the result of [65]. Conversely, if a global solution exists it can be shown to converge to an equilibrium in Z j for some j ~< k or to the trivial solution. If p < pk then there is no equilibrium in s j ~< k, so one has
v(., s) IIv(', s)llnJ
-]'-q)j
in H~ as s --+ ec
for some j / > k + 1. This contradicts the fact that z(v(., s)) <<,z(v(., 0)) - k. The above reasoning is made rigorous in [65] (also for p - pk) and the restriction on the fast decay of u0 is removed in [66].
Blow-up in nonlinear heat equations
739
2. Beyond blow-up 2.1. Global L l-solutions We begin with the definition of L l-solutions of the problem
(P)
ur--Au+ f(u), u--O, u(., O) -- uo,
x6X2, t>O, x eOX2, t > O , x ~ S2,
here I2 is a bounded domain in RN. DEFINITION 2.1. By an L l-solution of (P) on [0, T] we mean a function u 6 C([0, T]; Ll (S2)) such that f ( u ) c L l ( Q r ) , Q;r := I2 x (0, T) and the equality
[u Oltr d x -
f'f= r
uqJ, d x ds -
f'f
(uAq* + f ( u ) e P ) d x ds
holds for any 0 ~< r < t ~< T and O E C 2 (Q;r), q / = 0 on O$2 x [0, T]. By a global L l solution we mean an L l-solution which exists on [0, T] for every T > 0. Next we recall some facts about global unbounded L 1-solutions of Problem (P) with f (u) = u p and f (u) = e". For f ( u ) = u p it was shown in [70] that a global unbounded (in L ~ ) positive L lsolution exists if I2 is starshaped and ( N - 2)p ~> N + 2. We now explain the reason why such a solution exists. By the Pohozaev identity, there is no positive equilibrium. The equilibrium u = 0 is stable. If we choose an initial function u0 6 C (S-2), u0 ~> 0, u ~ 0 and denote by u(., t; ku0) the solution of (P) with u(., 0) = ku0 then
k* := sup{k > 0: u(-, t; ku0) is a global classical solution such that u(., t; ku0) --+ 0 as t --> oo } is positive and finite since for k large the solution u(-, t; ku0) blows up in finite time. Now, u(., t; k ' u 0 ) cannot be global and bounded since otherwise its w-limit set would have to contain a nonnegative equilibrium. But the only nonnegative equilibrium is zero and its domain of attraction is open in any reasonable topology. Hence u(., t; k ' u 0 ) cannot converge to zero as t --+ oo. On the other hand, if we take a sequence {k,, } such that k,, / k* then u (., t; k,,u0) is a monotone sequence of global solutions and the Monotone Convergence Theorem can be used in order to pass to the limit in a suitable weak formulation of (P) and show that u(., t; k ' u 0 ) is a global L l-solution (see [70] for more details). For a long time it had been an open problem whether or not u(., t; k ' u 0 ) is classical for all t > 0 and becomes unbounded only as t -+ oo. An answer was given in [33] in the case when S-2 = BR(O) := {x E RN: Ixl < R} and uo is radially symmetric. It is shown in [33]
740
M. Fila and H. Matano
that if (N - 2)p = N 4- 2 and u0 is radially decreasing then u(., t; )~*u0) is indeed classical for all t > 0; while for N4-2
N-2
6 <14- N-10
if N > 10)
(2.1)
u(., t" ~.*u0) blows up in finite time and continues to exist globally only as an L 1-solution. It is not known whether the L l-continuation beyond blow-up is unique. It was shown in [54] that for the equation ut -- A u 4- e u
in II~3 there exist (classical) selfsimilar solutions in (-cx~, T) and (T, oe), T 9 IR, which coincide at the time T where single point blow-up occurs and thus can be glued together to form a "peaking solution" over IR. The uniqueness of the continuation was left open. The above construction of a global peaking solution yields an L l-solution. It was shown in [33] that a similar construction can be done for ut -- A u 4- u p
in I~ N provided N > 2 and p satisfies (2.1).
2.2. C o m p l e t e b l o w - u p Under several circumstances a solution that blows up at a finite time T cannot be continued as an L l-solution beyond T. This phenomenon is called complete blow-up. To describe it, let us recall a result from [3], as applied to Problem (P) with f ( u ) - up. Let f,, (u) := min{u p, n}. Let Un be the unique global classical solution of (u,,)t1/111 - -
Au,, + f , , ( u ~ ) ,
O ~
u , (., O) - uo >>O,
x9 x9
t>O,
t>0,
x 9 ,f-2, uo 9 L~(S2).
Suppose that one of the following holds: (a) u0 E W0 'l (S2) and Au0 + f ( u o ) >1 0 in D'(S2), (b) (N - 2)p < N 4- 2. Assume that the solution u ofProblem (P) with f ( u ) -- u p blows up atthe time T 9 (0, cx~). Then u blows up completely in the sense that (i) l i m , , ~ u,,(x, t) = u ( x , t) for all (x, t) 9 ~2 • [0, T], (ii) l i m n ~ u n ( x , t) = ~ for all (x, t) 9 S-2 x (T, ~ ) . It was shown later in [33] that if Y2 = BR(0) and u0 is radially symmetric then blow-up is complete in the above sense also for p = (N 4- 2 ) / ( N - 2), N > 2. Other results on complete blow-up can also be found in [33,53,58], for example.
741
Blow-up in nonlinear heat equations
The notion of complete blow-up can be defined in the same manner for a more general nonlinearity f (u), including the case f (u) = e", see [58]. A different but equivalent definition of complete blow-up was used in [53]. There, Problem (P) was reformulated by seeking the minimal solution to the integral equation
u(x, t) -- )~
f0'f
G ( x , y, t - r) f (u(y, r)) d r dy +
G ( x , y, t)uo(y) dy,
(2.2)
where G is the Greens function for the heat equation in 12 with the Dirichlet boundary condition on 0 I2. The true time of existence tc (to ~> T) is defined as tc "-- sup{t" u (., t) < ~ almost everywhere in $2}. Blow-up is then called complete if t,. = T. It was observed in [3] that lim,,_,~ u,, is the minimal solution of (2.2). The following proposition explains the relation between complete and non-complete blow-up, where non-complete blow-up means that an L l-continuation exists. PROPOSITION 2.2. Consider Problem (P) with f (u) = e u. Assume $2 is convex. Suppose a solution blows up in finite time but can be continued globally as an L l-solution. Then the following holds: (i) For any {to with 0 <<,fi0 < uo, the solution with initial data [to exists globally f o r 0 <~ t < cx~ in the classical sense. (ii) For any [to with {to > uo, the solution with initial data {to blows up completely in finite time. This follows from [53, Theorem 2] where a more general nonlinearity is allowed.
2.3. Complex-time-continuation beyond blow-up Even if blow-up is complete in the sense of [3,53], there may exist a way of continuing the solution beyond blow-up in a manner different from what we described in Section 2.1. Masuda discussed such a continuation in [59] using a complex time variable, as we explain below. Consider the problem ut=Au+u
(P2)
2,
x6S2,
Ou -~v -- O,
x E OS2,
u (., O) = uo,
x ~ S2,
M. Fila a n d H. M a t a n o
742
Im t
D1
1_-
•
-
Fig. 1. D o m a i n D1.
here ,(-2 is a smoothly bounded domain in R N . Assume that u0 ~> 0, u0 ~ 0, and u0 e W2(I2), p > N. It is easily seen that the solution blows up in a finite time, say T, where T~< a1 and a'=
1 fs2 uo(x) dx.
I~l
Let us now extend the domain of definition of u(., t) over the complex time plane. To do rr 1 so, for an arbitrary choice of O e (0, g) and 6 e (0, S) define Dl'-and D2
"-- f)l-
{
t~C:
largt[
-O<arg
It is shown in [59] that if
(
t---i6tanO a
t
a-2lluollwp(S2) is sufficiently
/
small then there are
precisely two solutions ul, u2 which are analytic in t 6 Dl, D2 (respectively) as W2(12) valued functions. They converge to uo as t -+ 0, l arg t[ < 69, in the norm of W2 (,(2). If t e Dl AD2 with Re t < T - 6 then ul(., t) =u2(.,t).ButifRet > T +6 then ul(., t) u2(., t) unless uo is constant. Furthermore, it can be shown that both Ul, u2 --+ 0 as Re t --+ oo. This result implies that there is a (non-unique) continuation for a solution of (P2) beyond the blow-up time T.
3. Connecting equilibria by blow-up solutions 3.1.
Preliminaries
In the qualitative theory of one-dimensional parabolic equations, much effort has been devoted to the study of the connection p r o b l e m - determining which equilibria are connected
Blow-up in nonlinear heat equations
743
by heteroclinic orbits (see [ 17] for recent results and references). By a connection from an equilibrium 4~- to an equilibrium 4~+ we mean a classical solution u(., t) which is defined for t 9 ( - e ~ , oo) and satisfies u(., t) ---> 4~+
as t --> -+-cx~.
(3.1)
Here we shall discuss a new concept of L l-connections which was introduced in [ 19]. By an L l-connection we mean a function u(., t) which is a classical solution on the interval (-cx~, T) for some T 9 R and blows up at T but continues to exist in an appropriate weak sense for t 9 [T, cx~) and satisfies (3.1) in a suitable sense. Hereafter we consider the problem
(E)
ut - A u + Xe", u --0,
u(x,o)-uo(ixl),
xeBl(O),t>O,
x 9 OBl(0), t > 0, x 9 B1 (0),
where Bl (0) = {x e ]I~N: IX[ < 1 }, U0 is a continuous function on [0, 1] vanishing at r = 1, ~. is a positive parameter and 3 <~ N ~< 9. This initial-boundary value problem has attracted much attention in the research literature. It is relevant as a model of solid fuel ignition (see [5]) and has also gained theoretical significance due to a variety of interesting phenomena exhibited by its solutions (see [5] for references and a survey of results obtained up to 1988). Our presentation in Sections 3.23.4 will follow [19] closely. The material in Sections 3.5-3.7 is taken from [18]. Throughout this section, X stands for the state space for problem (E); in other words, X is the space on which (E) defines a dynamical system. Although the specific choice of X is not very relevant, due to the smoothing effect of the parabolic equation, it will be convenient to have X chosen such that it is imbedded in C l (Bl (0)):
x ~ cl(~(o)). Thus we define X to be the space of all radial functions in a fractional power space Y~ associated with the Laplacian on y0 = LP (B1 (0)), under Dirichlet boundary conditions (cf. [39]). If p > N and 1 > ~ > ( N + p ) / ( 2 p ) , we have the desired imbedding, that is, (E) is well-posed on X and all functions in X vanish on OB l (0).
3.2. Structure o f equilibria The stationary problem corresponding to (E) is equivalent to
(SE)
N-1 ~,.,. + - - q ~ , . r
~r(O) --0,
+ s ~ --O,
r 9 (0, 1),
~(~) --0.
Note that even without the assumption on the radial symmetry of u0, Problem (SE) describes all equilibrium solutions of (E), since they are positive hence radially symmetric due to the general result of [35].
744
M. Fila and H. Matano
Let us first recall some known results on the structure of solutions of (SE): PROPOSITION A [34,49]. Denote by S the solution set of the parametrized problem (SE): S -- {(q~, X)" )~ 6 R + and 4) is a solution of(SE)}. There exists a smooth curve
s ~ (r
R+ ~
x • R+
such that ~(s) is the solution of(SE) with X = X(s), sup ~ ( s ) ( x ) = 4~(s)(0) = s xeBl(O) and S -- {(4~(s), ~.(s)): s > 0}. Moreover, the following holds: (a) X(s) is a Morse function, that is if X' (s) = 0 then ) n (s) =/: O. (b) The critical values of s properties:
form an infinite sequence )~j, ~2 . . . . with the following
)~1 > ~,3 > "'" > ~ - 2 j + l ~
)~oo = 2 ( N
- 2),
~,2 < ~,4 < " ' " < ~-2j+2 / 7 ~,oo-
(c) For each )~ <~ ~,l define ~)~ -- ~ (Si)
(i -- O, 1 . . . . ),
where so < sl < ... is the sequence of all points s with X(s) = ~.. This sequence is finite if X ~ ) ~ and infinite if )~ = ) ~ . In the latter case we have ~(r)
--> ~boo - - I n
2 ( N - 2) ~,r 2
in Clloc((0, 1]).
Next we recall a result on stability properties of equilibria. PROPOSITION B [67]. (a) q~(s) is hyperbolic if and only if X' (s) =/=O. (b) Let m(q)(s)) denote the number of negative eigenvalues of the problem
(L)
~rr+ Cr(0)-
N r
1
~r+
~(1)=0
)~e~(r ) 7t
+u~--O,
r E (0, 1),
Blow-up in nonlinear heat equations
745
I I
A2
A~
A3
A1
A
Fig. 2. Bifurcation diagram for equilibria.
with ~ = dp(s) and )~ = )~(s). Then m(dp(s)) = #{g ~ N +" ~ < s and i/(g) -- 0}. In other words, m(dp~.) = j for any j such that d@. is defined. In particular, dp~ is asymptotically stable for )~ < )~1 and it is the only stable equilibrium of (E). Finally, for the number of intersections of two equilibria the following holds: PROPOSITION C [19]. If )~ < )~l and k > j are such that 4)~ and 4)~ are both defined then
c~ - ~ has exactly j + 1 zeros in [0, 1], all of them simple.
3.3. Global solutions We shall need the following result. PROPOSITION D [19]. l f the classical solution u(]xl, t) of(E) is global, then it is uniformly bounded, that is
sup lu(r, t)l < c~. t>O, r~[O, 1] Recall that global classical solutions exist only if )~ <~ 1.1, see [51], where ~.1 is the last bifurcation value of (SE). The basic ingredients of our method include the properties of the "zero number". For a continuous function ~p defined on an interval J, we put Zj(~)-
#{r E J" ~ ( r ) - - 0 } .
746
M. Fila and H. Matano
The subscript J is usually omitted if J - [0, 1] or if it is clear from the context which interval we have in mind. PROPOSITION E [13]. Let w(lxl, t) be a (classical) solution o f the linear equation w,-
Aw + a(Ixl, t ) w ,
t ~ (T,, T2), x ~ S2,
where I2 is either a ball, S-2 = B R(O), or an annulus, S2 -- B R(O) \ BRo (0) with R > Ro > O, and a is a radial continuous function on I2 x (Tl, T2). Assume that w(r, t) is not identical to zero and that it is continuous on J x (TI, T2) (J - - [0, R ] f o r the ball and J - - [ R 0 , R] f o r the annulus) and either w ( R , t) ----O
(t ~ (T1, T2))
w ( R , t) ~ 0
(t e (T,, T2)).
or else
In case ~ = BR (0) \ B Ro also assume that either w(Ro, t) - - 0
(t ~ (T1, T2))
w(Ro, t) ~ 0
(t ~ (Tl, T2)).
or else
Then the following properties are satisfied: (i) z j ( w ( . , t)) is finite f o r any t ~ (Tl, T2), (ii) t --+ z j ( w ( . , t)) is monotone nonincreasing, (iii) (Diminishing Property) if wr(ro, to) = w(ro, to) --O f o r some ro ~ J, to ~ (7'1, T2), then
z (m( , ,))
s))
(r,
,
,o < s
In order to avoid possible confusions, we make the convention that a solution (with no adjective) always refers to a classical solution; it will be explicitly specified if a solution is to be understood in the L 1-sense. Sometimes, when we need to stress the dependence on the initial condition, we write u(., t; u0), t 6 [0, T(uo)) (T(uo) <<.ec), for the maximally defined (classical) solution of (E). As a preparation for the main results of this section, we introduce some notation and recall a few known properties of domains of attraction. Let DA denote the domain of attraction of r that is DA -- { uo E X" the solution of (E) is global,
bounded and converges to 4~0 in X }, and let 0 DA be the boundary of DA in X.
Blow-up in nonlinear heat equations
747
LEMMA 3.1. A s s u m e )~ < )~l. Then DA is open in X and the f o l l o w i n g properties hold: (al) For uo, (to ~ X the relations {to <~ uo and uo ~ DA imply (to ~ DA. (a2) v ~ ODA if a n d only if v q~ DA and there is an increasing sequence {vn} C DA such that v, --+ v in X.
(a3) ODA is positively invariant under the f l o w o f (E): if uo E ODA then the solution u(., t) of(E) satisfies u(., t) E ODA as long as it exists (in the classical sense). (a4) I f v, w E 0 DA are distinct then they are not ordered: neither v <~ w nor w <~ v. PROOF. The fact that DA is open is a standard consequence of asymptotic stability of 4~0. Property (a l) follows from the comparison principle and the fact that 4~0 is globally asymptotically stable below (any solution below 4~0 converges to 4~0). Property (a2) is a consequence of general results on domains of attraction for equations admitting strong comparison principle (cf. [71], for example). In fact, any point in the boundary can be approximated from below or from above by a monotone sequence of elements of DA. In the present case, approximation from above is prevented by (al), thus we have (a2). Positive invariance of DA follows easily from the obvious positive invariance of DA and continuity with respect to initial conditions. Property (a4) is a consequence of positive invariance and the strong comparison principle (see the proof of [71, Theorem 5.4(v)]). 7q LEMMA 3.2. A s s u m e )~ <~ )~l. I f uo ~ ODA and {uo.,,} C DA is an increasing sequence such that uo., --+ uo then
u(r,
t) =
lim
u(r, t; uo.,,)
!l---+ O 0
is a global L l-solution.
PROOF. The result is proved in [54], although it is not stated there in this form. Using the estimates for global solutions from [54, Theorem 2.3], one checks easily that parts (i) and (ii) of the proof of [54, Theorem 2.5], repeated without any change, prove Lemma 3.2. D PROPOSITION 3.3. I f )~ <~ )~l a n d u is a global L l-solution as in L e m m a 3.2 then it is a classical solution of(E) on (Bi (0) \ {0}) x (0, cx~) a n d there is a 99 ~ C2((0, 1]) with the f o l l o w i n g properties: u(., t) --+ q),
in Clloc((0, 11) as t --+ oc,
N-1 q)rr nt- ~ q g r
+
~.e~~= 0,
r ~ (0, 1),
(3.2)
q)(1) = 0 a n d either q)(O) = ec or q9 ~ C l ([0, 1]) a n d q),.(O) = O.
PROOF. The proof uses a Lyapunov functional argument and it is very similar to the proof of (0.10), [69, Theorem 4]. Therefore, we only give a brief sketch of it.
M. Fila and H. Matano
748
Let {U0,n} be a sequence as in Lemma 3.2. Let T* be the blow-up time of the solution of y' = )~ey, y(0) = minu0. Then for t > T* and all n sufficiently large we have Ur(r, t; U0,n) < 0 for r E (0, 1), see the proofof [68, Theorem 1]. Since u is an L 1-solution, we have that u(r, t; uo) is finite for r 6 (0, 1] and t > T*. Also, u(r, t; uo) is a classical solution of N-1 Ut = Urr -~- - - U r
u(1, t ) = 0 ,
-~-)~eu,
t > T*, r E (0, 1), t > T*,
and ur(r, t; u0) < 0 in (0, 1) x (T*, cx~). Further, u(r, t; uo) is bounded for (r, t) 6 (r0, 1) x (T*, cx~), r0 E (0, 1) (cf. (3.1)-(3.6) in [69]). Following (3.7)-(3.12) in [69], we obtain that for any sequence t,, --+ c~ there exists a subsequence, denoted again by tn, and a q9 E C2((0, 1]) such that
u(., tn" uo) -->" q~(')
in Clloc((O, 1]),
where ~0 satisfies (3.2) and qg(1) = 0. Next, either qg(0) = cx~ or ~0(0) < cx~. In the latter case, q9 must satisfy qgr(0) = 0 by Lemma 1.1 in [69] and elliptic regularity [55]. Now, by standard arguments, the set of limit points of u(., t; u0) must be connected in C([r0, 1]) (r0 6 (0, 1)). Thus, as the solutions of (SE) are isolated in C([r0, 1]), there is exactly one limit point, hence the conclusion of the proposition follows. D LEMMA 3.4. Let k >~ 1 and let ,k be such that 49~ exists. Then (i) for any uo > dp~, the solution of(E) with initial data uo blows up completely infinite time; (ii) for any uo < ~ , the solution of (E) with initial data uo converges to 49~ as t --+ ~ . PROOF. Part (i) follows by combining the results of [51,53]. Part (ii) can be found already in [25] (without proof). We indicate a proof of it here. By the maximum principle, the solution is global and bounded. In particular, u(., t) < 4~ for t 6 [0, c~). Therefore, u converges (as t --+ cx~)to an equilibrium 4~{ such that 4~{ ~< q~. Since 4~ is unstable from below and 4~/z ~< 4~ only if i -- 0 or i - k (cf. Proposition C), one obtains that i -- 0. D
3.4. Classical connections For problem (E) it is rather easy to establish the existence of classical connecting orbits: PROPOSITION 3.5. Let )~ be different from all the bifurcation values )~l , )~2 . . . . . j, k, a connection from dPk to dpj exists if and only if k > j.
Given any
Let us sketch an outline of the proof of Proposition 3.5. Using local analysis near the turning point ()~k, 4~k) (cf. Proposition A statements (a) and (c)) and reduction to onedimensional center manifolds one shows that there exists a connection from 4~k to 4~k-1 so
Blow-up in nonlinear heat equations
749
far as ~. lies in the region where 4~k e x i s t s - that is, either ~.k+l < ~. < ~k or ~.k < ~. < ~.k+l and is sufficiently close to ~.k. We will denote the connection by 4~k ~'* ~bk_ I. Due to the Morse-Smale structure of (E) (see [72]), the connection persists as ~. is moved away from ~k. This way one shows existence of the connections
-
4~k ~ 4~k-l ~ ' " ~
4~0,
provided that all these equilibria exist, or, equivalently, provided that )~k+l < )~ < ~k or ~.k < )~ < )~k+l. Existence of this chain of connections, in conjunction with the MorseSmale structure, further implies that the connection ~bk ~ q~j exists if k > j (cf. [17]). Finally, the Morse-Smale structure implies that the connection 4~k ~ q~j can exist only if the Morse index of q~k is greater than the Morse index of ~ j (cf. [ 1 7 ] ) , thus k > j is necessary (cf. Proposition B(b)).
3.5. Uniqueness results for classical connections In this subsection we will prove some uniqueness results for classical connections between equilibria. Those results will be used later in establishing L l-connections between some equilibria. Let us first recall basic properties of the unstable and stable manifolds of equilibria. The equilibrium solution ~bk exists for ~ 6 ()~k, ~.k+l] if k is even and for ,k 6 [~.k+l, ~.k) if k is odd. This equilibrium is known to be hyperbolic for ~. 6 ()~k,)~k+l) or for )~ 6 (2.k+1, ~.k); more precisely, if we denote b y / / j the eigenvalue of the linearized operator N-1 Ak "-- 0,.9- + - - 0 n
+ )~eq~k,
then / / ! > //2 > ' ' "
(3.3)
> //k > 0 > / / k + l > / / k + 2 > ' ' ' -
In particular, the Morse index of 4)k is equal to k (cf. Proposition B). Now let W ~(q~k) and W u (4)k) be the stable and the unstable sets of q~k, respectively, and define for j < k
W~' (~k)"- {w ~ W ~'( O k ) "
lim
t---~- - O G
e -nt
Ilu(, t
-
II - ol,
where # is an arbitrarily fixed constant satisfying//j < // < / / j + l a n d / / j S a r e as in (3.3). It is not difficult to see that Wy (4~k) is independent of the choice of # 6 ( # j , # j + l ) and is locally, near q~k, a j-dimensional submanifold of Wl"oc(4~k) whose tangent space Tc?kWy (ckk) is spanned by the j eigenfunctions of Ak corresponding to #1,//2 . . . . . / / j . The proof of these statements can be modelled in analogy to [39, Theorem 5.2.1], for modifications which are necessary cf. [10, Lemma 4.1] or [11, Theorem C.5].
750
M. Fila and H. Matano
In view of the fact that dim Wy (r
-- j,
codim W s (dpj_ l) -- j - 1,
one can suspect that the classical connection from Ck to q~j-1 lying on the set Wj (r forms a one-dimensional manifold provided that W] (r and W ~(r 1) intersect transversally, hence the connection is unique or it consists of a finite number of orbits. We will show that the connection is in fact unique: THEOREM 3.6. The classical connection from dpk to dpj-i lying on the set W~!(dpk) is unique up to time shift.
PROOF. As the tangent space Tck Wj' (r is spanned by the first j eigenfunctions of the second order operator Ak, the Sturm-Liouville theory shows that any nonzero element of TCk Wy (r has at most j zeros on the interval 0 ~< r ~< 1. It follows that any two different elements w, t~ E W)~(r \ {r satisfy
where z j is what we defined in Section 3.3. Similarly, as TCj_~ WS(q~j_l) is spanned by eigenfunctions of the operator A j _ ! corresponding to the jth, (j 4- 1)th, (j 4- 2)th . . . . eigenvalues, any two different elements w, ~ ~ W s ( r \ {r satisfy
j. Consequently, if u(r, t) and 5(r, t) are two different classical connections (different even after an arbitrary time shift) from Ck to Cj_ l lying on the manifold Wj (r one has z j (u(., t 4- r) - 5(., t)) -- j,
t6R,
(3.4)
for any choice of a constant r 6 •. Now, since both of the values u (0, t) and 5 (0, t) connect the values r (0) and Cj_ 1(0), one can find to, r E IR such that u(0, to 4- r) = 5(0, to). It follows from Proposition E(iii) that the value z j ( u ( . , t + r) - fi(., t)) drops at t - to, contradicting (3.4). This contradiction shows that the classical connection from r to Cj_ l lying on Wy (r is unique. D
3.6. L l-connection from Ck to r
k ~> 2
In what follows we set Ik -- (~.k,)~k+l], Ik -- 0~k, ~.k+l) if k is even and lk - [~k+l, ~.k), ]k -- (~.k+l, ~.k) if k is odd. As mentioned in the previous subsection, the equilibrium r
Blow-up in nonlinear heat equations
751
exists if and only if )~ 6 Ik, and it is hyperbolic if and only if )~ E i k . We will denote by Cz (k, j ) and Lz (k, j ) the union of orbits for the classical connections from 4~ to 4~jz. and that for the L l-connections from 4)~- to 4~., respectively. DEFINITION 3.7. By an L l-connection from 4~ to 4~. we mean a function u(r, t) such that: (i) u is a classical solution of
ut = Au + )~e",
b/~0,
xEBI(0), -~
(ii) u blows up at t = T; (iii) u is a global L l-solution (as described in Lemma 2.2) of (E) with u0 = u(., r), r~2 and for any ~ Elk there exists an L l-connection from ~ to ck~. In other words, Lx (k, O) :/: 0. PROOF. For any )~ E lk, the operator Ak has precisely k positive eigenvalues #l > #2 > 9"" > #k, all of which are simple. Here and in what follows we denote 4~) by 4~k as we have done before. Let W~' (~bk) be as defined in the previous subsection. Since W~' (4~k) is a two-dimensional C 1-manifold near the point 4~k, and since the dynamics on W~~(4~k) near 4'k can be well approximated by a linear flow, we can find a neighborhood U of 4~k in X having the following properties: (a) A := U A W~' (~k) is homeomorphic to a two-dimensional disk; (b) F := OU A W~' (Ckk) is homeomorphic to a circle; (c) each orbit on W~~(OSk) except {4~k} intersects F at a single point; (d) there is no equilibrium point in A U F except 4~k. Considering that the eigenvalue #l of the operator Ak has a positive eigenfunction, one sees that there exist v+, v_ E A satisfying v+ > 4~k > v_. By Lemma 3.4, v+ leads to complete blow-up and v_ leads to convergence to 4~0. Therefore there exists a point w+ E F which leads to complete blow-up and a point w_ E F which belongs to DA. Since w+ does not belong to DA by Lemma 3.2, we easily see that the boundary of F (1DA with respect to the induced topology of F - denoted by O(F A DA) - contains at least two points, say w l, w2 (see Figure 3). Solutions of (E) with initial data w l and w2 can be continued globally in the L l-sense as described in Lemma 3.2. And by virtue of Proposition 3.3, both of the solutions converge to some equilibrium as t --+ cx~, say ~bjl and 4~j2, respectively. Hence
wi E C(k, ji) U L(k, ji)
(i=1,2).
752
M. Fila and H. Matano O() 11)+
F
~k ODA
11)_
DA
~0 Fig. 3. Sketch of W~(4~k).
As we have shown in the proof of Theorem 3.6,
zj(lloi -- ~bk) ~ 2
(i = 1, 2),
zJ(~ji - q~k) ~<2
(i=1,2).
therefore
It follows from Proposition C that ji = 0 or ji = 1. Hence {tOl, tO2} C C(k, O) U L(k, O) U C(k, 1) U L(k, 1).
However, tOi does not belong to C(k, 0) since it lies on ODA. And it cannot belong to L(k, 1) since for solutions lying on L(k, 1) the number of intersections with 4~k drops at least by one (see Section 3.7), which is impossible since z j ( w i -4~k) ~< 2 and zj(c~l 4~k) = 2. Therefore {tOl, W2} C L(k, O) U C(k, 1).
By Theorem 3.6 and the property (c) above,/-" n C (k, 1) is a singleton. This implies that at least one of the points w l, 11)2 belongs to L (k, 0). D
Blow-up in nonlinear heat equations
753
3.7. Conjecture and open questions As we have mentioned earlier, a classical connection from 4'k to ~bj exists if and only if k ~> j + 1. As for L 1-connection, we have the following: THEOREM 3.9. If there exists an L l-connection from 4)k to ~j, then k >~ j + 2. For the proof, we need several lemmas: LEMMA 3.10. Let u(r, t) be a global L 1-solution as in Lemma 3.2. Then ZJ (b/(', t) - ~bk) is monotone nonincreasing in t ~ IR. PROOF. This l e m m a follows from Proposition E and a certain limiting argument. The details are omitted. V] LEMMA 3.1 1. Let u(r, t) be an L l-connection from c/)k to ~j. Then there exists to ~ R such that j+2 z j ( u ( . , t ) - - d p k ) >~ k + 1
(if j < k), (if j > k)
f o r all t E ( - o c , to). PROOF. By Proposition C, we have
zj(qbj - qbk) --
j+l k + 1
(ifj k).
Considering that u (., t) converges to ~bj as t ~ that j+l z j ( u ( ' , t) - el)k) >~ k + 1
e~ locally uniformly in 0 < r ~< 1, we see
(ifj k)
for all large t. Hence, by L e m m a 3.10, the above inequalities hold for all t E R. This proves the l e m m a for the case j > k. Now, if j < k, then ~bj (0) - q~k (0) < 0. In view of this inequality, and with a close look at the argument which derives z j ( u ( . , t) - 4~k) ~> j + 1, we see that for each t 6 IR there exist points 0 < rl (t) < rz(t) < ..- < rj(t) < 1 such that ( - 1 ) i (u(ri(t), t) - c/)k(ri(t))) > 0
(i -- 1,2 . . . . . j).
Since u(r, t) blows up in finite time at r - 0, there exists to E R such that u ( 0 , to) - 4~k (0) > 0.
(3.5)
754
M. Fila a n d H. M a t a n o
This, together with (3.5), implies that there is another zero of u(., to) - 4~k lying between 0 and rl (to), hence
z j ( u ( . , t o ) - ckk) ~ j + 2. This and L e m m a 3.10 complete the proof.
D
LEMMA 3.12. I f u is a global L l -solution o f Problem (E) which blows up at t - T then >
if,,
< r <
here I - (0 1] and ck~ - In 2(N~2) is the singular solution o f the equation '
)~r-
N-1
qbr r -Jr- - - d p r
+ )~eck -- 0
f o r r > O.
PROOF. We show first that ZI(H(', tl) -- ~b~) > 0 for tl < T. This is obvious if ~ ~> 3.~ = 2 ( N - 2). A s s u m e that ~. 6 ( 0 , ) ~ ) . Let 7r(r) = In
2(N-
2)
,k(r -Jr-e) 2
= 4 ~ ( r + e),
e > 0.
A straightforward calculation yields N-1 ~rr @ ~ V
lPr @ ~.e ~p < 0
for r > O,
2 ~ r (0) - -
< O. 8
Also, ~p(1) > 0 if e is small enough. Hence, 7r is a supersolution of (E). If ZI(U(', tl) -4 ~ ) = 0 and e is small enough then u(-, t l) < 7r, therefore, u is a global classical (bounded) s o l u t i o n - a contradiction. Similarly as in L e m m a 3.10, we have that z1(u(., t) - ck~) is nonincreasing in t. Set rl (t) -- min{r ~ (0, 11" u(r, t) - ~oc(r) - 0}. Suppose that Zl (U(', t) - 4 ~ ) does not drop at t - T. Then rl (T) -- l i m t ~ r rl (t) > 0. If Ur(rl (T), T) -- ck~ (rl (T)) then Proposition E(iii) yields a contradiction. ! If ur(rl (T), T) > ck~(rl (T)) then there exist e, 6 > 0 such that
u(~, t) < ~p(~) u(r, T - 6) < ~ ( r )
f o r t e [ T - 6, T), f o r r 6 [0,6].
But then u ~< 7r on [0, 6] • [T - 6, T) since 7t is a supersolution. This means that u does not blow up at t - T - a contradiction. E] PROOF OF THEOREM 3.9. Let u(r, t) be an L l - c o n n e c t i o n from q~k to ~bj. Since u(., t) belongs to W u (~k) for t < T, where T is the blow-up time of u, we have ZJ(U(', t) -- q~k) ~< k
for t < T
Blow-up in nonlinear heat equations
755
(cf. [10]). This and Lemma 3.11 imply that either k ~> j + 2 or k = j. If k = j then z1(u(., t) - cp~) is constant for t 6 IK what contradicts Lemma 3.12. U] We suspect that the converse of Theorem 3.9 is true: CONJECTURE. An L l-connection from dPk to dpj exists if and only if k ~ j + 2. To study the above conjecture, we need to know more about the nature of global L l_ solutions of (E). Among other things, the following fundamental questions still remain open: QUESTION 1 (regularity). Suppose a global L 1-solution of (E) blows up at t = T. Then which of the following three situations occurs? (a) The solution remains singular for all t ~> T; (b) the solution remains singular for a while but eventually becomes classical again; (c) the solution becomes classical immediately after T. QUESTION 2 (continuous dependence). Do L l-global solutions depend on their initial data continuously even beyond the blow-up time? As for Question 1, the self-similar peaking solution constructed in [54] has the property (c), but this solution does not satisfy the right boundary condition. So far we do not know which of the possibilities can actually occur. As for Question 2, an affirmative answer seems unlikely if we consider any global L l-solution satisfying the integral identity given in Definition 2.1. We even do not know whether or not the continuation beyond the blow-up time is unique. However, if we only consider global L l-solutions that are defined as in Lemma 3.2 (which gives minimal solutions among all possible L l-continuations), we suspect that some sort of continuity holds. Note that lower semi-continuity follows immediately from the definition. NOTE A D D E D IN PROOF. It has been shown recently in M. Fila, H. Matano and E PolL6ik, Existence of L l-connections between equilibria of a semilinear parabolic equation, Preprint, that the conjecture from Section 3.7 is true.
Acknowledgment Part of this work was done while the first author visited the University of Tokyo. He was partially supported by VEGA grant 1/4190/97.
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[60] J. Matos, Convergence of blow up solutions of nonlinear heat equations in the supercritical case, Proc. Roy. Soc. Edinburgh Sec. A 129 (1999), 1197-1227. [61] J. Matos, Self-similar blow up patterns in supercritical semilinear heat equations, Comm. Appl. Anal. (to appear). [62] F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math. 45 (1992), 263-300. [63] F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type ut = Au + [ulP-l u, Duke Math. J. 86 (1997), 143-195. [641 F. Merle and H. Zaag, Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), 139-196. [65] N. Mizoguchi and E. Yanagida, Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation, Math. Ann. 307 (1997), 663-675. [66] N. Mizoguchi and E. Yanagida, Critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation,//, J. Differential Equations 145 (1998), 295-331. [67] K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation - A u = )~eu on circular domains, Math. Ann. 299 (1994), 1-15. [68] W.-M. Ni and P. Sacks, The number of peaks of positive solutions of semilinear parabolic equations, SIAM J. Math. Anal. 16 (1985), 460-471. [69] W.-M. Ni and P. Sacks, Singular behavior in nonlinear parabolic equations, Trans. Amer. Math. Soc. 287 (1985), 657-671. [70] W.-M. Ni, P.E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear pat'abolic equations, J. Differential Equations 54 (1984), 97-120. [71] P. Polfi6ik, Domains of attraction of equilibria and monotonicity properties of convergent trajectories in parabolic systems admitting strong comparison principle, J. Reine Angew. Math. 400 (1989), 32-56. [721 P. Polfi~ik, Transversal and nontransversal intersections of stable and unstable manifolds in reaction diffusion equations on symmetric domains, Differential Integral Equations 7 (1994), 1527-1545. [73] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-up in Problems for Quasilinear Parabolic Equations, Nauka, Moscow (1987) (in Russian); English translation: W. de Gruyter, Berlin (1995). [741 D. Tzanetis, Asymptotic behaviour and blow-up of some unbounded solutions for a semilinear heat equation, Proc. Edinburgh Math. Soc. 39 (1996), 81-96. [75] A. Vanderbauwhede, Centre manifolds, normal forms and elementary, bifurcations, Dynamics Report. Ser. Dyn. Syst. Appl., Vol. 2, Wiley, New York (1989), 89-169. [76] J.J.L. Velfizquez, Local behaviour near blow up points for semilinear parabolic equations, J. Differential Equations 106 (1993), 384-415. [77] J.J.L. Vel~zquez, Classification of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338 (1993), 441-464. [78] J.J.L. Vel~izquez, Higher dimensional blow up for semilinear parabolic equations, Comm. Partial Differential Equations 17 (1992), 1567-1696. [79] J.J.L. Vekizquez, Estimates on the ( N - 1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42 (1993), 445-476. [80] E Weissler, Single-point blowup for a semilinear initial value problem, J. Differential Equations 55 (1984), 204-224.
CHAPTER
15
The Ginzburg-Landau Equation in Its Role as a Modulation Equation*
Alexander Mielke Mathematisches Institut A, Universitiit Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany E-mail: mielke @mathematik, uni-stuttgart.de http://www, mathematik, uni-stuttgart.de/~ mielke
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Ginzburg-Landau formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Spectral assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Formal derivation of modulation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Transformations and scalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Symmetries of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. A simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. A few other amplitude equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Applications in hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The Rayleigh-B6nard problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The Couette-Taylor problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Sideband instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Special solutions and dynamics of the Ginzburg-Landau equation . . . . . . . . . . . . . . . . . . . . . 4.1. Plane waves and their stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The real Ginzburg-Landau equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Special solutions for the complex Ginzburg-Landau equation . . . . . . . . . . . . . . . . . . . . . 4.4. Global existence for the complex Ginzburg-Landau equation . . . . . . . . . . . . . . . . . . . . . 5. Attractors for large and unbounded domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Existence of attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Comparison of attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. An example with different limit attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Justification of the Ginzburg-Landau formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Kirchgfissner reduction, spatial center-manifold theory . . . . . . . . . . . . . . . . . . . . . . . . 6.2. The approximation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. The attractivity of the set of modulated patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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*The research was partially supported by D F G - S P P "Dynamische Systeme" under Mi 459/2 and by VolkswagenStiftung under I/71016. H A N D B O O K OF D Y N A M I C A L SYSTEMS, VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 759
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6.4. Shadowing by pseudo-orbits and global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Comparison of attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Comparison of inertial manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. Results on the Couette-Taylor problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abstract The Ginzburg-Landau equation (GLe) ~tA = div[(I -k- i S ) V A ] -+- p A - ~'IAleA, with A(t, x) 9 C, t >~ O, x 9 R d, appears in many different contexts, e.g., nonlinear optics with dissipation or the theory of superconductivity. In addition, it plays an important role as modulation equation and it serves as a simple mathematical model for studying the transition from regular to turbulent behavior when, for ~ > 0 fixed, the dispersion parameters (S, IMP') = s (I, - 5 ) with s >> 1 are considered. The purpose of this work is to review the latter two aspects of the GLe. In a variety of pattern-forming systems the nonlinear modulations of the basic periodic pattern can be described by the solutions of the GLe. We discuss this for three classical hydrodynamical situations: the Rayleigh-B6nard convection, Poiseuille and the Taylor-Couette problem. Indeed, the GLe should be seen as a normal form or the lowest order expansion of a bifurcation equation in the context of a weakly unstable system when continuous spectrum moves over the imaginary axis upon changing an external parameter. In particular, this theory gives a natural and simple approach to the theory of sideband instabilities. We explain the classical derivation of the GLe as a modulation equation of an original partial differential equation 9tu = Lu + N ( u ) and give the abstract setting which allows for the application of the Ginzburg-Landau formalism which is based on the ansatz
u(t, x) ,~ U A (t, x) = sA(~2t, s(x - Cgrt))ei(C~
-+-c.c.,
where e2 is the distance of the external parameter from its critical value. We then study the mathematical properties of the GLe which are relevant for the justification of the formalism. This includes a variety of special solution classes, a global semigroup theory in function spaces containing L ~ (II~d) as well as the construction of the global attractor. Finally we review the results which prove that the solutions A of the GLe inserted into the above ansatz provide good approximations for solutions u of the original system. In particular, the direct implications to hydrodynamical problems are discussed.
820 822 824 827 829
The Ginzburg-Landau equation in its role as a modulation equation
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1. Introduction
The purpose of this work is to review recent developments in the theory of the GinzburgLandau equation (GLe) as a modulation equation. While the theory of modulation equations is a standard tool in physics and mechanics, its dissemination in mathematics is very restricted. The main goal here is to explain the physical and mechanical background of this theory in mathematical terms to enable researchers in the fields of dynamical systems and partial differential equations to familiarize with the main questions and results. Clearly, we can neither give all the relevant results here nor it is possible to give all the mathematical details which are needed for the full understanding of the phenomena. In this introductory work we can only present the main ideas, motivations and techniques to display the general methodology and the simplest versions of the results associated to the field. Hopefully this survey is a stimulating guide to the recent literature, where more details and results can be found. The mathematical results in this field are roughly divided into two areas, (i) the analysis of the GLe in itself and (ii) the relation between the solutions of the GLe and the original PDE. Since the number of mathematical papers on the GLe published per year exceeds 100, we have to restrict our view, in area (i), to developments which are related to the questions in area (ii). In particular, this relates to questions on the solution behavior on unbounded domains. In Section 2 we give the general setup for the theory of modulation equations which was developed to describe modulations in space and time of a basic periodic pattern which appears in a system when certain control parameters become critical. If we consider the physical domain S2 = ]KJ x 27, where 27 is a bounded cross-section, with x e IKa and y e Z, then the basic periodic pattern takes the form p(t, x, y) -- E(t, x)~o.k0 (y),
where E(t, x) = e i(c~176
(1.1)
Here o9 e R and k0 6 R a denote the temporal frequency and the spatial wave vector of the basic pattern, respectively. A modulated pattern is a function of the form
u(t, x, y) = B(r, ~)p(t, x, y) + c.c.
(1.2)
where the amplitude function B(r, ~) e C depends on a slow time variable r and a large spatial variable ~. In our context we will have r = e 2t
and
~-s(X-Cgrt),
where Cgr is the group velocity of the basic pattern. The amplitude IB(r, ~)1 gives the intensity of the basic pattern while phase changes of B give rise to changes in the temporal frequency and in the spatial wave vector of the pattern. In the theory of modulation equations one starts with a partial differential equation
Otu = L ( # , Ox, Ov)u + N(lz, Ox, Or, u)
(1.3)
A. Mielke
762
which is posed on S2 = IRd • 2? together with suitable boundary conditions on 0 s This systems is supposed to have u -- 0 as the trivial solution for all values of the small parameter #. Moreover, # = 0 is taken to be the threshold of instability, that means that u -- 0 is asymptotically stable for/z < 0 and unstable for # > 0. At criticality/z - - 0 the linear operator admits only the basic pattern p as a neutral mode which is neither decaying nor growing. The aim of modulation theory is to understand the evolution of modulated patterns in the form (1.2). This is done by deriving a partial differential equation for the modulation function B. More precisely, one inserts # -- pe 2 and the multiple scaling ansatz
u(t, x, y) = UA (t, X, y) +
O(62)
with UA (t, x) -- eA(eZt, e(x - Cgrt))p(t, x, y) + c.c.
(1.4)
in (1.3) and equates equal powers of the terms e JE n to 0. The desired partial differential equation, called modulation equation, is then found as a solvability condition for the equation associated to e3E. Our interest lies in those systems for which the associated modulation equation is the GLe
OrA = div~ [AoV~A] + PXo, 1A -~'IA]2A.
(1.5)
Here )~0,I, ~" 6 C, and A0 6 C d • is a symmetric matrix whose real part is positive definite. The precise assumptions of this theory and the formal calculations in the general case are given in Section 2 together with a first simple example. There we also discuss questions of different scalings and of symmetries of the GLe. We will call the procedure described above the Ginzburg-Landauformalism, since it is a formal procedure which derives from the original problem (1.3), via the multiple scaling ansatz (1.4), the associated modulation equation. In our cases this is the GLe, and we refer to Section 2.6 for a few other modulation equations. The general philosophy of modulation theory can be found in the books [92,17] or in the review [ 110]. If A is a solution of the GLe, the function UA is called the Ginzburg-Landau approximation for the solution u of (1.3) which has the same initial data u(0, .) - UA (0, .). The major question in the justification of the GLe as a modulation equation is whether UA (t, .) remains close to the true solution u(t, .) over a sufficiently long time scale. Certainly this time scale has to be of order (_9(1/82) since only then the evolution of A (r, .) with r - e2t has any impact on UA (t, .). This justification question will be treated in Section 6. In Section 3 it is shown how the Ginzburg-Landau formalism can be applied to the three prototype problems in hydrodynamics, namely the Rayleigh-B6nard problem, the Couette-Taylor problem and the Poiseuille problem. In all three problems we have an open flow region with either one or two unbounded directions. At a critical Reynolds number the basic flow becomes unstable and experiments clearly show modulated periodic patterns. Historically these problems mark the origin of the GLe as a modulation equation. In fact, it was the need to explain large scale effects in these hydrodynamical problems which led to the development of the modulation theory in the late 60s and beginning 70s, see [ 111,137, 35,112] for the Rayleigh-B6nard problem and [139,69,32,140] for the Couette-Taylor and the Poiseuille problem.
The Ginzburg-Landau equation in its role as a modulation equation
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A mathematical theory for strictly periodic patterns was developed in the late 50s and 60s by using the Liapunov-Schmidt reduction, see, e.g., [83]. However, this theory could not explain the pattern selection which was observed in experiments, since many periodic patterns are stable with respect to perturbations of the same spatial period but unstable with respect to perturbations with slightly different spatial period. First explanations of this phenomenon of sideband instability were given in [47,86]. This led already to a linear theory of modulated patterns. In order to understand also the nonlinear evolution of modulated patterns the Ginzburg-Landau formalism was invented. As a by-product the theory of sideband instabilities obtained a simple form, see Sections 3.4 and 4.1. More general large scale effects which are of importance in the Couette-Taylor problem are discussed in [24, Section VII]. In Section 4 we study the GLe in its own right. The huge popularity of the GLe arises from the two facts that it is rather simple and explicit and that it appears in a many different applications, like for instance in superconductivity [62,45,23,14], in laser optics [80,103] and in reaction-diffusion systems [63,85,73,131,121]. In all these fields special solution classes as well as special properties of the GLe are of importance. There is a great distinction between the real GLe (rGL), where X0.1, ~" E R and A0 -- I, and the general complex GLe (cGL). In Section 4.1 we shortly discuss the rotating plane waves A(r, ~) = r e i(r176 together with their stability. This gives the relation between the classical field of sideband instabilities and the modulation theory. In the one-dimensional case the rGL has rather special nodal properties which can be used to study the solution behavior intensively. Moreover, for the GLe in one space dimension there are numerous results on standing, rotating and traveling pulses and fronts. For the two-dimensional GLe it is possible to construct rotating waves which reduce to the Ginzburg-Landau vortices in the case of rGL. At the end of Section 4 we develop a general semigroup theory for the GLe for space dimension d = l, 2, 3 for bounded as well as unbounded domains. For the case Re~"> 0 we show that the semiflow on LeC(R d) is well defined (under the usual conditions on the dispersion parameters if d = 3). Section 5 is independent of the theory of the GLe although the theory and methods were developed mainly for this equation. It is shown that the semiflow on Lvc (R d) has a global attractor when compactness and attractivity is taken with respect to a weighted norm. We use ideas similar to [7,55,54]. The reason for considering the GLe on unbounded domains is motivated by the modulational theory. On the full space it is easy to accommodate the multiple scaling ansatz (1.4) for any e > 0, since we do not have to fulfill boundary conditions. If one wants to work on bounded domains Y2e = ( - g , g)d x r for the full problem (1.3) then the length f has to be chosen at least of order 1/e otherwise the modulations due to A (s2t, s x ) would not be effective in ~2e. Using weighted norms we develop a theory which allows us to compare the global attractors A s?,' for different domains Y2, tending to the full space R d with the attractor A Rd associated to the semiflow on L ~ (Rd). We are able to show upper semicontinuity to this limit but lower semicontinuity is false in general. If the limit Y2,, ~ R d and the limit r -+ ec in the definition of the attractor are done simultaneously we obtain a generalized limit attractor which actually coincides with A RJ . Section 6 treats the question of the mathematical justification of the Ginzburg-Landau formalism. This theory was developed only within the last 10 years starting from [26, 142,
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A. Mielke
87]. Somewhat earlier a series of papers [74,72,73,2] used the Kirchg~issner reduction (cf. [84]), also called spatial center-manifold reduction, to derive the steady one-dimensional Ginzburg-Landau equation for some hydrodynamical problems, see Section 6.1. The following subsections treat the full time-dependent problem. By now it is not possible to prove that the full problem (1.3) can be reduced to the GLe (1.5) in the sense of dynamical systems. This would mean that there is an invariant manifold in the phase space Y of (1.3) and the reduced flow on this manifold is given by (1.5), perhaps after adding higher order corrections. The main difficulty in proving such a result arises from the fact that the continuous spectrum of the linearization at u = 0 does not have a spectral gap which would allow us to separate the critical modes from the stable modes. Certain attempts to avoid this splitting are given in [96,125,94]. Thus, the best one can hope for is to prove approximation of the solutions u(t, .) by UA (t, .) over a time interval [0, rl/E 2] for rl > 0 fixed. Under suitable assumptions such an approximation result can be shown. Additionally it is possible to show that all small initial data u0 will evolve such that after time t -- 1/e 2 the solution u(1/e 2, .) --Sl/eZ(UO) is a modulated pattern, i.e., there is a function A0 such that u(1/e 2, .) -- Uao ('). Thus after this transient time, the solution can be well described by a solution of the GLe according to the above approximation result. This fact is called the attractivity of the set of modulated patterns, [48,124,125]. Iterating the above-mentioned approximation and attractivity property it is clear that the true solution u(t, .) can be approximated by pieces of Ginzburg-Landau approximations Uan (t, .), t ~ [n/e 2, (n + 1)/e2), where we have to allow for jumps at the times tn -- n/e 2. A delicate analysis controls these jumps showing that they are in fact relatively small. Hence, we say that the true orbit is shadowed by a pseudo-orbit of the GLe. As such Rd pseudo-orbits are attracted into a small neighborhood of the global attractor .AGL (with respect to the weighted norm), it can be shown that the global attractor A Rd• ~ of the full -JRd problem (1.3) has a relatively small distance to the image of r under the mapping A w-~ UA. For the one-dimensional Swift-Hohenberg equation on the interval 12e -- ( - e / e , g/e) with periodic boundary conditions an even better convergence statement can be shown for the comparison of the flows on the inertial manifolds of (1.3) and (1.5), see Section 6.6. The major problem with the present status of the justification of the Ginzburg-Landau formalism is that it only gives information over long but not infinite times. So it is not possible to use standard methods of dynamical-systems theory to transfer results from the 'reduced system' (the GLe (1.5)) to the full system (1.3). For instance, the existence of pulse or spiral waves in cGL does not give automatically the existence of the corresponding solutions in the full problem. A classical counterexample is the existence of the downstream pulses in the two-dimensional Poiseuille flow, see the discussion at the end of Section 3.3. Another example of different long-time behavior is indicated in the discussion at the end of Section 6.6. Similarly the question of stability arises; for instance, if we consider the rotating plane waves, then it is clear that they correspond to space-time periodic patterns of the full problem. The stability of these waves in the GLe is easily investigated as shown in Section 4.1, however, there is no direct way to infer the same stability information for the associated pattern in the full problem. One has to repeat the analogous calculations for the spectral problem associated with the linearization of the full problem at this pattern, see
The Ginzburg-Landau equation in its role as a modulation equation
765
[97,99,100,117] for such an approach. Similar problems arise, if we try to transfer dynamic phenomena like phase slips, diffusive stability or diffusive mixing from the GLe to the full problem, see [51,133,107]. In this situation, the best we can do is to learn from the analysis of the GLe to be able to make good guesses for the full problem. Then we have to transfer the analytical methods used to study the GLe to the according methods for the full problem. Hopefully we are then able to prove the desired result for the full problem directly.
2. The Ginzburg-Landau formalism In this section we describe the general formalism how to derive the modulation equation from a general PDE. We give precise assumptions on the linear part and discuss certain scaling and symmetry effects. We consider general PDEs on unbounded domains S-2 in the form s -- IRd x Z', where Z C IR" is a bounded domain with Lipschitz boundary. We write z = (x, y) for spatial points in I-2 = IRd x Z . The underlying PDE is written abstractly in the form Otu = L ( # , Ox)u + N(lz, Ox, u),
(2.1)
where u :X-2 --~ IR,,1 denotes the unknown function and # is a real parameter. The linear operator L(/z, 0x) is a partial differential operator on I2 including boundary conditions on 0 f2. It involves the derivatives 0x as well as 0,,, but we only display the former ones as x plays a distinguished role as unbounded variable. The nonlinear terms N ( # , Ox, u) are assumed to satisfy N ( # , 0x, e u) 0 ( 6 2 ) for sufficiently smooth u :S-2 --+ IRm. =
2.1. Spectral assumptions In such a situation u = 0 is the trivial homogeneous solution of (2.1) and we are interested in the onset of instability. Throughout we assume that the system has u --= 0 as an asymptotically stable state for # < 0 and u = 0 is unstable for # > 0. The question that is answered by modulation theory is which patterns are seen when the parameter # is close to the instability threshold # = 0. As in finite-dimensional dynamical systems this is determined by the eigendirections associated to those eigenvalues passing the imaginary axis. In our situation of a PDE with x 6 IRa the spectral problem is solved by using Fourier transform in the x-direction. We insert the ansatz u(t, x, y) = e zt ei(x'k)q) (y) into the linear part of (2.1) to obtain the spectral problem X4) = L ( # , ik)4~
(2.2)
which is now posed on the bounded cross-section Z . Thus, for typical problems L ( # , ik) has a discrete spectrum s p e c L ( / z , k ) . Note that s p e c L ( l z , - k ) = s p e c L ( # , k ) since L ( # , 0s) is a real operator. In fact, for the Ginzburg-Landau formalism we only need the
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A. Mielke
eigenvalue family )v(/z, k) with largest real part to be isolated for/z ,~ 0. More precisely, the general situation leading to the GLe as a modulation equation can be characterized as follows. Basic spectral assumptions. There exist 6, Y > O, ko ~ lRd and eigenvalues )~ : ( - 6 , 6) • Bko (6) --+ C such that f o r all /z ~ ( - 6 , 6) the spectrum of L(/z, k) satisfies the following: (1) F o r k q~ Bko(a) U B_ko(6) we have spec(L(/z, k)) C {Reef ~< -Y}(2) For k ~ Bko(a) we have spec L(/z, k) -- {~.(/z, k)} tO Ss(/z, k)
with Ss(/z, k) C {Reef ~< -?,},
where )~(/z, k) is an algebraically simple eigenvalue with the expansion X(/z, k) - Xo(/z) + (Xl (/z), k - ko) - (A(/z)(k - ko), k - ko)
+ o ( I k - k013).
(2.3)
(3) We impose Re ~0(0) = 0, (d/d/z) Re ~.0(0) > 0, Re )~l (0) = 0 and Re A(0) is a positive definite matrix. Here Ss(#, k) constitutes the stable part of the spectrum of L ( # , ik). The conditions on )~(#, k) guarantee Re )v(0, k) ~ 0 with equality only for k = k0. Moreover, Re)v(#, k0) has the same sign as #, which exactly corresponds to our definition of the threshold of instability. Further on we use the abbreviations d )~0,1 -- ~ , 0 ( 0 )
E C,
Cgr -- -i)vn (0) 6 IRd
and
A0 -- A(0),
where Cgr is called the group velocity of the basic pattern. These assumptions lead to three different generic cases which either lead to the real or the complex GLe. There might be other cases, but we only treat these ones as they associate to generic cases without degeneracies. CASE 1. ko Vk 0 and )v(/z, k) 6 lR. This situation typically appears if the system has an additional reflection symmetry x --+ - x and hence )v(/z, k) = ) v ( # , - k ) = )v(#, k). The diffusion part of GLe will be real and further symmetry properties determine whether the nonlinear term is real or complex. CASE 2. ko ~ 0 and )~(0, k0) = ico with co > O. This is the general situation for which we obtain the anisotropic cGL. CASE 3. k0 = 0 and )~(0, k0) = ico with co > O.
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In this situation system (2.1) has a spatially homogeneous Hopf bifurcation when # passes the origin. If (2.1) is rotationally invariant we have additionally X(#, k) = X(#, Ikl) and we obtain an isotropic cGL. We will refer to these cases regularly in the subsequent discussions.
2.2. Formal derivation of modulation equations We now want to show how the GLe
3rA--div;(AoV;A)
(2.4)
+ pXo, I A - ~"IAI2A
is obtained from the original system (2.1) by a multiple scale expansion. At this stage the procedure is formal since it assumes that the original problem has solutions which are modulations of the basic periodic pattern over sufficiently long time intervals. The mathematical justification of this assumption is studied in Section 6. To simplify the notations we omit the cross-sectional variable y, however, the approach is fully general. 2.2.1. The linear part. As ,k(/z, k) is an eigenvalue of L(/z, ik) with the eigenfunction 9 ~.k we see that v(t, x) -- e~(rt'k)t+i(k'x)~u.k solves
Otv -- L ( # , 3x)V,
(2.5)
which is the linearization of (2.1). For (#, k) ,~ (0, k0) we use the parameter scaling /,t -- e2p
and
k -- k0 + etc
with 0 < e << 1.
In most cases one uses only p = l, however, for consistency and for a possible treatment of subcritical bifurcations (~ < 0) we have introduced the fixed constant p 6 [ - 1, 1 ]. Via (2.3) this scaling leads to ,k(e 2p, k0 + e x ) -- ico + e 2p X0.1 Ei (Cgr, a: ) -- e 2 ( AOK, tr ) frO(e3). The above solution v of (2.5) can be written as -
v(t, x)
-- e (p)~~176
-
e i(wt+(x'k~
) ~e2p,ko+ez.
(2.6)
This solution is a product whose second factor is approximately the basic periodic pattern p(t, x) - e i(~176 q~0,k0 underlying the whole modulation theory. For # - 0 this pattern and its complex conjugate are the only patterns which are not damped. The first factor in (2.6) depends only on the slow time variable and the large space variable
r -- e2t
and
~ - e(x - Cgrt).
(2.7)
In fact, if we define the linear parabolic equation 3T B -- div ~ (Ao V~ B) + p Xo, l B
(2.8)
A. Mielke
768
for B(r, ~) 9 C, then it has the general solution /,
B(r, ~) -- [ ei(X'~)e(pz~176 JKGR d
f (x) dm(K)
where m is a complex-valued Radon measure on IRd. Thus, we expect that the modulation ansatz
v(t, x) ~ UB(t, x) -- B(e2t, e(x - Cgrt))p(t, x) 4- c.c., with p(t, x)
--
e i(c~176
CI)O,ko,
(2.9)
approximates solutions of (2.5) whenever B satisfies (2.8). We say that the amplitude function B modulates the basic periodic pattern p. In particular, IBI modulates the amplitude of p and arg B modulates its phase. Unfortunately, the ansatz (2.9) is not a sufficiently good approximation in the case when r really depends on k. In this situation it is necessary to control things in Fourier space. We introduce the Fourier transform .Tx[U] and 9t'~[B], then we find
.Ux[UB](t, ko 4- etr
= e i(~-)te
-d ( ~ [B(e2t,
")]) (x)q~O,ko 4- c.c.
However, using the operator-valued exponential mapping e tL(tt'k) we find the general solution of (2.5) with v(0, x) -- vo(x) in the form .T'[v(t, .)](k) = etL(u'ik).T'[vo](k). By our spectral assumptions we have f'[v(t, . ) ] ( k ) = eZ(U'k)tB'(k)q~u,k + O ( e - • for t --+ oo. If we compare with the above formula, we see that the expansion of the eigenvalue is matched properly for solutions B of (2.8), however, the difference ~u,k - r may introduce errors of order Ik - k01 = elxl. Thus, for exact results the approximation V8 in (2.9) should be replaced by
v(t, x) ~ Vs(t, x)
= m;l
Ira,
)] (k-'<~
+ ~176}] (x -
(2.10)
where X+k0 :R d ~ [0, 1] is a smooth cut-off function which is equal 1 for I k - k01 or Ik 4- k0[ ~< 6/2 and 0 for Ik - k01 and Ik + k0l/> 6. Here B is again a solution of (2.8). We refer to (6.19) and (6.20) for the exact mathematical implementation of these Fourier transform methods. 2.2.2. The nonlinear part. Since the structure of the nonlinear terms in GLe (2.4) is clear from the outset, we can derive its coefficient without using modulations. It suffices to compare constant solutions of the GLe with periodic patterns in the original problem (2.1). The basic periodic pattern is p(t, x) = E(t, x)q00,k0 where we use E = E(t, x) = e i(c~176
769
The Ginzburg-Landau equation in its role as a modulation equation
The nonlinear terms generate higher harmonics E j of the basic (t, x)-pattern E. If the amplitude of E is of order e, then the nonlinearity produces amplitudes of E j which are of order 8 Ijl for j ~ 0 and of order 8 2 for E ~ -- 1. This leads to the ansatz 1
u(t, .) -- e[A(r)Eq~o,ko + c.c.] 4- e 2 E E2j O2j(r) j=-I 1
4- e3 Z
E2J+' tP2J+' (r) + O(64),
(2.11)
j=-2
where the only x-dependence occurs through E = E(t, x). The unknowns A and Oj just depend on the slow time r and on the cross-sectional variable y in the latter case. Recall that we do not account for slow spatial modulations in this restricted ansatz. To insert (2.11) we need to expand L ( # , Ox) and N(#, Ox, u) as follows:
L(e2p, Ox)[E j tp] -- E j L(e2,o, ijko)tp, -
+ e3 EJ, l.,,=-j EJ+l+"T(j.l.,,)[qJJ, ~', q",,] +
0(64) 9
Here the bilinear operator/3 and the trilinear operator 7- are symmetric with respect to interchanging the order of the arguments together with the associated indices. Substituting ansatz (2.11) into (2.1) and equating the coefficients of the terms eJE l of lowest order to 0 we find a l E l" e2E 2" e2E ~ :
icoA~o.ko
-
-
icoA~O,ko,
2ico~2 -- L(0, 2iko)~2 + A2B(l.l)[~O.ko, ~0,ko], 0qJo -- L(0, 0)~o + [a[Z2B(l.-1)[~O.ko, q~0,ko],
e3E l " OrA~O,ko + icoqJl -- PXo. IA~O.ko + L(0, iko)qJl + a 2B(l.0)[q~0.ko, q~0] + A-2~(2,-1) [q't2, ~0,ko] + Ial 2A 3~1, l.-l)[q>0.k0, ~0.ko, ~0,ko]. We have omitted the equations for 8JE -! as they are just the complex conjugates. From our spectral assumptions we know that qJo and ~2 can be calculated from the second and third relation as ~o(r, y) -]A(r)lZWo(y) and qJ2 (r, y) - A(r)ZW2(y) with Wo - 2L(0, O)-ll3(l_l)[CbO.ko, ~0.k0], W2 -- (2icol - L(0, 2iko)) -1B(I. l)[q~0.k0, q)0.ko]Thus, the fourth equation takes the form [L(0, iko) - i c o l ] qJl -- (OrA - pXo.lA)CbO.ko - ] A I 2 A ~ .
(2.12)
A. Mielke
770
w
with q*. = 2B(l,O)[OO,ko, Wo] + 2B(2,-I)[W2, O0,ko] -+- 3"T(1,1,-1)[O0,ko, qb0,ko, r iw is an algebraically simple eigenvalue of L(0, iko) this equation is solvable if and only if the right-hand side is orthogonal to the eigenvector O~,ko of the adjoint of L(0, iko). We choose O~,ko such that (OO,ko, O~,ko) -- 1 where (.,.) is the duality pairing. Because of our spectral assumption in Section 2.1 the solvability condition for (2.12) is
O=OrA--p~o, IA+~[AI2A
(2.13)
with ~'-- - ( ~ , , O*) ~ C.
For generic problems the cubic coefficient ~" is nonzero. Joining the linear and the nonlinear result we obtain the GLe (2.4). Of course, (2.13) is the lowest order expansion of the reduced ODE on a two-dimensional center manifold for (2.1) when the solutions are restricted to spatial periodicity, see, e.g., [71 ]. This equation is often called Landau equation and takes the form d - - a -- g ( # , a) dt
= )~(#, ko)a --I-~'(/z) ]a 12a -+-g'(/z) ]a 14a + O(]a]7),
a E C.
(2.14)
2.3. Transformations and scalings The linear term in cGL can be considered as a parameter for most applications. On the one hand the real part of a -- p)v0,1 can be chosen via the bifurcation parameter # -- e2p. On the other hand the imaginary part of a can be adjusted via replacing A(r, ~) by e -'~ A(r, ~). Then, A solves cGL with the linear term a -- p)~0,1 - i~. From our spectral assumption we have Re)~0, l > 0, and thus by choosing p, ~ 6 IR suitably we can generate every a E C. Below we will restrict a to be real, namely equal to p. The general form of GLe we derived so far is Or A -- div ~ (A~ A) + )~O,lpA + c"[A 12A where the matrix A ~ ~ C d• is symmetric and has a real part which is positive definite. Setting .,.~
T--rr,
~ = K~
and
, 4 - - s e i~rA
with r, s > 0, ~ 6 R and an invertible K 6 Rd •
0~,4-- ldiv"((KtA~ r
With A ~
1/r K tA 0 K
)+
~.0,1
(2.15) we find the equivalent form A
m
iw A + _ I z I 2 x . ,.~
r
C
rs 2
Sl "1t- iS2, Sj 6 IRs~Xm d we choose K .v/TSTI/2Q with Q t Q _ I, then -- I q- i QtS11/2 $2 $71/2 Q and a suitable choice of Q allows us to diago-
nalize the imaginary part. Next, we choose ~ -- p Im)~o, 1 and r - Re )vo, I. For s we either t a k e s - _v/l l/r or, if Re " < 0, we can also t a k e s - y / - R e ~ / r . Then, after dropping the tildes we arrive at
d OrA -- E ( 1
j=l
+ iotj)O ~j 2 A + pA + d"'(fl)lAl2A
(2.16)
771
The Ginzburg-Landau equation in its role as a modulation equation
where either dA(fl) - - e i~ or d'(/3) -- - ( 1 + ifl), respectively. We could also reduce to the case p 6 { - 1 , 0, 1} by choosing r - - I p l k o , l, however, it is sometimes better to keep p as a parameter which may change sign. The parameters 0/1 . . . . . Old are called linear dispersion parameters, and /3 in d ( f l ) -- ( 1 + i/3) is the nonlinear dispersion parameter. GLe is called isotropic if all f f j a r e the same. Taking the complex conjugate of the equation we can interchange the signs of all these dispersion parameters simultaneously. If all otj have a large modulus and have the same sign, e.g., Otj ~ 1 and if ] Im~"l >> [ Re~'l, then (2.16) can be considered as a perturbation of the nonlinear Schr6dinger equation 8T A -- iAA • iIAIZA. By rescaling accordingly we obtain A
OrA -- i
j--l
1---
Otj
~J
A
+ pA -
~
+isignfl
IA A
"
(2.17)
For 1/C~j, IO, 1/fl --+ 0 we obtain the nonlinear Schr6dinger equation; it is the focusing case if sign/3 = - 1 and the defocusing case for sign/3 = + 1. In the global existence theory in Section 5 this difference will be important. Steady solutions or rotating waves of the form A (r, ~) -- e l~ A (~) solve an elliptic problem. In the isotropic case c~ - otj the equation is (1 + iot)AA + (p - i~)A + d ( f l ) l A ] 2 -- 0 which transforms via an appropriate scaling into AA-(I+i0~
)2 A + ( 2 + i 0 2 ) ( I + i 0 2 ) I A I 2 A - - 0
(2.18)
A
as long as d ( f l ) , p - i ~ r (1 + iot)[0, oo). This form allows for the study of pulse solutions, i.e., solutions which decay exponentially, namely like e -I~1. For instance, for d - 1 and 01 - 0 2 there is the explicit pulse A(~) -- (cosh~) -(l+i01) see Section 4.3.1 for more details. For 01 -- 02 -- 0 we obtain the real GLe (rGL), and for d - 2 there exist the steady Ginzburg-Landau vortices of degree n c Z. They have the form A (x) r , ( l x l ) ( x l + i x z ) ' / l x l " with r, "[0, oo) --> [0, oo) satisfying r,,(s) ~ s" for s --> 0 and r,,(s) --+ l / x / 2 for s --+ e~, see Section 4.2.1.
2.4. S y m m e t r i e s o f the p r o b l e m Here we treat the question how symmetries of the full problem (2.1) reduce to symmetries for the associated GLe, where symmetries of GLe are related to restrictions on the coefficients A0, k0, l and ~". Before doing this we recall the general symmetries of GLe, namely invariance under time and space translations (T and ~) as well as the phase invariance A ~ e i4~A, ~ E S 1. The first two arise from the original problem which is also invariant under time and space translations (t and x). However, the phase invariance has to be understood as a normal form symmetry; we refer to [74,94] for a discussion. For a usage of the symmetries of cGL in the study of bifurcations and chaos on finite domains we refer to [6] and the references therein. We now return to the original question about further symmetries and first treat the case k0 ~ 0. Depending on the symmetries of the original problem the derived GLe also has
772
A. M i e l k e
symmetries. Assume that the original problem has a reflection symmetry 7-4" (x, u) w-~ (Rlx, Rzu) with R 2 -- IRe and R 2 -- IR .... Reflecting the basic periodic pattern p ( t , x ) gives the reflected pattern ~ ( t , x ) - R2p(t, Rlx) --ei(C~176 . We have either -- + p or ~ - 4-~. This follows since our assumptions imply that, up to complex multiples a 6 C, there are only two basic patterns; and since ~ is a reflection, we have a - -+-1. Another way to see this is to use 7~L(#, Ox) -- L ( # , Ox)T~ which gives L(#,ik)-- R21L(#,iRtlk)R2
andthus
)~(#,k) - ) ~ ( # , Rtlk).
(2.19)
From ~ -- 4-~ we conclude co - 0, Rtl k0 - - k 0 and R2 q~0,k0 -- -+-~0,k0 9With the general fact )~(/z,-k) - )~(/z, k), (2.3) and (2.19) we obtain ~ l ( # ) -- --Rl~,l ( # ) ,
Zo(#) - ~o(#),
Ao = RI-AoRtl.
This implies ~.0, l 6 R and Cgr -- R lCgr. Thus, GLe is symmetric with respect to the reflection (~, A) w-~ (RI~, + A ) and the coefficients A0, )~0,1 and ~" satisfy
RIAoRtl----Ao,
2.0.1ER,
~'ER.
From this we cannot conclude that A0 is also a real matrix except if it is scalar. However, generically the present situation is associated to the above case 1 with )~(#, k) 6 R for Ik - k01 ~< 6, and then we know A0 6 IRdxd. The case ~ -- -+-p implies Rtl k0 -- k0 and RzqO0,k0 -- -+-qO0,k0, and similar to the above we obtain R l C g r - Cgr and RI AoRtl -- A0. GLe is now invariant under the reflection (~, A) w-~ (RI ~, + A ) which has no consequences for the coefficients )~0.l, ~" 6 C. This is the generic situation described in case 2 above. Case 3 corresponds to k0 - 0 where the basic periodic pattern p does not depend on x. Since p depends on t (co ~ 0), any reflection gives ~ - i p . We obtain the same conclusions as for case 2. However, in this situation the set of reflections is no longer restricted by Rtl k0 -- k0 and it is possible to consider all reflections in the space R d. Then we have an isotropic situation and obtain A0 -- bIRe with b ~ C.
2.5. A simple example To illustrate the above method we study a scalar problem on •2, namely OtU - -
--A2u
--
202u - u + (a, Vu) + (b, VAu) + # u
-+- yl u3 + (F, Vu)u 2 nt- y2ulVul 2.
(2.20)
Here u(t, xl, X2) E I[~ and Ol -- O/Oxl, moreover Yl, Y2 E • and a, b, F ~ ]I~2 are fixed coefficients while # is the small parameter. The operator L ( # , ik) is simply a scalar and thus it coincides with the eigenvalue ~(/~,k)--Ikl
4 + 2k 2 - 1 + i [ ( a , k ) - I k l 2 ( b , k ) ] + #
The Ginzburg-Landau equation in its role as a modulation equation
773
We find k0 -- (1,0) t and the expansion
&(e2p, ko -+- etc)
- ico - 6i(Cgr, K) -+- 62(p _ ( A o K , to)) + (._9(63)
4 - 6ibl with c o - al - bl, Cgr -- b + (2bl, 0) t - a, Ao -- \ - 2 i b 2
-2ib2 ). 2 - 2bl
Since the problem is scalar and the nonlinearity is cubic the coefficient ~" 6 C can be calculated simply by inserting the periodic pattern p(t, x) - E ( t , x) - e i(c~ into the nonlinearity and identifying the term associated to EJ with j - 1. This gives the complex coefficient of the cubic term
~ " - 3gl + iFl + g2. The last two formulae show that we can generate a rich class of GLes. We are particularly interested in versions which have a reflection symmetry with respect to either R(1)'(xi,x2) w-~ ( - x l , x 2 ) o r R(2)'(Xl,X2) ~-+ ( X l , - - x 2 ) . In the first case the coefficients have to satisfy al - bl -- Fl -- 0. This implies co -- 0 and ~" 6 R. Moreover, the group velocity Cgr is orthogonal to the wave vector of the periodic pattern, namely Cgr -'(0, b2 - a2) t. The diffusion matrix A0 may have purely imaginary off-diagonal coefficients. Using the Ginzburg-Landau formalism we find approximate solutions of (2.20) in the form u(t, X) ~ UA(t, X) -- 6A(62t, 6 ( x l , x2 -- (b2 - a 2 ) t ) ) e ir' + c . c . ,
where A solves the associated GLe. In the case with reflection R (2) the coefficients must satisfy a2 -- b2 --/-'2 -- 0. As a consequence co -- al - bl may be different from 0 and Cgr - - (3bl - a l , 0) t is the group velocity. The diffusion matrix is diagonal but with nonreal entries. The approximative solutions take the form
u(t,x) ~ UA(t,x) --eA(e2t, e(xl - ( 3 b , - a l ) t ,
x2))e i((a'-b')'+r') + c.c.,
where A solves the corresponding GLe.
2.6. A few other amplitude equations The theory of modulation equations is much more general than presented in this work. In many different contexts it is possible to employ a multiple scaling ansatz in the form given above. Equating equal powers to 0 one finds an equation for the desired amplitude function. This applies also to cases where several spectral bands become unstable. For each band we obtain an amplitude equation, and they are coupled nonlinearly, see, e.g., [93,40] and the references therein. Typically the coupling of these equations becomes nonlocal in ~ due to the different group velocities associated to the different spectral bands. The question of the mathematical justification of such systems is treated in [ 129].
774
A. Mielke
In hyperbolic systems, where the whole spectrum lies on the imaginary axis, the nonlinear Schr6dinger equation 0 r A - - i [ A ~ A + pA + yIAI2A]
(2.21)
appears as a modulation equation (p, ~, 6 ]K). The most important application in this case is the theory of water waves where modulation equations are used to study the nonlinear interaction of wave packages, see for instance [68,148,31 ]. Mathematical justifications in the sense of Section 6 for such kind of problems are given in [77,87,130,132]. We return to dissipative problems where the associated modulation equations are parabolic. Typical applications occur in hydrodynamics (with viscosity) or in chemical reactions. A first variant of the GLe occurs in the case when the linear theory is exactly as above, but the coefficient ~" of the cubic term vanishes. Such a situation occurs typically in multi-parameter systems when the additional parameter is chosen appropriately. In the Landau equation (2.14) the cubic coefficient satisfies
C"(~) --C" + #d'+ O ( # 2)
for # ---> 0.
We are now interested in the case ~" -- 0 and 3"# 0. With/z -- ps 2 and the scaling a(t) -sl/ZA(sZt) the Landau equation (2.14) produces the scaling limit (d/dr)A = p)~o, lA + pd'lAlZA + ~'(0)[AI4A. If modulations are also taken into account, then the linear part is the same as in the standard case by our spectral assumptions. The modulation ansatz takes the form u ( t , x ) - sl/ZA(sZt, s(x -Cgrt))E(t,x)~o.ko + c.c. and leads to the so-called
generalized GLe OrA -- div~ (AoV~A) + p)~o,,A + p'd[A[ZA + g'(0)lal4a + (VI, V~A)IAI 2 --]-(V2, V~A)A 2,
(2.22)
where Vl, V2 6 C d. This equation is sometimes also called the derivative GLe; in the case V1 = V2 -- 0 it is also called the cubic-quintic GLe. An analysis of this equation is given mainly for the spatially one-dimensional case, see, e.g., [12,49,38,144,46,80,82,119]. The mathematical justification of the generalized cGL is established in [128,118]. In many physical models there is an additional conservation law (e.g., conservation of mass) which implies that the operator L(#, 0) has an eigenvalue 0, which implies that for L(#, k) there is an eigenvalue family )~(0)(#, k). If this family is real and has the form )~(0) (#, k) = oe(/z)lkl 2 - fl(/z)lk] 4 + (Q(lkl 6) where fl(0) > 0, c~(0) = 0 and or' (0) > 0, then the ansatz u(t, x) = 8rv(84t, 8x)~o,o and # = ps 2 yields a modulation equation of CahnHilliard or Kuramoto-Sivashinsky type:
OrV -- -fl(O)A~v - pt~'(O)A~ v + f ( v , V$v, V~v), where f is homogeneous with f ( s r v , sr+lv, s r + 2 w ) = 8r+4f(v, V, W) which determines r. In the Cahn-Hilliard equation we have f = - A ( v 3) and r = 1; and in the
The Ginzburg-Landau equation in its role as a modulation equation
775
Kuramoto-Sivashinsky equation we have f = IVy] 2 which gives r -- 2. We refer to [92, 143,17] and the references therein for derivations of these amplitude equations. The question of justification is treated in [ 134] for the Kolmogorov flow problem in an infinite strip. For rotationally symmetric problems with x E R 2 the eigenvalue curve Z(#, k) is necessarily rotationally symmetric, i.e., we have Z(p, k) - Z(#, Ikl). If k0 :/: 0 then Re)~(/.t, k) -0 along a whole circle. Assume k0 = (r, 0) t and Z(#, k) 6 R, then it is natural to expand k = k0 + (etcl, el/zx2) and we obtain
zk(pe2, ko + (eKI, e l / 2 x 2 ) ) - e2(ZO.lP- A~
+ K2) 2) + O(e3),
with A ~ > 0. The parabola tel -- - x ~ / ( 2 r ) approximates the critical circle {k: Ik] - r} in the point ko. The associated two-dimensional modulation equation is called the NewellWhitehead equation [ 111 ] and reads
OTA -- A~
-i0~2)2A + ZO. lpA -~'IAI2A.
(2.23)
An analysis for the approximation properties of this formal equation is given in [ 126]. In hydrodynamical problems with two unbounded directions, like the Rayleigh-B6nard problem or the Poiseuille problem, there always appear additional modes due to the mean flow. To lowest order this additional equation couples an order parameter ~r 6 R (associated to the mean pressure) with the equation for the amplitude parameter A 6 C. This additional equation is typically elliptic, since the pressure in the Navier-Stokes equation does not appear with a time derivative. In [13,33] the following system is derived for the Rayleigh-B6nard problem between parallel plates:
OrA-
(0~, - i0~2)2A
+ pA -[A[
2A + i a 0~2 ,
(2.24)
0 - a~2~ + yla~, O~21AI2 + y2a~2 Im(AO~2A ). In Section 3.3 we repeat the arguments in [69,32] for the perturbations of the Poiseuille flow close to the threshold of instability. We obtain the coupled system
OrA -- clO~ A + c2O~2A + pc3A + c4IAI2A + csA O~, ~, O-
div~ V ~ O + V
0
'
with cj E C and y 6 R, where Re ct > 0 for I = 1. . . . . 4. See (3.5) for the exact values of these constants. The mathematical justification of this system was given in [ 18].
3. Applications in hydrodynamics Historically the GLe was first derived in the context of hydrodynamics where the modulated periodic patterns can be seen in lots of experiments and in nature. Famous examples
776
A. Mielke
are cloud streets, water waves and the Taylor vortices of a flow between rotating concentric cylinders. First papers in this contexts were published around 1970, see for instance [ 111,137,35,112,69,32]. For this reason it is suggested in [92] to use the name "HockingStewartson equation" (after [69]) rather than "Ginzburg-Landau equation". In fact, the latter name is not derived from the present context of modulational theory but from its usage in the theory of superconductivity, see [62,45,23]. The connection between superconductivity and the modulation theory arises from the fact that in both cases the rGL plays a central role. In the following three subsections we present the basic cases in hydrodynamics where the Ginzburg-Landau theory is applicable. There are many other cases where the method of amplitude equations gives useful physical insight (e.g., in reaction-diffusion problems), however, due to the limited space we restrict ourselves. The fourth subsection introduces the theory of sideband instabilities which was one of the major driving forces for the development of the theory of modulation equations. 3.1. The R a y l e i g h - B d n a r d problem We consider a two-dimensional fluid layer of an incompressible viscous fluid between two rigid plates which are kept at two fixed temperatures. (In the three-dimensional case a more complicated amplitude equation appears, see (2.24).) After nondimensionalizing the velocity u = (u l, u2), the pressure p and the temperature deviation 0 from the linear profile between the two walls, we find the system Otu + (u . V)u + V p -
RO[~)'"" - A u = 0 , | div u -- 0, P(OtO + (u . V)O) - Ru2 - AO -- O,
f o r t ~> 0, ( x , y ) 6IR x (0, Jr),
(3.1) together with the stress-free boundary conditions OyUl - - U 2 --- 0 - - 0 for y 6 {0, Jr }, see [111,35,96,13,99]. These boundary conditions are used only for mathematical simplicity, the more physical conditions u l -- u2 = 0 -- 0 can be done similarly, however the calculations can no longer be done analytically, see [33]. Here the parameter P is called the Prandtl number and R 2 is called the Rayleigh number. The latter parameter is considered as bifurcation parameter as it is proportional to the temperature difference at the two boundaries whereas P is a material parameter. For the analysis we insert the ansatz Ul(t,x,y)
with
l
ofj E C
{OtlCOs(ny)~
u2(t, x, y) O(t, x, y)
= eZ,,t+ikx lot2 s i n ( n y ) / sin(ny)/
p ( t , x, y)
~kOt4cos(ny),]
into the linearization at (u, 0) = 0. We obtain the relation
+
+,,2 +
+
The Ginzburg-Landau equation in its role as a modulation equation
777
from which we derive that the critical parameter is R -- R0 = 3x/3/2, that is for R ~< R0 we have )~n(R, k) <~ O. The critical eigenvalue occurs for n -- 1 with wavenumber k0 = l/x/2, it is real and has the expansion
~.(,O~ 2 k0 -Jr-6K') -- 62[ '
42]
4Rop 9(P+
1)
P+I
K
"+- 0 @ 3)
where R -- R0 + log2. The associated eigenvector ~0,k0 has the form q:'o.ko (Y)= (i~/2 cosy, siny, v/'-3sin y , - 3 cosy) t. Yet, there is another critical mode occurring for n = 0 and k = 0, which derives from the fact that the Navier-Stokes equation is invariant under the addition of a constant to the pressure, or equivalently, from the freedom of superimposing a mean flow. In the case of one unbounded direction this degeneracy can be projected out by using the fact that the mass flux Q(t, x) --
f
Tr
u l (t, x, y) dx
=0
is independent of x which follows easily from div u = 0 and the boundary conditions. Thus fixing Q(t, x) =_ 0 in addition, the formal derivation of the Ginzburg-Landau formalism is valid and we obtain the following rGL as amplitude equation
Or a --
4 P + 1
o~a +
2p
p2 a - ~lalZa. ~/-3(P + 1) 8(P + 1)
(3.2)
With ~0,ko from above the approximations of the solutions to (3. l) read
{ . cosy O(t, x, yy)| )] _ .2(t,
(ezt' ex)eiX/'/2 !|/ v"-3 siny sin y
p(t, x, y) ]
+ c.c. +
(..9(e 2)
\ - 3 cos y
where A solves (3.2) and e is defined via R -- Ro + pe 2. The first mathematically rigorous justification theory for the Rayleigh-B6nard problem is provided in [ 123].
3.2. The Couette-Taylor problem For this problem the Navier-Stokes equation is considered between two infinitely long concentric cylinders which are rotating with different speed.
A. Mielke
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Following [24, Chapter II. 1.2] we denote by rin, rout, COin and COout the radii and rotational velocities of the inner and outer cylinder and by v the kinematical viscosity. The problem can be made nondimensional by introducing three similarity parameters: -- coout/coin,
r l - rin/rout E (0, 1),
- rincoin (rout - rin) / v.
The Navier-Stokes equations for the velocity v E N3 take the form
Otv +7~(v. V)v + V p -
Av = 0 , | div v -- 0,
!
in s
-- R x So,
(3.3)
where Z~ -- {y E R 2" r~ < [y[ < 1} with r~ - r//(1 - 77) is the cross-section in form of an annulus. The boundary conditions are v (x, ro cos 4), ro sin 4~) - (0, - sin q5, cos ~b)t
and
v(x, cos ~b, sin q~) - ~ / ~ ( 0 , - sin~b, COS q~) t for all 4~ E [0, 27r). The basic Couette flow Vcou is such that fluid particles stay on circles concentric and orthogonal to the axis y - 0. It has the explicit expression
Vcou (x, r cos ~b, r sin 4~) --
/ ,r0,sin]
A
with ~"(r) = f i r § b/r,
~'(r) c o s ~ ] A
where fi', b E R are chosen such that ~'(r,7) -- 1 and ~'(1) = ~/r/. Like in the Rayleigh-B6nard problem above we also impose that the mean flux through the cross-sections is 0, i.e., fz,7 vl (t ' x ' y) dy --- 0. The Ginzburg-Landau formalism is now applied to the perturbations u -- v - Vcou of the basic Couette flow. We will obtain a onedimensional GLe, since the problem has only one unbounded direction. As the problem has the additional reflection symmetry x ~ - x we know by Section 2.4 that if the GinzburgLandau formalism applies with k0 :/: 0 then it leads to a real GLe. The parameters r/and the rotation ratio ~ E]K are usually fixed and the Reynolds number T~ is used as the bifurcation parameter. Numerical computations (cf. [24]) show that our assumptions on the leading unstable eigenvalue are always fulfilled if ~ > G(r/) for some ~(r/) < 0. The critical Reynolds number is then denoted by ~ 0 = Tr and we have k0 > 0. The associated eigenvalue is simple, and hence it is real. Moreover, the associated eigenvectors 4~r,,k must be rotationally invariant. Thus, for 7r = ~ 0 § Pe2 the GLe reads
OrA - - FlO~A + F 2 p A - F3]A]2A, where the real coefficients yj are all positive, the first calculation of these coefficients appears to be in [83] using the Liapunov-Schmidt reduction as in Section 2.2.2. We refer
The Ginzburg-Landau equation in its role as a modulation equation
779
to [74] and [24, Chapter VII] for the first rigorous derivation of the steady part of the GLe in the context of the Couette-Taylor problem. Since the eigenvector q}u,k is rotationally invariant it follows that all solutions which are constructed via the Ginzburg-Landau formalism
v(t,x, y ) - - Vcou + eA(e2t, ex)@o,ko(Y) +c.c. + O ( e 2) are also rotationally invariant. It was established in [135,136] that every perturbation of Vcou approaches the set of rotationally symmetric functions exponentially fast. Moreover there is an associated local attractor ,A in the set of all rotationally symmetric functions which attracts all solutions starting sufficiently close to Vcou, see the above references or Section 6.7 for more details.
3.3. The Poiseuille flow In contrast to the Rayleigh-B6nard problem and the Couette-Taylor problem the Poiseuille flow does not play such an important role in the theory of modulation equations for mainly two reasons 9 First, by now it is experimentally impossible to reach the theoretical threshold of instability which is at Reynolds number 57729149Typical experiments see a stable Poiseuille flow for Reynolds numbers below 2500, while for larger Reynolds numbers the domain of attraction of the Poiseuille solution is too small to be realized by experimental conditions. The second reason involves the bifurcations which are predicted 9 It turns out that the real part of the cubic coefficient ~" in the cGL is positive rather than negative 9 Hence the associated bifurcations are subcritical and lead to unstable phenomena 9 Nevertheless the Poiseuille problem was historically important (cf. [69,32]) and it still serves as a problem on which one can test numerical and analytical methods. We mainly include it here as it is one of the few hydrodynamical problems which leads to a cGL with two unbounded directions. The Poiseuille flow occurs between two plates when a pressure gradient drives the system. Consider in nondimensional form the flow domain 12' = R 2 x ( - 1 , 1) with two unbounded directions x l and x2 and one bounded direction y E ( - 1 , 1). The velocity v(t, x, y) is now in R 3 with components vl, V2 and v3 in the direction xl, x2 and y. The basic equations are the Navier-Stokes equation with no-slip boundary conditions
3tv + (v 9 V)v + V p -
-R- A v -- 0 ' v = 0
divv--0
in s
(3.4)
on3~.
Here R = Uoh/v is the Reynolds number, where U0 is the typical velocity, h the height of the fluid layer and v the kinematic viscosity9 The basic Poiseuille solution is Vpois(t, x, y) -- (1 - y2, 0, 0) t which is associated to the pressure p(x, y) = - 2 x l / R which gives a constant pressure gradient in x l-direction. We are now interested in small solutions around the basic Poiseuille flow. Since Vpois is independent of x E IR2 the classical ansatz u(t, x, y) = e ~t+i(/''x) ~ ( y ) can be inserted into the linearization at Vpois. The classical Squire theorem (cf. [32,2]) shows that instability
A. Mielke
780
first occurs with a wave vector parallel to the x 1-direction. In fact, numerically one finds that the critical Reynolds number is R0 ~ 5772.2 and the associated wave vector is k0 (1.0206, 0) t. The eigenvalue expansion can be found numerically and it reads ,k(R - R0, k) - ico + ,k0,1 (R - R0) - i(Cgr, k - k0) - ( A o ( k - ko), k - ko) + h.o.t. with co ,~ - 0 . 2 6 9 , ,k0, l ~ (0.168 + i0.811) 10 -5, r ~'~ (0.383, 0) t, A 0 ~ diag(0.187 + i0.0275, 0.00466 + i0.0808), see [32]. The associated eigenfunction can be expressed by the eigenfunction of the associated Orr-Sommerfeld equation ~ 0 : ( - 1 , 1)--+ C as ~0,k0 (y) - ((d/dy)qg(y), 0 , - i ( k 0 , el)cp) t, see [2]. However, as in the other two cases above the mean flow has to be taken into account. Since we have two unbounded directions we cannot restrict the mean flow of the solutions. Thus, one either has to reduce the problem to a problem with only one unbounded direction, e.g., by making it periodic in one of the directions, or we have to generalize the GinzburgLandau formalism. We prefer the latter idea, as it allows us to find the other cases also by reinterpreting the obtained modulation equation accordingly. To stay consistent with the normalization in [32] we let R = Ro + pe 2 with p = 1/ Re ~0,1, then the ansatz for the solutions u = v - Vpois of (3.4) takes the form
u ( t , x , y) -- eA(e2t, e(x - C g r t ) ) E ( t , x ) ~ o , k o ( y ) + c.c. + h.o.t., 2xl p(t, x, y) = po(t) -R- + e ~ ( e z t , e(x - Cgrt))E 0 + h.o.t., and equating the coefficients for el E m to 0 one arrives at the coupled system
Or A - A~ O~, A - A~
~0,1
- A -- Cl IAI2A - c2A 0~ ~ -- O, Re ~o, 1
[
div ~ V~ ~ 4- t'
(z )l 0
(3.5)
= O,
where Cl ~ 29.5 - i144, r '~ --28.0 4- i642 and y ~, 0.0453. The appearance of a second equation for the pressure perturbation r (which does not have a time derivative in it) is typical for all Navier-Stokes problems with two unbounded directions, see Section 2.6. We stayed within the frame of two unbounded directions as it serves as a model where the anisotropic cGL appears (argA~ --fi argA~ In fact, the formal approach here was made mathematically rigorous, in the sense of Section 6, in [ 18] for the full Poiseuille problem with two unbounded directions. It should be noted that flow problems with unbounded directions are not well-posed without suitable conditions at infinity. We either have to specify the mean flux or the mean pressure gradient to control the flow at infinity, see [5] for more details. For (3.5) this means that we may either prescribe an additional mean pressure gradient qJ E •2 via 7r(~, t) - (~, ~) + ~(~, t)
with ~(., t) bounded o n R 2
The Ginzburg-Landau equation in its role as a modulation equation
781
or an additional mean flux U e ]1~2 defined by 1 ~ [ (IA(se0t)12)] U = lim V~ 7z(~, t) + y d~. e--,~ 4~-y- ~(_e,e)2 Of course, we might as well prescribe certain components of tp and .T" and we may also make these quantities time-dependent. We can reduce to the case of one unbounded direction by restriction to special solution classes and thus obtain a classical one-dimensional GLe. First we consider solutions which are independent of x2 and hence the solutions of (3.5) are independent of ~2. If ,f'l is prescribed, the second equation can be integrated to 0~ 7z = - y I A ] 2 -4- f'l and inserted into the first to obtain
OrA-- AOIO~,A +
~.0.1 -Jr- f ' l c 2 ] A + (Cl - yc2)IAleA. Re)~0.1
(3.6)
We see that the additional downstream flux .T'l amounts in the same as redefining the Reynolds number. Note that prescribing tPl instead of f l leads to the condition tPl = ~0~, 7z~ - ~ l - ylal2~, where ~f]] --lime~c~ I fe_e/(~') d~. This gives 0~,7z - tPl + y~lA[2~ - ylAI 2 and we are lead to the nonlocal equation
O T A - AOIO~,A +
~.O, 1 Re,~o.l
-~-tPlc21A -f- (Cl
- yc2)IAI2A + yc2
~IAI2~A. (3.7)
Second, we may follow [2,4] and consider solutions of (3.4) which are periodic in downstream direction Xl with a period close to that of E. This amounts to looking for solutions of (3.5) of the form A(r, ~) -- e it~ A(r, ~2) and 7z(r, ~) -- ~ ( r , ~2) and we find
'~ f ~0,1 Or A -- A0220~2.4+ L R ~ o , 1
A~
X + c, IXl 2 A.
(3.8)
Here we have prescribed tPl = 0, and the associated problem with fixed f'l leads again to a nonlocal equation similar to (3.7), see [5]. Below we will show that for (3.6) and (3.8) there exist pulse solutions of the form A (r , ~j) -- dl eiVr [cosh(r/~j )]d2 with d l, d2 E C, v E JR, r / > 0 and Re d2 < 0. However, there is a major difference between the two cases with respect to the full Poiseuille problem (3.4). To prove the existence of such pulse solutions for the full Poiseuille problem one needs to use the reflection symmetry ~j ~ --~j which is present in (3.6) and (3.8). However, the full problem (3.4) is only invariant under the reflection x ~ (x l, - x 2 ) but not with respect to x ~ ( - x l, x2). As a consequence it is only possible to prove existence of reflectionally symmetric pulse type
A. Mielke
782
solutions for the case associated to (3.8), see [3,4]. The case associated to (3.6) was treated in [36] where existence of nonsymmetric pulses is shown by introducing a travel speed as an additional parameter.
3.4. Sideband instabilities Historically the theory of sideband instabilities in hydrodynamics was one of the major motivations for the development of the modulation theory. Here we want to explain this phenomenon and relate it to the stability treatment of plane waves in Section 4.1. We consider the full problem (2.1) for/z > 0 and assume that the cubic coefficient ~" in the GLe satisfies R e F > 0. For the linearization L (#, 0x) at u =-- 0 there is already an open set of wave vectors unstable:
/C(#) -- {k e R a" Re,k(#, k) > 0, (k, ko) ~> 0}, where/.(/z, k) is the critical eigenvalue of L(/z, ik) in (2.3). Under our spectral assumptions K~(#) forms a smooth region of nearly elliptical shape with center in k0 and diameter of order 4r~. Using the Liapunov-Schmidt reduction it is easy to show that for all k e K~(#) there exists a periodic solution u(t, x) = L/(#, k, t, x) of the nonlinear problem (2.1) in the form
U ( # , k , t , x , y ) - ~;u,k((, Y) -- )2u,k(( + 2re, y)
with ( = ~ t + (k,x)
where ~ ~ co is a new parameter (cf., e.g., [99]). Such solutions are called plane traveling waves as they only depend on one coordinate in the (t, x)-space. The first approximation of V~,k is given by Be i~ 4~u,k (y) + c.c. where B e C \ {0} satisfies the algebraic equation
i ~ B - - g ( # , k , IBI2)B
with g ( # , k , s ) - - X ( # , k )
- ~ ' s +h.o.t.,
cf. (2.14). This algebraic equation is of course compatible with the GLe since the lowest order term of ~u,k can be written as UA with A(r, ~) -- e i((~-c~176 B. By the principle of exchange of stability it is known that the solutions L/(#, k, t, x, y) are stable with respect to perturbations in the same function class, namely those which only depend on ((, y) with period 2zr in (. However, in experiments not all of these periodic patterns are observable, since some of them are unstable with respect to perturbations having a slightly different wave length. Thus, one is led to study the spectral properties of the linearization around the solution L/(#, k, t, x, y) in general function spaces:
Otw -- L(#, Ox)W + DuN(/z, O x , l g ( # , k , t , x , .))[w]. This leads to a differentialoperator with spatially periodic coefficients. Using the BlochFloquet ansatz w(t, x) -- e zt+(~'xl W((, y) the problem reduces to the spectral problem for a family of Bloch operators
~(~,#,k, Oc)W-~W
The Ginzburg-Landau equation in its role as a modulation equation
783
posed on the bounded domain S i x r . The vector cr E Ii~d is called the sideband vector of the perturbation, since W ( ~ ) = W ( ~ t + (k,x)) already contains the underlying wave vector k. By construction we know that ~(0, #, k, 0C)0r = 0 such that ~(o-, #, k) = 0 for o- = 0 E ]1~d. Moreover, ~ has discrete eigenvalues which can be expanded in cr close to 0. Define the set of unstable sideband vectors S(#, k) - {or 6 R Cl" It~(cr, #, k, .) has an eigenvalue ~" with R e ~ > 0} associated to the solution Vrt,k. If S(#, k) is nonempty then the periodic pattern Vrt,k is unstable. If S(#, k) intersects every neighborhood of cr = 0, then H is said to have an unstable sideband. (Note that 0 -J: S(#, k) if L/is stable with respect to periodic perturbations in r A general analysis of the Bloch-Floquet ansatz in hydrodynamics is given in [ 117]. Here we want to reveal the interplay with the Ginzburg-Landau formalism. In fact the solutions H(/z, k, t, x, .) constructed above have their counterparts in the plane waves Awave(r, se) = r e i(ar+(K'8))
with r 2 =
1 [pRe)~o.,-((ReAo)x,x)].
Re~"
The condition r 2 > 0 defines an ellipsoid which is the blowup of the set /C(#) = {k: Re)~(lz, k) > 0}. The stability of these plane waves as solutions of the GLe can now be studied much easier due to the phase invariance A w-~ ei'~A of GLe. Writing perturbations in the form A (r, ~) = A wave (1", ~ ) [ 1 + B (r, ~) ] we obtain a parabolic equation for B which is homogeneous in space-time. Hence, the spectral problem can be solved algebraically by using Fourier transform, see Section 4.1. In most physical papers the stability or instability of a periodic pattern H ( # , k,-) is directly inferred from the stability or instability of the associated plane wave for GLe. Yet, the given connection between the sideband instabilities of the periodic patterns H in the full problem and that of the plane waves in GLe is formal so far. However, in [97,99,100] it is shown that this connection can be made rigorous in many cases by using the principle of reduced instability and thus avoiding the Ginzburg-Landau formalism.
4. Special solutions and dynamics of the Ginzburg-Landau equation In the whole of this section we only consider the GLe without any reference to the full problem. This chapter is completely independent of the others. Hence we may use t ~> 0 and x E R ~t to denote the time and the space variable. We will also use the abbreviation Oj - - O/Oxj. Thus the GLe in its form (2.16) reads now d OtA -- Z ( 1
j= 1
+ iotj)OjA + pA - ( 1 + ifl)lAI2A.
(4.1)
A. Mielke
784
For the mathematical theory we introduce a few function spaces. The main point here is that we need appropriate spaces which enable us to study general nondecaying functions such as spatially periodic or quasiperiodic solutions as well as fronts, pulses or solutions with different periodic behavior at infinity. For domains s C ]1~n we define the weighted norm
IIAIIL~ -
(jo
(4.2)
w ( x ) l A ( x ) [ p dx
and with LP(s we denote the associated Banach space. The weight w" I~d ~ (0, cx~) is assumed to be integrable over IRd and satisfies the estimate [Vw(x)[ ~ C w ( x ) . Throughout it will be sufficient to consider the weight w ( x ) = e -Ixl . The basis of the theory for unbounded domains are the uniformly local Sobolev spaces p Lul (s which are defined as follows. Let
"P Lul(S'2 ) -- {u
E
Lloc(S-2)" [[Ul[LupI <(X)}
with [[U[[LuP~ -- sup [ [ w ( . -
y ERd
y)ullL .
The final uniformly local space is defined as LuPl(s
= closure of C~d (s
in LuPl(s
where Cb~ (a"-2) is the space of all C ~176 functions whose derivatives are bounded in s Note the embeddings Cbdd,unif(ff2) C Lql(S'2) C LuPl(a'2) for 1 ~< p < q < c~ with norm estimates ]]U][LuPl ~ c(q-P)/(qP)llUllLq ~ cl/Pl[u[lLOO where Cw - fRd w ( x ) d x . k,p The uniformly local Sobolev spaces Wul (s
are defined as usual"
P 9D a u e Lul P(#2) forotE N o, n lal~k}. WkuiP(#2)- {u e Lul(#2) For the Hilbert spaces we write Hkul(s space Wkuip (s
= Wulk,2 (a"2). In the case of a bounded domain the
and its topology coincide with the usual Lebesgue space W k'p ($-2).
4.1. Plane waves and their stability We have indicated in Section 3.4 that the plane waves and their stability properties are of central interest in the theory of pattern selection. In particular, they explain the phenomenon of sideband instability which is observed in hydrodynamical experiments on large spatial domains. Here we present the relevant algebraic calculations. Because of the rotational symmetry A v-+ e iy A there is a simple solution class given in the form A ( t , x ) - r e i(at+(K'x)). Inserting this into (4.1) and recalling the notion S -diag(ul . . . . . Otd) we find the explicit solutions A(t, x) = r e i(at+(x'x)) with r = V/p -[K[ 2 and 6 - ( S x , x) - fi(p
-[gl2).
(4.3)
785
The Ginzburg-Landau equation in its role as a modulation equation
These solutions play a central role in the theory of pattern formation. They correspond to space-time periodic solutions in the full problem whose frequency co 4- e6 and wave vector k = ko 4- ex are small perturbations of those of the basic periodic pattern p = E@0,k0. The question of pattern selection is thus intimately related to the stability of the solutions in (4.3). This stability analysis is classical in the real case, where it is called the Eckhaus instability, since it was first derived in [47]. The isotropic complex case was first treated in [ 109]. We repeat the main steps here since we want to generalize the theory to the anisotropic case. Stability is best tested by inserting the ansatz A ( t , x ) rei(~t+(K'x))(1 4- b l ( t , x ) 4 i b 2 ( t , x ) ) into (4.1) and to find the linearized equation for the perturbation b = (bl, b2) E R 2. This special ansatz and the rotational invariance lead to a linear problem with constant coefficients: _
1
--c~j 1
j=l
Oj Oj2b2 4- 2xjOjbl
- 2(p --IKI
9 flbl
We have to use a real representation as we now want to study the complex spectrum of this linear real operator. Inserting b ( t , x ) - eZt+i{rl'X)(Zl,Z2) t with Z, z l , z 2 E C, we find solutions if and only if )~ is an eigenvalue of
M ( p , to, 7) -
-1712 - 2i{Sx, 7) - 2(p - I x l 2) 2i{x, 7) - (So, 7) - 2fl(p - I K I 2)
-2i{x, 0) + (S0, 0) - I r / l e - 2i{Sx, r/) } "
The characteristic polynomial takes the form Z 2 + (al + ia2)Z 4- (a3 + ia4) -- 0 with al -- 210l 2 + 2r 2 a2 - 4 ( S x , 0) a3 -- 1712 + (S0, 0) 2 --4{Sx, 0) 2 --4(x, 0) 2 4- 210]2r 2, a4 -- 411712 + r2](Str 0) - 4 ( x , 0)(S0, 7). The two roots ~-1 and ~2 of this quadratic polynomial have nonpositive real part if and only if al~>O,
a 2 + a 24-4a3>~0,
a2a3 + a la2a4 -- a 2 >/O.
The first condition is always satisfied and the other two lead after the substitution 0 - r~ to a necessary and sufficient condition for (linearized) stability: 4({x, ~)2 _ {SK, ~)2) <~ r2(5l~l 4 + lOl~l 2 + 1 + (S~, ~)2 4- 2fl{S~, ~))
and
4{x ~)2 ~ r 2 (1 + l~12)2[l~l 4 + 2l~l 2 + {S~, ~)2 4- 2fl(S~, ~)] ' (1 + l~12) 2 4- (/3 4- (S~, ~))2 for all ~ 6 R d.
(4.4)
A. Mielke
786
We arranged the terms such that the left-hand side is quadratic in tc while the righthand side is linear in r 2 --/9 - IKI2. In the nonisotropic case the analysis of these conditions is rather complicated, however, we may derive a necessary condition for stability by considering those terms in the second condition which are quadratic in ~, namely 4(x, ~)2 ~ 2r211~12 _jr_fl(S~, ~)]/(1 + f12) for all ~ E IRd. This provides the anisotropic
necessary condition for stability d
1 + otjfl >~0
for j -
1. . . . . d
and
K2
r2
J ~ 2(1 +/3 2) . Y~" 1 + lYjfl
(4.5)
j=l For the isotropic case the conditions simplify, see [44]. In particular, it is possible to show that the first condition in (4.4) is a consequence of the second. Defining (1 + y2)2[y2 -I- 2 -I-ot2y 2 + 2otfl] v(ot, fl) -- min
VClR
(1 + y2)2 _+_(fl --I-oty2) 2
we have stability if and only if 4[KI2 ~< r2v(ot, fl). Thus, stable rolls can only exist for v(ot,/3) ~> 0 which holds if and only if the Newell criterion [109] 1 + c~/3 >~ 0 is satisfied. Moreover, the rotating wave which loses stability last is the one with tc - - 0 . We arrive at the isotropic stability criterion 1 + c~fl ~> 0
and
Ix I2 ~ p
v(+,/~) 4 + v(c~,/3)
9
(4.6)
Since v(0, 0) - 2 we find the Eckhaus criterion ]K] 2 ~ p/3 for rGL, [47]. The ansatz b(t, x) -- e )'te i<~'x) (z l, Z2) t shows instability with respect to perturbations in the space L2ul(iRd). From the theory in [100,117] it follows that this implies instability in the classical space L 2 (iRd). However, in the case of stability there is a slight difference: in LZul(IRd) we have the perturbation b(t, x) = (0, Z2) t which does not decay whereas perturbations b(t, .) with b(0,-) ~ L2(IRd) decay to 0 for t --+ oo, namely lib(t, .)ILL2 --+ 0 and lib(t, .)IIL~ ~< ct-d/4. The stability of the rotating plane waves in the nonlinear problem is an even more delicate question because the spectrum of the linearization is continuous and contains a point on the imaginary axis. Thus, we can only expect marginal stability without exponential decay. In the present situation it can be shown that the rotating waves are diffusively stable in the sense that initial perturbations which are spatially localized (e.g., small in H l (IRa)) will decay in L ~ (R d) like t -~ for some ot > 0, see [20,30,78]. The nonlinear stability with respect to perturbations in L ~ (iRd) is an open problem.
4.2. The real Ginzburg-Landau equation In our notation the real GLe is the GLe for the complex order parameter A(t, x) E C when the equation has real coefficients. It can always be made isotropic and we normalize the
The Ginzburg-Landau equation in its role as a modulation equation
787
coefficients to 1"
OtA- AA § A -IAI2A.
(4.7)
Sometimes the name 'real GLe' is also used for the real part of our equation, namely the scalar parabolic equation Ot~P -- A 7r § ~p -- ~p3 which is often called Allen-Cahn equation. If rGL is considered on a bounded domain I-2 C R d with suitable boundary conditions, then it is a gradient system with respect to the Ginzburg-Landau energy functional 1 12 - ~[A 112 E ( A ) = f s 2 ( ~[VA
1 14) dx § f + ~[A
~/1l A l -, d a .
(4.8)
1-2
We use r/E R to generate the Robin boundary conditions (VA, v) + oA - - 0 , which includes the Neumann case 0 -- 0. For Dirichlet boundary conditions and for periodic boundary conditions we also let 0 - 0 and choose the appropriate function space. (In fact, also the isotropic complex GLe (4.1) with Otj - - / ~ and suitable boundary conditions has E ( A ) as a strict Liapunov function.) Thus, we can conclude that the global attractor consists of steady states and heteroclinic connections between them, see [64,8,141 ]. However, this theory does not give any information on the case of unbounded domains. There still exists a global attractor (cf. Section 5), however, most solutions have infinite energy and thus need not settle down to an equilibrium, see, e.g., our example of diffusive mixing in Section 4.2.3, the coarsening dynamics discussed in [52], or more general the discussion in Section 5.3. Recently in [ 138,59] the notion of extended gradient flows was invented to handle the situation of infinite energy solutions for systems which would be gradient-like systems on bounded domains (cf. [115]). Using the local energy density e R, the energy flux f e R d and the dissipation rate - g ~< 0, as defined in (4.19), it is possible to show that the solutions in the attractor of (4.7) in the full space IIUl have to satisfy certain energetic restrictions. In particular, for d ~< 2 it is possible to show that there are no time-periodic solutions. Moreover, the co-limit sets with respect to the weighted norm [1 9 [[L~. (see (4.2)) always contain an equilibrium. Another important property involves the maximum principle which usually only works for scalar parabolic equations. The maximum m(t) of [A(t, x)[ over Y2 satisfies the scalar differential inequality th ~< m - m 3 . This follows by the maximum principle if we write rGL with A -- re i4~"
Otr -- A r -[V~b[2r + r - r 3,
2 Ot~ -- Aqb + -(V~b, Vr). r
As consequence we find that the global attractor A, as constructed in Section 5, lies in the ball {a ~ L ~ (S-2)" [Ia IIoc ~< 1}. Second we may consider nodal lines of Re A or more general of u(t, x) -- Re[e i• A (t, x)] for any ?' E R. If A is a solution of (4.7), then u solves the scalar linear parabolic equation
Otu--Au+(1-1A(t,x)[2)u.
(4.9)
Clearly, new nodal lines cannot be created. If for instance u(t, x) > 0 for t 6 [to - e, to) and Ix - x0[ ~< e, then u(to, xo) -- 0 is impossible because of the maximum principle. This
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concept will be especially useful in the one-dimensional setting since the nodal lines are just points, see Section 4.2.2. 4.2.1. Ginzburg-Landau vortices. Strictly speaking Ginzburg-Landau vortices are steady states of the real Ginzburg-Landau equation on very large two-dimensional domains, see [14,76,113,90,75]. For simplicity we state the problem only o n •2 and refer to the references for the case of a domain which is large with respect to the natural length scale of the vortices. However, one still speaks of Ginzburg-Landau vortices when the solution looks locally like a steady state which moves slowly in space. The existence theory for single steady vortices can be found in [63] or the references in [ 14,145]. For each n 6 Z there is a unique vortex (up to translations and phase invariance) of degree (or vorticity) n. It can be written in the form V,(x) - oe,,(Ix I)e i'~~ where qg(x) - arg(xl + ix2) such that Ixlne in~(x) = (xl -I- ix2)", a,,(s) > 0 with or(s) ~ for s --+ cx~ and
1
n2
1
0 -- ot~ + -o~ ~ -- c~3 s= z +ce. s For this we use the fact that Aq)(x) = 0 and Vqg(x) = i x i - 2 ( - x 2 , x l ) t for x =/=0 which implies (Vet,, (Ixi), Vq)(x)) =--0. Moreover, we have an(S) = CnS I'i + O(slnl+')s-,O for some c,, > 0 and an(S) = 1 + O ( e - S ) s ~ o c . The case n = 0 is, of course, trivial with Vo(x) -- 1. It was shown in [145] that these steady states are unstable if in] ~> 2 and that they are linearly stable for ]n l ~< 1. In the unstable case the vortex generically breaks apart into In l vortices all having the degree sign(n). Interaction of vortices can be studied if we place a finite number of stable vortices in the plane R 2, say at the locations (j, and ask for the behavior of the solution A (t) = Gt (A0) of rGL with initial condition N
Ao(x)- I-I
(x-
(o)).
j=l
The locations x j ( t ) of the vortices for t > 0 are then defined via A(t, x j ( t ) ) = O. The interaction can be studied well if the initial conditions are such that N vortices of degrees nj - + 1 are spaced far apart from each other, namely z(0) >> 1 where z(t) = min{l~'j(t) - ~'m(t)l" j --/: m}. It turns out that the solutions remain approximately of the above form, namely N
A(t, x) -- H Vnj (x - r j=l
+ O(z(t)-')
for z(t) --+ co.
(4.10)
The Ginzburg-Landau equation in its role as a modulation equation
789
The locations xj (t) of the vortices are approximated by ~'j (t) which are obtained by solving an ODE for r = (~'l . . . . . ~'X) E R 2N. It is given as the gradient flow associated to the renormalized energy H (~_) -- - ~
n j nm log I~'j - ~'m I-
j#m
Thus the ODE ~" = - r e H(~') takes the form m
n in
~j -- -VCy H(~_) -- 2nj Z If j - Cml 2 (r - r m:/:j
j = 1 . . . . . N.
(4.11)
We refer to [76] for a mathematical justification of this approximation result in the case of large bounded domains. It remains only valid in the limit z(t) >> 1 which excludes collisions of the vortices. Typically a vortex of degree + 1 and and a vortex of degree - 1 attract each other such that z(t) goes to 0 in finite time, the two vortices of opposite degree collide and annihilate each other, i.e., they disappear into a trivial vortex of degree 0. A similar analysis can be made for the two-dimensional cGL with otl = cg2 = f l , which we write in the form OtA -- ( i + 6)(AA + A - I A I 2 A ) , where 6 ~> 0. In the case 6 = 0 we obtain a form of the nonlinear Schr6dinger equation. It is again possible to show that solutions of the form (4.10)exist, but now the locations ~j(t) satisfy the ODE
(
-n j
--6
VCj H (~_).
(4.12)
In particular, this system is Hamiltonian for 6 = 0, see [113,90,75] and the references therein. 4.2.2. One spatial direction. The one-dimensional rGL can either be considered on the real line with general initial data or on an interval (0, ~) with suitable boundary conditions. In the latter case we may have a gradient system with respect to the energy given in (4.8). In the first case all stationary solutions, which are solutions of the four-dimensional ODE A" + A - IAIZA = 0, can be classified (up to phase and translation invariance) as follows, see [57]: (a) real solutions A ( x ) = r ( x ) e IR where r" + r - r 3 = 0 (periodic or homoclinic r ( x ) = tanh(x/~/-2)); (b) rotating plane waves A ( x ) -- x/1 - t c 2 e iKx with IKI <~ 1; (c) quasiperiodic solutions A ( x ) - - r ( x ) e i4~(x), where r(.) and 49'(') are periodic with the same period, see [21,39];
A. Mielke
790 (d) homoclinic solutions
A(x) -- y//2(x 2 +
C2
tanh2(cx))e ir
where c - v/(1 - 3tc2)/2, 0 < K 2 < 1/3 and r - x x + arctan[ c tanh(cx)]. Each of these solutions is unstable except for the rotating plane waves with x 2 ~< 1/3, cf. [57]. If some of these solutions are compatible with the boundary conditions of the finite interval (0, ~), then they can be stable as the set of perturbations is reduced as well. It is an interesting question to ask which of these steady states are connected by heteroclinic solutions. One of these results is obtained in [51 ] by considering rGL with the quasiperiodic boundary conditions A(t,g)=ei• on the interval/2 = (0, g). The phase space is
Ze,• -- {A ~. Hll(~, C)" e-i•
is periodic with period t};
(4.13)
and rGL is a gradient system with respect to the energy E in (4.8). In addition to A = 0 there are families of steady states (e i~ UKj)fESI where
Ux(x)-v/1-K2e
ixx
and
xjg.-ymod2rc.
The energy of these steady waves is E(UK) =--t~(1 --x2)2/4. In [51 ] it is shown that there exists a solution A : R --+ Ze,• starting at UKI for t -+ - - ~ and converging to e ir U t ( 2 for t --+ oo, if I(1 and x2 satisfy certain restrictions. First, UK~ has to be unstable with respect to perturbations in Ze,• Second, there must be a solution UK2 which has lower energy, i.e., E(UK~) > E(UK2). This gives K2 > (~2 _jr_2zr2)/(3~2) and x 2 > tc2. The third condition is the most difficult but it makes the proof straight forward. It is assumed that except for the family e i~pUK2 there are no other steady states B of rGL in Ze,• with E ( B ) < E(UKI). The following result is obtained by choosing K2 = tel - 2re/g, see [51 ]. THEOREM 4.1. Assume that Kj r (--1, 1) satisfies the relations 1 < xl - K 2 < X/6KI -- 2. Then there exists a solution A" R x Hlul(~, C) of rGL and a lk ~ S 1 such that lim A(t, x) -- UKI (x) t-----~ -- O0
and
lim A(t, x) -- e i~ UK2(x), t---~ e~
uniformly in x e R. The assumptions of this result imply tel ~ (1 / ~/2, 1) and x2 < 0. The solution A obtained in the theorem is not the one which is typically observed in computer experiments, when
The Ginzburg-Landau equation in its role as a modulation equation
791
is taken much larger than 27r/tCl. In most cases a solution starting near UK~ with tel > l / x / 3 converges to UKj for some xj 9 [0, 1/if3). But there is no proof of such a statement. There is a rather special property for rGL which generalizes the concept of "lap numbers" for scalar parabolic equations. We discuss this now for a bounded interval but it is clear that it can be transferred to the real line, if sufficient control over the behavior at infinity is obtained. Consider solutions A of rGL on (0, ~) and assume for the moment that A ( t , x ) =/=0 on (tl, t2) x (0,~) then the angle (k(t,x) in A = r e i~ is well defined up to multiples of 2zr. The key observation is that if 4~ has a local maximum at a point (to, x0) then the maximal value will decrease in time. Indeed, consider u(t, x) = Im[e-i4)(t~176 x)] which solves (4.9) and has a double zero at (to, x0). By the theory for scalar parabolic equations (cf. [22]) this zero will disappear, which means 4~(t, x) < 4~(to, xo) for t 9 (to, to + e) and Ix - xol < e with sufficiently small e. The analogous statement holds for minima of 4~. However, new minima and maxima can appear when the solution undergoes a so-called phase slip, which means that A(tl, x l) 9 C is 0. This occurs only at discrete points and typically O~A(tl,xl) =/=O. In such a situation 05 develops a jump of size o-zr with o- 9 {1, - 1 } before the phase slip and has a jump of size -or 7r directly after the phase slip. In the case of Dirichlet or Neumann boundary conditions for the interval (0, ~) it is shown in [22] that the torsion function O(t) = max qS(t, x) - min 4~(t, x) .r~(O,~) .r~(0.(~) is either constant along solutions or strictly decreasing, as long as no phase slips occur. As a consequence it can be shown that all solutions converge to a planar steady state (i.e., 4~ = const). For quasiperiodic boundary conditions we have the boundary condition 4~(t, g ) = 4~(t, 0) + ), + 2zrN(t) with N(t) 9 Z. Moreover, the above arguments show that the total variation Var(oS(t, . ) ) - s u p { ~ - ~ l d p ( t , x j ) - ~ ( t , j- 1
Xj_l)l" O<~xo <Xl < . . . < x n ~<~}
decreases along solutions of rGL even if phase slips occur. However, this does not imply that the function I4~(t, e) - q S ( t , 0)l = I~' + N(t)l decreases, in fact Theorem 4.1 gives examples where 7 + N(t) changes sign. More information on phase slips can be found in [27,51,22]. Another important solution class is formed by traveling waves and oscillating fronts which take the form A (t, x) = a (x, x - vt) where v 9 IR is the velocity of the front and a(x, z) converges to steady states for z --+ + o c ; lim a ( x , z) -- UK_ (x)
Z---+ - - ~
and
lim a ( x , z) --
~.---+ OC
e ioUK+ (x).
In the case that x+ = 0 one can use the ansatz a(x, z) = e iK-x f ( z ) and finds the ODE f " + (v + 2 i x _ ) f ' + (1 - x 2 ) f - [ f [ 2 f
--0.
792
A. Mielke
It is easily shown that for any v > 0 there is a unique solution (up to phase and z translation) /
satisfying f ( - o c ) -- V/1 - tc2 and f(cx~) - 0 . The general case is much more difficult and only partial results are known, see [50]. 4.2.3. D i f f u s i v e mixing. Here we are interested in solutions which behave like one stable pattern at x ~ + o e and like another one at x ~ - e e : A (t, x) - e i/~• UK+ (x) ~ 0
for x ~ + c e ,
(4.14)
where tc2 < 1/3. Clearly, if the initial condition satisfies these asymptotic boundary conditions, then the solution does for all times. Our concern is the dynamics of the system in the intermediate regime. The structure of (4.7) becomes clearer when using polar coordinates A = re i4~ giving rt = rxx + r (1 - r 2 - dp2),
dpt -- q~xx + 2 r x ~ x / r .
The underlying idea of diffusive repair and diffusive mixing is that for t --+ ec the derivatives rt and rxx tend to 0 in suitable norms, see [27,58]. In particular, this implies that asymptotically the amplitude r is slaved to the phase derivative q~x such that r 2 -- 1 - cb2 + O ( t - l / 4 ) . Hence, asymptotically the so-called phase diffusion equation ckt -- a ( ~ x ) ~ x x
with a ( s ) = (1 - 3s2)/(1 - s 2)
(4.15)
is relevant. The boundary conditions (4.14) imply ~ x ( t , x ) --+ x+ for x --+ -+-cx~.Hence, for r/(t, x) = ~ x ( t , x ) we obtain from (4.15) the problem rh -- (a(O)OX)x,
rl(t, x ) --+ x+
for x --+ l e e .
(4.16)
It is shown in [58, Theorem 3.1 ] that (4.16) has, for all x+, x_ E ( - 1/x/3, 1/x/3), a unique similarity solution r l ( t , x ) - " ~ ( x / x / 7 ) where ~ is monotone and attains its limits x+ and x_ faster than exponential. It satisfies the ODE [a(~'(~))~"(~)]'+ ~
(~)
for~ ~ IR.
(4.17)
With ,~(~) - x_~ + f ! ~ [ ~ ( s ) - x_] ds we define the time-dependent profile U(t,x)
-- V/1 - ~ ( x / x / r t ) 2 e i`/Tff(x/`/7).
(4.18)
The mathematical theory of diffusive mixing consists in showing that this profile is a good asymptotic approximation for the limiting behavior of suitable solutions A of rGL satisfying the boundary conditions (4.14). A first result was obtained in [20] for the case x+ ~ 0, the general case with x 2 < 1/3 is treated in [58] and provides the following result.
The Ginzburg-Landau equation in its role as a modulation equation
793
THEOREM 4.2. Let x+, x_ ~ (--1/x/3, 1/x/3) with x+ =/=x_. Then there exist to > 0 and s > 0 such thatforall Ao ~ H2ul(lR) satisfying IlA0-U(t0, .) IIH2(•) ~ s, the unique solution A (t) c H2ul(R) of rGL with A (0) -- Ao satisfies, for t --+ (x),
IlA(t)- ~(t0 + t,
IIIA(t)I-
and
I (,0 + ,, )111
9
Note that the assumptions use the nonweighted classical H2-norm while the results are stated in the uniformly local norm. This is due to the diffusive character which transports initially localized perturbations out to infinity. The second result in the theorem manifests the fact that the slaving of the amplitude r(t, x) -- lu(t, x)l occurs faster than the diffusive mixing which is mainly driven by the phase diffusion which has the decay rate 1/x/T. The above result says that the limit behavior of the solutions for t --+ cx~ is completely determined by the time-dependent profile U. From its explicit form we find the relation
I
-'/2)
f o r t ~ c~.
Here we again have to use the weighted norm, since a similar estimate in L2ul(IR) cannot hold due to the different limit behavior for x --+ oo of U ( t , x ) and eit~uK,. The values x, and 4}, are given via x, -- ~(0) and 4}, = N (0) ~: 0 and hence are well-defined functions of the limit values x+ and t<_. As a consequence we have uniform convergence on each bounded interval ( - x l, x l ). The results of Theorem 4.2 mean that the solutions A (t) converge to the circle S-
{e i~ UK, ~ L2ul(~.): }6 E ~}
of equilibria in the sense that dist~,(A(t,-), S) --+ 0 for t --+ oo. The solutions u(t) do not converge to a particular point in S but they slide along S with speed l~b,t-1/2 and thus never settle down completely. It is interesting to consider the flow of the local energy density
1
1
1
e ( t , x ) - - - ~ l v a ] 2 - ~lal2 + 4 ] a [ 4
1
]22
1
1
~ [[Vrl 2 + r 2 [V~b ] - 2r2 -Jr- -r44
associated with the Ginzburg-Landau functional (4.8). Here we have given the multidimensional case for later reference. As the total energy E(A(t, .)) of the solution is not defined, the energy density can be used to study the energy dissipation. Moreover, we may compare the relative energy
Erel(A(t))- fR[e(t,x)-
e(O,x)]dx,
which is finite as the boundary conditions are time-independent for all times. As a result we will find that this relative energy tends to -cxz like - x / t for t --+ oo.
A. Mielke
794
The local energy density satisfies a conservation law with a dissipative term: 0t e + div x f = - g ,
(4.19)
where the energy flux f ( t , x ) E R d and the dissipation rate g(t,x) >>.0 are given by
f (t, x) -- - Re[(AA + A - I A I 2 A ) V A ] = - ( A r - rlVr 2 4- r - r3)Vr - (r2Ar + 2 r V r . V r 1 6 2
g(t,x) -- 10,Zl 2 - - ] A A -k- A - I Z l 2 Z ] 2 = (Ar - rlVr 2 + r - r3) 2 +(rAr162
2.
Returning to the one-dimensional case of diffusive mixing we employ the asymptotics derived in Theorem 4.2 and (4.18) and obtain the expansions (1 - 3~2)2~, 2
e(t,x) = h('~) + O(1/t),
g(t,x) = t(1 _~'2)
+0(1/te)
where h(r/) - - ( 1 - 72)2/4 and ~" and its derivative are evaluated at ~ - - x / x / 7 . Using (4.17) we find d
--Erel(A(t)) -- ~ Ote(t,x)dx dt
-- - f.r
oR
,f
g(t, x) dx = - - 2
eR
dd~ h (~(~))d~ + 0 ( 1 / t 3 / 2 ) .
Defining r/oc (~) = x+ for -1-~ > 0 we obtain via integration by parts d dt E r e l ( A ( t ) ) - - B(K_, 2~/'[x+) + 0 ( 1 /t3/2) ,
B(x_,x+)-
f
with
- h(~(se))] d~.
Clearly B(x_, x+) is strictly positive for x_ # x+. The energy density in the middle region tends to h (x,) which always lies below the average of h (x_) and h (x+), since the function h is convex on the 0-interval ( - 1 / x/3, 1/ x/3). In a situation with h (x_) < h (x,) < h (x+) the energy flux has to transport energy from the region x >> 1 via the region x ~. 0 to the region with x << - 1 . The same result could be obtained in a simpler way (but with less insight) by considering the relative energy fR [e(t, x) - h (r/~c (x))] dx. The leading term of this integral is scaling invariant and gives -x/-iB(x_, x+).
The Ginzburg-Landau equation in its role as a modulation equation
795
4.3. Special solutions for the complex Ginzburg-Landau equation There is a very rich literature on different aspects of cGL, however, we focus here on solutions defined on R a, in particular traveling and rotating waves. In the one-dimensional case one can look for solutions having a fixed profile which rotates in phase with frequency and travels along the x-axis with speed v, namely A(t, x) = e i~ta(x - vt). This leads to the ODE (1 + iot)a" + va' + (p - i ~ ) a - b " l a 12a --0.
(4.20)
Because of the explicit algebraic structure of this equation there is a variety of explicit solutions in addition to the family of rotating plane waves discussed in Section 4.1. One of these families are the so-called Bekki-Nozakki solutions [12], which are surveyed and generalized in [144]. These solutions are heteroclinic connections between two rotating plane waves at infinity. However, the solution is not structurally stable, see [37]. This is in contrast to the pulse solutions considered in the next subsection. Since (4.20) is a four-dimensional ODE with phase invariance, it can be reduced to a three-dimensional ODE and studied in detail. The real variables p, ~ and v serve as parameters. Particularly the nearly integrable case (i.e., arg(p - i~) ~ argO" ~, arg(1 + iot) and v ~ 0) is studied in great detail, see, e.g., [70,38,36,81,37,79,3]. 4.3.1. One-dimensional pulse solutions. Pulses are considered mainly in the onedimensional case. The simplest pulses are rotating waves in time and decay exponentially for Ix l --+ ec. To make the pulse a stationary solution we write cGL in the form
OtA = (1 + iot) [0.~A - (1 + i01)2A + (1 + i02)(2 + i02)IAI2A].
(4.21)
If 01 and 02 are near zero, then this equation can be seen as a perturbation of both, the real GLe if ot ~ 0 and of the nonlinear Schr6dinger equation if Iotl >> 1. It was first observed in [ 139] that this equation has an explicit steady solution for 0 -- 02 -- 01, namely
Ho(x) -- ei• [coshx] -(l+i~
(4.22)
There are more general steady pulse solutions of (4.21). In particular, in applications of cGL to laser optics (cf. [ 103,80]) one is interested in multi-pulse solutions, i.e., solutions with ]A(.)I 2 having several well-distinguished maxima. Such solutions were found in cGL in [81] for the case 0 < [021 << 1 and 101 -021 << 1021. This work was generalized in [3] and then applied to the Poiseuille problem in [4] where 02 between 4 and 30 is relevant. The application to the Poiseuille problem involves the justification of the Ginzburg-Landau formalism by the help of the Kirchg~issner reduction, see Section 6.1. For the construction of multi-pulse solutions one analyzes the stationary problem. This is a four-dimensional ODE which is reversible because of the reflection symmetry x ~ - x of cGL and is SO(2) invariant. After factorization, the problem can be treated as a threedimensional problem which is singular at the origin. We fix 0 - 01 and treat ~ = 02 - 01 as a small bifurcation parameter: A"-
(1 + i0)2A + (1 + i(0 + e))(2 + i(0 + e))IAI2A = 0 .
(4.23)
796
A. Mielke
The explicit pulse (4.22) exists for e - 0 and it breaks apart for e -~ 0 if the pulse is nondegenerate with respect to the parameter e. The analysis shows that nonde~eneracy holds for all 0 6 R except for an at most countable set {0, • . . . . } with Oj --+ cx~. Numerical calculations give 01 ~ 8.032 and 02 ,~ 9.51, cf. [3]. Using appropriate local and global Poincar6 maps it is possible to study the flow in a tubular neighborhood of the primary pulse solution. One obtains existence of shift dynamics as well as reversible and nonreversible periodic orbits for all small e. It is clear that the existence of a homoclinic orbit means that the stable and the unstable manifolds of (A, A') -- 0 E C 2 ~ ]1~4 (both are two-dimensional) coincide, since homoclinic orbits occur as one-parameter families. Such a homoclinic solution is called n-pulse solution if it has the form I l l
A(x) ~ ~ m=
e iYm
Ho(x - Ym)
1
with the phases Fm 6 ~l and shifts Ym 6 ]~. Here we have Ym+l - - Ym >> 1 which then justifies the name 'multi-pulse'. We do not call these solutions n-homoclinic as they do follow n different homoclinic orbits rather than just one. Denote by E C ( - e 0 , e0) the set of all e such that (4.23) has a homoclinic orbit. Then we have E - - Un~__l E n where E n is the set of e for which (4.23) has an n-pulse solution. The results obtained in [81,3] can be summarized as follows" A
A
THEOREM 4.3. Assume that 0 ~ {0,-t-01,4-02 . . . . }, then the following holds. (a) Both sets, E 2 and E 3, consist o f two sequences converging monotonically to e -- 0 with rate e -zr/l~ one sequence from above and one from below. Moreover the two sequences separate each other. The associated homoclinic orbits are transversal again. (b) For all n the sets E n are infinite and contain e -- 0 in their closure. (c) l f n = pq with p, q ~ 2 then E q C closure(En). Considering 11o as a stationary solution of cGL in the form (4.21) we may ask about the stability of these solutions. Clearly, it can only be stable if the asymptotic state A = 0 is stable as well. Since the continuous spectrum of the linear operator A w-~ (1 + ic~)[A" ( 1 + i0) 2 A ] is given by {-l+2ot0+02-s
2•
+ l-O2)],
s ER},
we obtain the necessary condition for stability - 1 + 2or0 + 0 2 ~ 0. However, there are also discrete eigenvalues, in particular we obtain always two eigenvalues 0 due to translation and rotation invariance. In the real case ot = 0 ---0 with Ho(x) -- 1/(coshx) it is easy to see that there is also one unstable eigenvalue since the eigenvalue problem for A = a l -k- ia2 takes the decoupled form a'( - ( 1 - 6 H 2 ( x ) ) a l -- Xa,,
a~2' - ( 1 - 2 H 2 ( x ) ) a 2 -- Xa2.
These are two selfadjoint operators with real spectrum whose continuous part is (-cx~, - 1]. The second equation has the lowest eigenvalue 0 with eigenfunction Ho, however, the first
The Ginzburg-Landau equation in its role as a modulation equation
797
equation has the second eigenvalue 0 with eigenfunction H6 (which has one zero). Hence, by Sturm's nodal theory for eigenfunctions the first eigenvalue for the second equation is strictly positive. For small c~ and 0 the pulse solution remains unstable. In fact, the numerical calculations in [ 16,15] show that this positive real eigenvalue persists in a large region of the (oe, 0)-plane. Moreover, it can be shown analytically that for - 1 + 2oe0 4- 02 = 0 the eigenvalue 0 is in fact threefold. It is easy to check, that (4.21) with c~ = (1 - 0 2 ) / ( 2 0 ) has a family of rotating pulse solutions: 1
V• (t, x) -- --e i6t [cosh(vx)]-(l+i0) 9/
with gr - (g2 _ 1) (1 + 02) 2 2O
For V = 1 we find the standing pulse and the derivative with respect to V at ?, - 1 gives that (1 + 02) 2 t
-
0
V1 (x) - W ( x )
with W ( x ) - [1 + (1 + i O ) x t a n h x ] V l ( x )
solves the linear problem arising from (4.21) by linearizing around Vl. As Vl is an eigenfunction to the eigenvalue 0 we have found the generalized eigenfunction W. The numerics in [16,15] suggests that this additional real eigenvalue passes through 0 from left to right upon leaving the region - 1 + 2oe0 4- 02 ~< 0. Further off this region the new real unstable eigenvalue meets the other unstable eigenvalue to form a complex pair which then may move back into the left complex halfplane. However, the pulses remain unstable due to the continuous spectrum. By now it is an open problem to show that all these pulses are unstable. Stable pulses are known to exist only in the generalized GLe (2.22), see [79,80,82]. 4.3.2. Spiral waves. Spiral waves are solutions on R 2 whose spatial pattern (e.g., Re A (t, .)) looks like a spiral, while the whole pattern rotates with a fixed angular velocity 7 around a fixed center x,. We insert the ansatz A ( t , x ) -- B ( R ( g t ) [ x
- x,])
with R ( s ) - (_c~
s
cosSinS)s
into the isotropic two-dimensional cGL OtA -- (1 + iot)AA + XO, l p A - ~ ' I A I 2 A and find the elliptic PDE (1 + ioe)AvB + v ( V , . B , J y ) + X o . l p B - ~ [ B I 2 B = 0 ,
(4.24)
where y = R ( g t ) [ x - x,] and J - (0, o ' ) " The phase invariance of cGL allows us to construct spiral wave solutions using ODE methods by searching B in the form B ( y ) - b(lyl)e i''~~
where q)(y) = arg(yl + iy2).
798
A. Mielke
Such solutions are called n-armed spirals since the real part of A has n positive and n negative regions when we circle around the center of the spiral. Problem (4.24) reduces to the nonautonomous ODE
I
1b~- n2 b ] 4- ()~o,ip 4- i v n ) b
(1 4- ic~) b" 4- s
7
-FlblZb --0
s-
lyl E [ 0 , ~ ) .
This construction is a straightforward generalization of the Ginzburg-Landau vortices considered in Section 4.2.1, however, now the coefficients and b(s) are complex, and we have the rotation velocity ~, as an additional parameter. After division by 1 4- iot we arrive at the system 2
lb, n b" 4- -s - -s-~ b 4- ~ b - ~]b[Zb - 0
(4.25)
where )~ - ()~o.ip + i y n ) / 1 + ic~ and 7 - F / ( 1 + ic~). Note that it is always possible to make )~ real by choosing the appropriate rotation speed y. However, U is fixed due to ot and ~" which can not be varied through the bifurcation parameters. The case of real ~. and 7 in (4.25) is well understood. In fact, for )~, 7 > 0 we obtain exactly rescaled versions of the Ginzburg-Landau vortices, see Section 4.2.1. For the case ~., ~' < 0 there are infinitely many solutions for each n which decay exponentially for s -+ cx) and have k E No simple zeros, see [ 120]. By perturbation arguments it is then possible to prove that all these solutions persist for small imaginary part of ~', in the sense that there is an appropriate rotation speed y such that (4.25) still has a solution, see [ 120,121 ]. Existence results for spiral waves in general reaction-diffusion systems (which do not have the phase invariance of cGL) are given in [ 121 ]. Stability questions and further bifurcations of spiral waves are treated in [ 116].
4.4. Global existence f o r the complex Ginzburg-Landau equation The local existence of solutions is classical on either bounded domains or unbounded domains when suitable boundary conditions are added. Here we discuss the main ideas typical for GLe and refer to the literature for the standard technical details of parabolic systems of PDEs, see [44,9,41,43,61,88,98,101]. 4.4.1. Bounded periodic domain. We consider the domain $2 - 12e - ( - e l , el) • ... • (--gd, gd), which has volume 2d~l ... ~d, and complement cGL d
OtA - Z ( 1
2 + iotj)OjA + p A - (1 + ifl)lAI 2
on 12e
(4.26)
j=l
by periodic boundary conditions. This will allow us to do integrations by parts without worrying about boundary terms.
The Ginzburg-Landau equation in its role as a modulation equation
799
It is well known that smooth solutions exist locally in time and moreover, if a solution stays bounded in L p (I-2) for some p > d, then it exists globally and is analytic in space and time, see, e.g., [41,88]. In certain cases even distribution-valued initial conditions can be allowed [89]. Thus, global existence is established if we find a priori bounds of an appropriate L P-norm. The standard energy estimate reads
[A[P-ZA- OtA dx
[A[ p dx - Re P
=-
d ~ QjP,l+io~(A) -Jrj=l
~' (A)--Reb fo
where Qj.b
[ plAIp - IAIP+2]dx
(4.27)
Oj([A[p-2-A)ajAdx.
Simple algebraic manipulations (cf. [98,101 ]) show that
QjPb(A) >~O,
if b E C satisfies Reb >~ 2 ~p/ p- - 2 1 [Imb[.
(4.28)
(In fact, under this condition the integrand RebOj(IAIP-2A)Oj A >~ 0 of QjP,b is pointwise nonnegative.) From this we derive a simple L p estimate if the linear dispersion parameters are not too large. Using
az
-
bz I+Y
~ ya([a/b] 1/• - z)
for all a, b, z, 9/> O,
we obtain the following result which implies that the LP-norm cannot blow up.
2N+l LEMMA 4.4. Assume 2 <~ p <~ ~ - l
d
2p
/2
--dt Ia(t)ll~'"~
for j = 1 . . . . . d, then we have the estimate
p
vol(s2)- IIAU)II,,,)
(4.29)
This settles the global existence questions in dimensions d = 1 and 2 since it is always possible to choose p > 2 >~ d in the above lemma. For d ~> 3 one obtains the restrictions [O/j[ < 2x/d - 1/(d - 2). There is another method involving cancellation properties in the nonlinear energy fs2 [~-[VA[ 2 + l[A[ 4] dx. This works especially well in the defocusing case (sign(O/jfl) > 0) but also in some limited regime of the focusing case. These estimates are also very useful in dimensions 1 and 2, when one wants to derive estimates on the Lee-norm of solutions, see Section 4.4.3. Here we give the main idea and refer to [88,98,101] for a more detailed
A. Mielke
800
analysis leading to a wider range of allowed dispersion parameters. The first derivatives are estimated via ld
d
2 ~dt [[OJAII2L2 -- -- Z
--
[[OjOIAIIeL2 4- PIIOjAII2L 2 -- Qj,14
+ifi
(A) ,
(4.30)
/=1
where we have used Re(14- iott) fs202~ 02A dx = II0j 01All 2L 2 which follows by integration by parts. It is an immediate consequence of this and the above Le-estimate that the HI-norm of A (t,-) cannot blow up if the nonlinear dispersion parameter fl satisfies I/~1 ~< 43, since 04, l+ifl (A) ~> 0 by (4.28) In the defocusing case we assume, without loss of generality, that cU, 13 ~> 0. By the above we may further assume 13 ~> x/~. We let F(A) = E d l TIIOjAIIL2 ~j 243j "-- (14- olj )/fl, and by adding (4.30) and (4.27) with p = 4 we arrive at
88
with
d F(A) <. - E ~jllOj OzAII2L2 4- P 6jlIOjAII2L2 4- IIAII44 j,l=l j--I
--
dt
d _
L6 -- E Q 4j,zj (A) j=l
IIAII 6
where zj -- 1 - iotj 4- 3j(14- ifl). The choice of 3j guarantees Q4J,Z,j (A) ~> 0. With 0 < 3, ~< 3j ~< 3" and 2 ~< IIAIIL2 w e obtain
IIOjAII2L IIOj2AIIL2
d
d
- F(A) <. - 3 , dt
d
EIIo2AII 2L2 + 2p * 1
4- 2plIAII 4L4
-
-
l
II
A
IIL2 IIA liE2
pF(A) -IIAII 6L6
<~ - p F ( A ) 4- dp23"2 3,
L2 --}-2plIAII44 IIAII2
(4.31) IIAII6L6
<. - p F ( A ) + p3Cvol(.C2), where C = 44-4d3/2(~*/3,) 3. Again we conclude that F(A(t, .)) cannot blow up and since F (A) controls the norm in L 4 and H l we obtain global existence also for d - 3. We summarize this result in the following statement and refer to [88,61,101] where more dispersion parameters are allowed and the generalized nonlinearity - ( 1 -4- ifl)IA 12~A is considered. THEOREM 4.5. We consider dimensions d 6 {1, 2, 3 } where in the case d - 3 additionally one of the following three conditions holds:
(i) Ir ~< J~, (ii) maxj=l ..... d Icql < ~/8,
The Ginzburg-Landau equation in its role as a modulation equation
801
(iii) minj=l .....j olj~ ~ O. Then the solutions A (t,-) of (4.26) exist globally in time. The above energy estimates also provide the existence of an absorbing ball, namely
lim sup]
A(t) I L; ~
pl/2[vol(f2)]
lim sup
lip ,
it----+ ~
IVAr IIzL=~
pC [vol(f2)]
1/2 ,
t-----> ~
(4.32) which follows from (4.29) and (4.31), respectively. 4.4.2. Unbounded domain. To generalize the existence theory to unbounded domains we follow an approach initiated in [26,25] and developed further in [105,98,101,54] using weighted norms. Alternative approaches are given in [7,61 ]. Here we restrict ourselves to the case s -- R d and refer to the given references for more general unbounded domains. The main idea is to obtain a priori bounds in the uniformly local norms LuPl(~d) via weighted energy estimates. We show this in the simplest case of the LZul-bound from which it will become clear that employing the weight will introduce only a few lower order terms in the energy estimates. Thus, the sign properties of the leading terms are not changed. Moreover, the volume of the domain will no longer be relevant but only the integral of the weight over the domain. Recall the weight w(x) = e -Ixl which satisfies the two important properties IVw(x)] ~< Cl w(x) and fRj w(x) dx <~Cw. With this and ~" = max{I 1 + iot[: j = 1. . . . . d} we have
2 dt -
J
wIA[ 2 dx -- Re
- ~,, wlval 2 +
,t
A OtA dx
ReA~-~(1 +
io~j)Ojw OjA dx
1
+ ~ j w[plAI 2 -
Ial 4] d~
<~~~t { - wlVAI2 +~IAIIVwlIVAI}dx + ~,l w[plA[2- IAI4] d~ <~~,~ w[C31AI2 -
IAI 4] dx
<~~jwC3(C3-lAl2)dx <~ C3[C3Cu,-~,twlAl2dx], where C3 = (ffC1/2) 2 + p. This implies HA(t)] 2 L~, ~< e_2C~t " ][A(O)][2L~,
-k- C3Cu,
802
A. Mielke
Now we can use the fact that the same estimate holds, if the weight is replaced by the translated weight w(. - y). Taking the supremum over y 6 R d we find the estimate
IIA(t)II' u, e-C ' IIA(0) Ilgwu,§ Similarly the estimates for IIA IILuP~ and []OjA][L2u~ c a n be obtained by generalizing the estimates with the constant weight given in the previous subsection. The following result is a special case of [98, Theorem 4.3]. THEOREM 4.6. Let 1-2 C R d be a general domain and impose either Dirichlet or Neumann boundary conditions for the solutions of GLe. Under the same assumptions as in Theorem 4.5 the GLe (4.26) defines a global semiflow on LuPl(g2)for all p > d. Moreover, this semiflow has an absorbing ball. The use of weighted spaces has several positive effects. First, we are able to capture phenomena which are typical in the theory of unbounded domains, namely spatially quasiperiodic solutions, fronts, and pulses. Second, as explained above, the energy estimates with constant weight 1 can easily be generalized to the weighted case. Third, we are able to find absorbing balls for GLe in LuP(ItU) whereas in LP(~ d) we only have a global semiflow where solutions, however, may grow to infinity. This is already seen in rGL (4.7) with d -- 1, since there are solutions A(t, x) .~ b ( ] x ] - 2t) for t >> 1 where b is the traveling front profile solving b" + 2b' + b - b 3 -- 0 with b(oc) -- 0 and b ( - o o ) = 1. The fourth advantage of weighted spaces is that they even allow for improvements on the estimates for bounded domains. We come back to this point in the next subsection. However, before this we state a result on the existence of the global attractor. For bounded domains, e.g., in the periodic case this theory is classical (cf. [64,8,141 ]). In Section 5 we present a theory which is directly applicable to GLe even for unbounded domains. It uses the weighted norm LP(g2). Together with the smoothing properties of parabolic systems 1,p the weighted norm generates compactness; viz., a bounded set in Wul (s is compact in LuP,(~). The following result is proved in [105,98] and is a special case of Theorem 5.1 below. THEOREM 4.7. Under the assumptions of Theorem 4.6 the semiflow of the GLe has a p global attractor . A - . A s~ c Lul(~) with the following properties: (i) .,4 is nonempty, bounded and closed in LuPl(.f2); and .A is compact in LP (g2). (ii) .A is invariant under the semiflow (Gt('))t~o of GLe, i.e., Gt(.A) - .A for all t >>.O. ) in the semidistance induced by the norm (iii) .A attracts all bounded sets in Lul($-2 P ][ 9 [[Lp, (see (5.2)). (iv) .A is invariant under all symmetries of GLe and the domain, in particular under the phase invariance A (.) ~ e iYA (.). If ~ -- R cl then .A - .ARJ is translationally invariant. The conditions (i) to (iii) are the definition of an (LuPl(g2), Lwp (s
803
The Ginzburg-Landau equation in its role as a modulation equation
4.4.3. Bounds, length scales and turbulence. The cGL serves as the simplest model to study turbulence, see [9,92,91,17]. The notions ' soft' and 'hard turbulence' are used to distinguish two different regimes of temporal chaos. In the case of soft turbulence the spatial behavior of the solutions remains rather simple whereas in the case of hard turbulence the solutions display rapid spatial changes like spikes and vortices which have much smaller length scales than the other typical structures. Often soft turbulence is also called 'weak' turbulence and hard turbulence is called 'fully developed' turbulence. A connection between the smallest length scale in turbulence and bounds in the L ~ norm for solutions is derived in [ 10] for the cGL and the Navier-Stokes equations. Hence, it is desirable to derive upper bounds for the solutions of cGL. Turbulence is related to the flow on the attractor, so the diameter of the attractor .A has to be estimated in dependence of the dispersion parameters: "D(p, Oil . . . . . Otd, 3, .(2) -- sup{ IIAIIL~" A 9 .As? (p, oel . . . . . ota, 3)}. A measure of turbulence is now obtained by the size of D. As follows from (4.29) with p = 2 we have the simple bound IIAIIL2 ~< [pvol(~)] 1/2 for all A 9 A s2. Hence, if 7? is significantly larger than p l/2 then the solutions must display spatially localized behavior (spikes), and we might expect hard turbulence. In the case of a finite domain with periodic boundary conditions the estimates (4.32) provide exponents V, 3 > 0 such that
7~)(p, Oil . . . . . O/d, fl, ,.Q) ~ (~(Otl . . . . . Otd, 3)p• [vol(K2)] ~. To this end the Gagliardo-Nirenberg estimate Ilu[l~ ~< Cllull ~w~i, Ilu IIL,,,~ ~-o with suitable 0 is employed. Additionally, the scaling (see (2.15)) (~', ~', ~', ,7) - (r~, r2r, r - 2 p , r -1 A) with vol(Y2) - r'lvol(s gives 1 - 2 ) / + d3 - 0 . Since ~ ) 0 we find g ~> 1/2. Indeed, when the weight w = 1 is employed, one often obtains g > 1/2, see [9,10]. The periodic solutions form a special subclass in Llu'l(IRa). Our a priori bounds in this space and the scaling invariance of cGL on all of R a immediately imply
D ( p , oil . . . . . Otd, 3, Nd) __ p l/2~(ot I . . . . . O/d, 3).
(4.33)
In [101] the constants C(ot . . . . . or, 3) (isotropic case) were estimated leading to the upper bounds
_
[ Co[(1 + ot2)1/4(1 + fl2)-1/12
d -- |" C(ot, fl) ~ /
-Jr- (1 Jr-fl2)l/6]
d - 2: C(ot, or, fl) ~<
for 1 + c~/3 + ~-1o~-/31 ~> 0, else,
f12 1/4
C0(1 + o~2)1/2(1 +
)
C0(1 -+- Or2) 1/6 x (1 + ot2 +/32) Co[(1 + 0/2) 2
x (1 + 32)] I~l+~/8
for 1 + ~/3 + ~lot - / 3 ] / > 0, else.
A. Mielke
804
For d = 1 the upper bound is polynomial in I~1 and I~1 since we are still in the subcritical regime. For d = 2 the upper bound in the focusing regime is exponential in I~1. It is interesting to note that, using perturbation arguments for the nonlinear Schr6dinger equation (nlS), there is certain evidence for an exponential lower bound. For the focusing nlS with d = 2 (that is (2.17) with oil = or2, - f l --+ cx~) one knows that solutions blow up in finite time. In [56] the nlS was perturbed by adding small dissipation and small linear forcing in the form
OtA - i ( A A + ylAI2A) + elA + e2AA - e31AI2A.
(4.34)
Using formal perturbation arguments along the blow-up solutions they establish that for small ej > 0 the solution behaves similarly to the blow-up solutions until the amplitude reaches a value approximately given by
IIAII~ ~
c0(62 -+-
263)-1/6ecl/(e2+2e3),
for some positive constants co and c l. After that value is reached the dissipation wins over the focusing effect and the solution defocuses. See also [91,146] for numerical simulations showing formation and decay of such spikes. The numerical evidence suggests that these spike-forming solutions lie on the unstable manifold of the spatially homogeneous solution A(t, x) -- (el/e3) Hence, these spike forming solutions lie in the global attractor A s2 . Inserting our definitions el -- p, e2 - 1/c~ and e3 - 1/fl this suggests a lower bound for D of the form
l/2eielt/e3.
C~
l/2 (ot + ]fl]~ ) l/6eClOelfll/(ot+lfll)
Recent numerical results in [ 147] support the exponential lower bound. In [11] it is shown under suitable assumptions on y, d and ej that the solutions Ae of (4.34) converge for e --+ 0 to solutions A0 of the nonlinear Schr6dinger equation on a suitable time interval. Another way to measure the chaotic or turbulent behavior of cGL is obtained through the dimension of its attractor r rather than its diameter. Of course, most estimates for the dimension also depend on good estimates on the size of the solutions. For bounded domains the finite-dimensionality of the attractor was proved in [60]. There it was shown that the dimension of the attractor is proportional to the volume of the domain when all other parameters are kept fixed, namely dimH(.A ~) ~ C(p, ~, fl)vol(S2). Another approach to the dimension of attractors on large and unbounded domains is given in [28,29,149]. There cGL was studied directly on R d. For e, s > 0 one defines N(e, ~) to be the minimal number of e-balls in L ~ ( ( 0 , s which are needed to cover the
The Ginzburg-Landau equation in its role as a modulation equation
805
set {Al(0.e/" A e .ARd}. Then one shows that the limit H ( e ) -- lime__,~c s -d logN(e, g) exists. Finally one can show that the following dimension per unit volume is finite: log H ( e ) dim.u.v(.A ) "Rd " - lim sup s~o+ l o g ( l / e ) It is interesting to note that this new approach does not use any volume forms in lower dimensions and thus does not need any Hilbert space structure. It is based on Kolmogorov's e-entropy (topological entropy), which is estimated by analyzing the influence of the initial conditions in one place on the solution at later times at distant places. One such estimate is given in (5.3).
5. Attractors for large and unbounded domains 5.1. Existence of attractors For this section we generalize our point of view and consider general parabolic PDEs ut = div ( D V u ) + f (u)
w i t h x e X 2 C I R d, u(t, x) e IR'",
(5.1)
on a large domain s C R d where 'large' means {x e ]]~d: iX I < ~} C ~ for some s >> 1. This also includes general unbounded domains. For a recent survey on the general attractor theory including attractors for parabolic systems on bounded domains we refer to [ 115]. Throughout we assume that the diffusion matrix D, the nonlinearity f , the boundary 0s and the boundary conditions 'bc' are such that we have global existence of P smooth solutions defining a global semiflow ,9~ (t, .) on Lul(s for a suitable p e (d, oc). Moreover, we assume that there exists a radius R which is independent of s such that Ilullp,ul ~< R} is an absorbing set for the semigroup (see [25,98,61, Babs -- {u E L u1(s p 101 ] for a construction of such bounds). Under these assumptions we have the following existence result for the global attractor ,Abe. The case of bounded s is classical [8,64,141] and that of unbounded Y2 is treated in [7,105,55,98]. For the formulation we recall the definition (4.2) of the weighted norm ]] " ][L~, a n d the associated weighted LP-space L~,(s THEOREM 5.1. Under the above assumptions the semiflow ,S'b~(t, .) has a global (LuPl(s L p, ($-2))-attractor AbF2c with the following properties: (a) it is flow invariant, i.e., $ ~ (t, ,Ab~) -- ,Ab~f o r all t ~ O, P (b) it is nonempty, bounded and closed in Lul(~2), and compact in L~,(S2), P P (c) it attracts bounded sets B in Lul(~) in the Lu,-distance, i.e., diStL~; (Sb~ (t, B), A ~ ) ---> 0 f o r t -+ oo where diStL~ (F, G ) -
sup{inf{ Ilu - VllLJ~" v e G}" u e F}.
(5.2)
A. Mielke
806
The important feature here is the usage of two different norms. The weighted norm localizes the distance measure and thus restores compactness whenever we are on an unbounded
l.p
domain, viz., bounded sets in Wul (12) are precompact in L p (S2). For 12 -- R d it is shown in [7,28] that the attractor ,A•d is typically infinite dimensional. There are other cases of attractors for reaction-diffusion equations on unbounded domains or for the Navier-Stokes equation on unbounded cylinders which are different from the theory above. There the nonlinearity is such that the solutions converge exponentially to u -- 0 if no external forcing is added. Now the system is forced by a function g (x) which decays exponentially for Ix l ~ ~ . Then it is shown that the semiflow has a compact attractor of finite Hausdorff dimension. The functions in the attractor decay all exponentially for Ixl ~ e~, see, e.g., [1,7,54] and the references therein.
5.2. Comparison of attractors There are two different cases of comparing attractors of PDEs. Either we study the same PDE on two different domains, which is done in this section, or we may study two different PDEs on the same domain. The latter point of view will be addressed in Section 6.5 where the attractor of GLe is compared with that of the full problem. If the same PDE is posed on two different domains, we want to compare the functions in the attractors on the intersection of these domains. To this end we simply embed LuPl(Y2) into LuPl(II~d) by extending all functions with 0 outside of S-2. Clearly this embedding preserves the norm. The basic comparison result between the flows on different domains or with different boundary conditions can be roughly formulated as follows, see [98,28] for more precise statements. Take two domains ff2j C ~d and associated boundary conditions (bcj) such
s/ Sb~ (t, .) are well-defined. Furthermore assume {x 6 IRd" Ixl < } C S'2j for j = 1, 2. Then, for each R > 0, there exists a constant C > 1 which is independent of S21 and S-22 such that that the semiflows
IIs2 (..)-
c. ~< c e C t ( l l U l -
u2llL~, + [s
'/p)
(5.3)
for all uj E LuPl(~Qj) with Iluj I1~ ~ R. Here the second term on the right-hand side contains all the influence of the different domains and boundary conditions whereas the first term is a standard Lipschitz condition. In particular, if U l(X) = u2 (x) for Ix l ~< r < ~ then the first term is less than C[rd-le-r]l/P which provides a good control over the difference of the solutions over even smaller balls. An L ~ - v e r s i o n of 5.3 is provided in [102]. Using (5.3) and certain variants of it, it is not difficult to show the following upper semicontinuity results, see [98]. THEOREM 5.2. Under the above assumptions for each s > 0 there exists an s > 0 such that for all domains 1-2 C R d with {x: Ix] < g} C 12 and all bc E {Dir, Neu, Per} we have distL~ ' (Abnc, A IU) ~< s.
The Ginzburg-Landau equation in its role as a modulation equation
807
The remaining open problem is under what conditions we also have lower semicontinuity of the attractor, i.e., diStL(;. (A Rd , A~c) ~< e for suitably large domains s In the next subsection we give a simple example showing that lower semicontinuity is in general false. In our example the limit of A(bc e'e) for ~ --+ oc exists for each bc E {Dir, Neu, Per}, but it depends on bc and is strictly contained in .AR. ,/4R d is a problem of interchanging the limit The problem of convergence of AS2,, ~bc,, to t --+ oc in the definition of the attractors with the limit 12,, ~ R d. The interchange of such limits would only be possible, if the attraction rates for the attractors As2,, ~bc,, are uniform in n" however, this is in general not the case. In [65] a generalized limit attractor was introduced which allows for double limits where t,, --+ oc simultaneously with the parameter e,, --+ 0. In our situation this leads to the following definition. Consider a sequence (bc,, I2,),er~ of boundary conditions and domains which approach IRd in the sense that {x: [x l < g,, } C I2,, with ~,, --+ oo. Now set A* -- {u E LuPl(IRd)" 3t,, with t,, ~
oo 3u,, E LuPl(s2n)
with sup Ilu,, IIp.u~ < oo such that n EN
IIs ~, b(,,,, cu,,), - , u II ,, (R,, -+ 0 for n -+ oo,|. The motivation to consider A* rather than w-lim ~as2,, is that for practical purposes (e.g., bcn in experiments or numerics) we always have to consider finite time and b o u n d e d domains. Thus, one should neither prescribe the limit t ~ oo before n ~ oo nor the opposite, see [65] for a further discussion. THEOREM 5.3. For all s e q u e n c e s (bc,,, K2,,),,EH as a b o v e we h a v e fit* -- A Rd. PROOF. We first show A Rd C A*. To this end consider u E A Rd . As the sequence ~,, --+ oo for n --+ oo is given as above, we can choose a sequence t,, with t,, --+ oo such that the right-hand side in (5.3) tends to 0 if we use t = t,,, ~ = g,, and u l = u2. By the invariance of A S there exist v,, E A R,I C L/~I(R d) such that S R'~ (t,,, v,,) - u. Defining u,, E Llu'l(S-2,,) via u,, = v, Is2,, and using (5.3) we obtain the desired result u --
w-lim Ss2" (t,,, !l ~ O0 bCn
U I1
)
since IIv,, - u,, IIz~,~, can also be estimated by C [ g . d - l e - g " ] l/d. For the opposite inclusion we note that A* is contained in Babs and hence it is a b o u n d e d set in L~'I(IRJ) 9 If we show that A* is also negatively invariant (i.e., A* c S R'' (t, A * ) for all t > 0), then we have the desired inclusion A* c A R' . Let u E A* and t > 0, then we have to find fi" E A* such that u - S R 'l (t, fi'). Take the sequence (t,,, u,,) with u -- w-lim v,, / / ----->OQ'
where
1)n
-- SS2" bcn (tn ' u,,)
808
A. Mielke
according to the definition of .,4". The sequence V'n = S s2''bc,'(tn - t, Un) is precompact in L p (IKd) and, after taking a subsequence if necessary, we have ~ ' - w - l i m n ~ s2" (t, v__) and (5.3) we conclude Vn ~ ,5'bcn
II " - s~' (,. ~)II.c
Vn. With
~< II. - ~" Vb~...(, v,.)I1.~" + II SO"( be.. , . "%') -- S~' ('. ~)II L~ ~< llu -- ",. II. ~; + Ce~'[ll"';.-
for n --+ c~ 9This gives the desired result u - ,SRJ (t, ~').
~'11. ~ + [e,; 'e-~"] '/'] > 0 U]
5.3. An example with different limit attractors We consider the scalar parabolic equation
ut - Uxx + f (u)
with f (u) = - u ( 1
-u) 2
(5.4)
for x e ( - g , ~) where g > 0. The essential feature of the nonlinearity f is that the ODE fi = f ( u ) has exactly two equilibria, namely u = 0 and u = 1. While u - - 0 is a stable equilibrium, the second equilibrium u = 1 is degenerate and semistable. If ~ < c~ we take one of the following boundary conditions: (Dir): u ( - ~ ) = u (e) = 0, (Neu): Ux(-~) = Ux(~), (Per): u ( - g ) = u(g) and U x ( - g ) = Ux(g). The initial value problem is well-posed and we have a global semiflow on L2ul((-g, ~)) which we denote by (S~c(t, "))t~>0. The associated attractors are denoted by .A~c. More general boundary conditions could be allowed, but this would complicate the notations and the analysis considerably 9 For ~ < c<~ we can study the attractor explicitly as the flow is a gradient flow with respect to the energy
E ( u ) -- F F(u)-
e
[ U2x / 2 - F ( u ) ] dx
fo ''
f(v)dv--(6u
with 2 - 8u 3 + 3u4)/12.
The standard theory for attractors for flows with a Liapunov function (see [64,141,8]) tells us that the attractor consists exactly of the equilibria and their heteroclinic connections. Denote by ge e the set of equilibria of the semiflow Sbec, then
g'~Dir--
{0}
and
g'~Neu-- El~er--
{0,
1}.
We obtain the following result for the global attractors.
The Ginzburg-Landau equation in its role as a modulation equation
809
THEOREM 5.4. For all 0 < ~ < oc we have A~i r _ - {0} a n d ANe u -- Aeper_-- A , where A , = {u c L P ( ( - ~ , ~)): 30 ~ [0, 1]: u ~ 0}. For (Dir) the result is clear since the only equilibrium u - - 0 is exponentially stable. For (Neu) and (Per) we use the fact that the linearization at u -- 1 has the eigenvalues - ( k r r / g ) 2, k 6 Z. There is a one-dimensional center manifold associated to the eigenvalue 0, which, in fact, is given by the x-independent functions. Thus, the only solution connecting u -- 1 with u -----0 is the spatially constant solution u ( t , x ) - - O ( t ) , where 0 = f ( O ) . From Theorem 5.4 we immediately obtain the limits w-limA~i r - { 0 } , ~-+~c
w-limA~e u - w-lim A~e r - A , . ~-+oc
~-+~c
(5.5)
Here w - l i m e ~ A(g) -- B means distL~' (A(g), B) + diStL_~ (B, A(g)) --+ 0 for g --+ oc. Finally we show that the attractor A ~ as defined above is larger than A , . This establishes the example where the limits of the attractors exist, depend on the boundary conditions and are strictly smaller than the limit attractor. Clearly we have A , c A ~ . However, there are more solutions, for instance traveling waves of the form u(t, x ) --z ( x - c t ) with z satisfying the ODE z" + cz' + f ( z ) = 0 . If c = 1/~/-2 then we have the explicit solution z ( x ) - e X / ' / 5 / ( 1 + eX/'/5). Moreover, for each c >~ 1/~/2 there is a unique solution Zc which is monotone and satisfies z~.(0) = 1/2, z c ( - e c ) = 0 and Zc(eC) = 1. For Icl < 1 / v ~ there are no nonconstant bounded solutions and for c ~< - 1 / v / 2 we have the solution z c ( x ) = z - c ( - x ) . This shows that the limit attractor A ~ contains at least the two-dimensional set
: A , u { z c ( . - y): Icl ~ c,, y ~ R}. In the space L~, (R) this set forms a closed subset which is homeomorphic to the closed unit disk since Iz~.(x)l ~< K / I c l implies that z,.(. - cy) converges for c --+ ec to a constant function which depends smoothly and monotonically on y. Together we have shown w-lim Aeir ~ w-lim A~e u -- w - l i m j4~e r ~ ~ C J4Cx~. We conjecture that in this special situation we have/3 = .A~c. A similar conjecture for the nonnegative solutions of the Fisher-KPP equation u t = ux.r at- u - u 2 does not hold. There, the infinite-dimensional unstable manifold of u -= 0 is intersected with the cone of positive functions. In [66] it is shown that the associated attractor is infinite dimensional.
6. Justification of the G i n z b u r g - L a n d a u formalism
Finally we treat the question how good the formally obtained Ginzburg-Landau approximation really is. We recall the full problem Otu = L ( l z , Ox)u + N ( # , Ox, u)
(6.1)
810
A. Mielke
from (2.1) where u(t, .) :S-2 --+ R m is the unknown function on the unbounded cylindrical domain 12 -- ~a x r with r being a bounded cross-section. We will denote the solutions by u(t) = $~ (uo) where (S~)t>~o is a (local) semiflow. With/z -- pe 2 the modulation ansatz takes roughly the form u(t,
x)
-- UA
(t, x) -~- O(E2) where
UA (t, x) -- eA(e2t, e(x - Cgrt))ei(~176
+ c.c.
(6.2)
Here ico~0,k0 = L(0, ik0)~0,k0 and A(r, ~) solves the associated GLe
OrA -- div~ (AoV~A) + P~,o, 1A -- ~"IAI2A.
(6.3)
The solutions of the GLe are denoted by A(r) = ~r(A0) where (~r)r~>0 is the (local) semiflow. The questions we want to ask are the following. (1) What informations do we obtain for the full problem when studying the GLe? (2) Do solutions A of the GLe generate, via the modulation ansatz UA, good approximations of solutions of the full problem? (3) Are the solutions obtained from the Ginzburg-Landau formalism the typical ones which are seen in experiments? (4) Do special solution classes for the GLe correspond to similar solution classes for the full problem? If yes, can we deduce stability information for the full problem? (5) Are inertial manifolds or the attractor of GLe related to those of the full problem? Here we are, of course, interested in rigorous mathematical estimates. Our aim is to show existence of solutions which can be described by the GLe. Moreover, we want to emphasize that the Ginzburg-Landau formalism allows us to prove global existence for certain solutions of the full problem, e.g., for the three-dimensional Navier-Stokes equations, see below. We insist that it is not enough for the justification procedure to take nice solutions of GLe, insert them in the modulation ansatz and then show that the error in the full problem is small. Without further assumptions this does not say that the full problem has a solution which really behaves like this. A warning example is given in [ 126] where it is shown that the Newell-Whitehead equation (2.23) is a formal modulation equation which does not correctly predict the dynamics of the full problem.
6.1. Kirchgiissner reduction, spatial center-manifold theory The first mathematically satisfactory justifications of the GLe were given for the spatially one-dimensional case and were restricted to either time-independent solutions or timeperiodic solutions for the case )~(/z, k) 6/t~ and the case )~(0, k0) = ico # 0, respectively.
The Ginzburg-Landau equation in its role as a modulation equation
811
The main idea is to consider the full problem (6.1) as a spatial dynamical system where x 6 IR plays the role of time. Thus, we solve for the derivatives in x-direction and write the system as d - - v = K ( # , O~., Ot)v + F ( # , v) dx
(6.4)
where F is the nonlinearity and K the leading linear operator. The new variable is constructed using u and some of its x-derivatives, e.g., v = (u, Oxu) t if (6.1) was second order. Because of the relation between the two linear problems Otu = L ( l z , Ox, O~.)u and ( d / d x ) v = K(l~, O~., Ot)v we know that Jk = i~ is an eigenvalue of L ( # , ik) if and only if ik is an eigenvalue of K (tz, O,., i~). We temporarily reintroduced the cross-sectional derivatives O~. to display the generality of the approach. But further on we will drop O~, again. The spectral assumption for L ( # , ik) in Section 2 immediately provides spectral assumptions on K (/~, ico), namely spec K (0, ico) A i R = {ik0, -ik0}. These eigenvalues are geometrically simple, however, their algebraic multiplicity may be higher. It was proved in [2, Theorem 2.1 ] that the multiplicity is m if and only if Jk has the expansion )~(0, k) - ico + )~m.o(k - ko) m + O ( [ k - k0[m+l),
for some Jk,,,,0 6 C \ {0}. Thus, in the real case (case 1 in Section 2) and in the complex case with k0 = 0 (case 3) we immediately obtain m = 2. In the latter case we use )~(0, - k ) = )~(0, k). In case 2 we typically have Cgr = (d/dk))~(0, k0) -~ 0 and the eigenvalues are simple. The spatial center manifold reduction was first developed in [84], therefore it is also called Kirchg~issner reduction. This method reduces the original elliptic partial differential equation on a cylindrical domain to an ODE with respect to the axial variable. The method was further developed in [95] for quasilinear or non-autonomous problems. Timeperiodic problems are treated in [85,72,73]. To apply it we choose an appropriate phase space X, e.g., a closed subspace of [LZ(X')] 2m in the time-independent setting and of [ L 2 ( r • S 1)]2'" for the time-periodic case. Here r C ~'~ is the bounded cross-section of the cylinder s and S 1 denotes the time axis modulo the period. Thus, in the time-periodic case the real time variable t takes the role of a cross-sectional variable while x 6 IR is considered as unbounded time-like variable instead. To apply the center manifold theory we split the phase space X into X l @ X2, where X l is the spectral part of K (0, Ot) associated to the eigenvalues on the imaginary axis whereas X2 associates with all the remaining spectrum. Thus we have dim X l = 4 in case 1 and case 3 and dim X l = 2 for case 2. If p i : x --> x j are the associated spectral projectors, then (6.4) can be written as d dx vl - Kl vl + F1 (#, co, vl + v2),
d m dx 1;2 ---- K21;2 -Jr- F2 (/A, o), V l -Jr- v 2 ) ,
where K j = K(O, a,)lxj, Fi(#, o), v) = Pj (K(#, 0,) - K(0, 0,) + F ( # , v)).
812
A. Mielke
Since the eigenvalues of K2 are bounded away from the imaginary axis, one expects that all small bounded solutions can be described by the critical variable vl alone. In fact, the spatial center-manifold reduction states that all small bounded solutions lie on a locally invariant manifold which is a smooth graph over a neighborhood of 0 6 X l: .A~ center -- {u1 - q - h ( ~ , c o , 1)1)" ]]u1]] ~ } .
This allows us to reduce the study of all small solutions which are bounded in x-direction to the ODE d dx
(6.5)
Vl - Kl Vl + F1 (#, co, Vl + h(#, co, Vl )),
where v2 was eliminated by the help of the reduction function h(/z, co, .) : XI --+ X2. We now consider the three spectral cases separately. CASE 1. )~(#, k) ~ R and ko :/: O. For this case dim X l = 4 and K1 has the eigenvalues +ik0 with a nontrivial Jordan block. Thus after choosing appropriate coordinates (A1, A2) E C 2 in X1 the ODE reads
dx
A2
(i 0 1) ()A,+ ( ,< AIA:AIA:> ) 0
ik0
A2
N2(#, AI, A2 AI A 2 )
(6.6)
'
where we have omitted the complex conjugate equations. Doing a normal form transformation up to cubic order we may assume that the nonlinearities N j have the form Nl(lZ . . . . ) -- ClAl
+O(IAll 3 + IA~I),
N2(#, .. .) -- c2AI -Jr-c3A2 + c41AI 12At -+- O( I A I 15 +1 A2A2 I + ]A2 13), where cj = Cj ( # ) are some complex constants with cj(O) ~---0 for j -- 1, 2, 3. As in the Ginzburg-Landau modulation ansatz we define B j ( x ) = - e - i k ~ and insert this into (6.6). The obtained ODE becomes x-dependent, however, only in the higher order terms:
dx
B2
c2
1 ) ( Bl ) c3 B2
+
(
) 2h.o.t. c4]Bl] Bl + h.o.t.
9
Now we can introduce the same scaling as in the modulation ansatz BI(e~) - - e A ( ~ ) , Bz(e~) = EZc(~) and # = pe 2 giving an ODE for (A, C) E C 2. Solving the first equation for C = (d/d~)A + O(e) we arrive at a second order equation for A (~): d~ 2 a - c'2(O)pa - c4(O)]A]ZA + g e, p, - A A 6'
'
'~-
A
'
-- O,
(6.7)
where g(e . . . . ) = O(e) for e --+ 0. Thus, after neglecting the small function g we find the steady one-dimensional GLe.
The Ginzburg-Landau equation in its role as a modulation equation
813
Of course, the final steps in this procedure are also formal like the derivation of the GLe itself. However, here we are only dealing with an ODE. By this way, neglecting certain terms of higher order can be justified more easily. However, the main purpose in using the Kirchg~issner reduction for the justification of the GLe is not to derive (6.7). We should rather say that (6.6) is the correct reduced ODE and this ODE can be studied rigorously by taking into account all higher order terms. For instance it is possible to show the existence of a family of periodic orbits which corresponds exactly to the family of rotating waves. Using the principle of reduced stability as discussed in [99,100] it is also possible to proof the stability results given in Section 4.1. We refer to [74] for more details and for a treatment of the Couette-Taylor problem as described in Section 3.2. A further discussion of the GLe as a normal form equation is given in [94]. There it is shown that the nonautonomous terms in (6.7) involving ~/e can be pushed to arbitrarily high order. CASE 2. ~.(0, k) = iw + i C g r ( k - k0) + h.o.t, with Cgr, k0 ~ 0. In this situation the center manifold is two-dimensional and the ODE can be written as d
--B--ikoB+N(#,~,B,B) dx on the center manifold and it cannot be directly related to the GLe, see [72, Section 4]. This system is SO(2)-invariant under the transformation B ~ e ir B as a consequence to the time-invariance of (6.4). We trivially obtain solutions of the form B(x) = r e ik with k ~ k0 which correspond to the rotating waves of GLe. CASE 3. k0 = 0 and ~(0, O) = iw. We again have dim Xl = 4. The operator Kl has an algebraically four-fold eigenvalue which is geometrically two-fold. After some normal form transformation the ODE can be written as
1 ) (
dx
ce(kt ~)
c3(#,~)
+
h.~89
c4(/z,~)lAl
Al+h.o.t.
)
"
(6.8)
Again the system is SO(2)-invariant under the transformation B ~-~ e i'k B as a consequence of the time-invariance of (6.4). We employ the scaling ~ -- ex, Al -- eA, A 2 - - ~ ' 2 C , / 2 - - p~,2 and ~ -- co + eo-, then we solve the first equation for C as a function of A and d A / d ~ to obtain the following second order ODE for A (~):
d2
d
(
dd
t
d---~A + bla--~ A + ( b2p + b3a2) A + b4I A I2A +"g e, p, a, A, A, --d-~A, -d-~-A - 0 . (6.9)
814
A. Mielke
This case was treated in [73] and applied to a reaction-diffusion system. In [2,4] the spatial center-manifold reduction was used to prove existence of periodic and multi-pulse solutions (cf. Section 4.3.1 above) for the Poiseuille problem as described in Section 3.3.
6.2. The approximation property We now return to the full time-dependent problem and want to study the approximation properties of the modulation ansatz when evaluated along solutions of the GLe. The general idea is to use the modulation ansatz as a first guess of the solution, insert the function u(t, x) = VA (t, x) + r(t, x) in the full problem and then try to show that the remainder remains small for all x 6 R d and for sufficiently long times. Here it is important to recall that the solutions A(r, ~) have to be rescaled via t = ezr when they are inserted in the full problem. To see the flow of the GLe acting on an initial condition A (0, .) we have to observe the solution for r in an interval [0, r0] with r0 > 0 independent of e. Hence, in the full problem the relevant time interval for t is [0, r0/e 2] which is a very large time scale. Inserting u = VA + r with VA = UA + O(e 2) in (6.1) we obtain the equation
Otr = LEr + N ( # , VA + r) -- N ( # , VA) -- Res(VA)(t, x), (6.10) with
ReS(VA) -- Ot VA -- L ( # , Ox) VA -- N ( # , VA),
where Le = L ( p e 2, Ox). The term Res(Va) is the residual which we obtain upon insertion of VA. Assuming that A is a solution of the GLe (6.3) and by including suitable corrections into VA -- UA + O(e 2) we can arrange things such that Res(Va) is sufficiently small, let us say o ( e m ) . Moreover, the nonlinearity N ( # , .) satisfies a Lipschitz continuity
IIN<#, u)- N(#, )II Cu[llull + 11113"Ilufor sufficiently small u, fi'. Here n -- 1 if N starts with quadratic terms and n -- 2 if it starts with cubic ones. The linear operator L~ generates a semigroup (etC~)t>~o which satisfies the bound
Ile' ll
~< C,e c2~2' f o r t ~>0.
With this and IIVAII = O(e) we try to estimate r by the help of variation of constants formula for semigroups
r(t) -- etL~r(O) +
f
t e(t-s)L(e)[N(#, VA(S) + r(s))
-0 - N(#,
VA (S)) -- Res(VA)(S)] as.
Letting mr(t) = sups~[o,t] Ilr(s)]l and similarly mRes and mVz we obtain the estimate
mr(t) ~ Cl e ~'~2' Ilr(o)I[ + ft
C,e c' e2(t_S)(mRes (s)
=o
"~ CN[mr(S) -Jr-mVA (S)] n Imr(S)II)d,.
The Ginzburg-Landau equation in its role as a modulation equation For times t 9 [0, z-1/82] with rl > 0 fixed and mRes(t) ~< CR 6j, mVA (t) <~ CA8 interval we arrive at
mr(t) <~ D,
Ir(o)l] + -D2 j (CR8j
815 on
this
+ CN[mr(t)+ CAe] n mr(t)).
It is immediately clear that this simple approach only works if n ~> 2. Indeed, to show that
mr(t) remains small for t 9 [0, r l / e 2] we need n >~ 2 since the prefactor Dze-ZCN(CAe) " must be bounded independently of e. These simple arguments can be made rigorous for the problems with cubic nonlinearities, see [26,87] and [105, Section 3.1]. However, for general problems, e.g., those in hydrodynamics, another approach is needed. This was introduced in [142] and further developed in [ 122,19,127]. Applications to several hydrodynamical problems are now available, namely in [ 123] for the Rayleigh-B6nard problem, in [18] for the Poiseuille problem and in [135,136] for the Couette-Taylor problem. The general observation which is behind the analysis is that the modulated solutions of the basic pattern have Fourier transforms which are strongly concentrated around the two points +k0. Through nonlinear interactions these Fourier clusters generate new modes around the points mko with m 9 Z. However, modes with [ml ~ 1 are not critical and hence they are exponentially damped. In particular, quadratic interactions of the basic pattern generate only modes with m 9 {-2, 0, 2}. Thus, the basic observation is:
Quadratic interactions of critical modes are noncritical.
(6.11)
So far the argument was only true under the assumption k0 ~- 0. However, the case k0 -- 0 with X(0, 0) -- i~o ~: 0 can be treated as well since a similar mode coupling occurs in spacetime, see [ 131 ]. We now give the mathematics in more detail. We consider (6.1) in suitable function spaces Y C X, which contain modulated solutions. Typically one takes Y and X in the form [W~in(Y2)] m with ky >~kx. Now assume that A ' [ 0 , rl] • ]Kd --+ C is a given solution of the associated GLe. Moreover, we use a modulation ansatz VA (t, x) -- UA + O(s:) such that the associated residual Res(VA), see (6.10), satisfies an estimate
IlRes(Va)( t,
")llx CR
fort 9 [0, r,/s2].
This can always be achieved by using the fact that solutions of GLe are arbitrarily smooth and then doing the formal expansion of Section 2.2.2 to the desired high order, see, e.g., [74,106]. The main functional analytic tool involves the splitting of the critical modes (also called Ginzburg-Landau modes) and the remaining stable modes. This cannot be done in a simple manner as we have to cut through the continuous spectrum of the operator L~. Hence we do not define an exact splitting using projections but use filters Ec and Es instead of projections Pc and P~. (Here 'c' and 's' are abbreviations for 'central' and 'stable'.) The filters are defined such that Ec + Es = Ix and, if we define the spaces X~ -- closure(E~ X) for ot 9 {c, s}, then E,~ X~ = X~. Note that Ec Es ~ 0, and hence Xc N Xs is still an infinite dimensional subspace containing some stable modes.
816
A. Mielke
The construction of Ec is done by the help of Fourier transform. We choose a small 6 > 0 independent of e and a smooth cut-off function X+k0 :R d ~ [0, 1] with X+k0 (k) = 1 if Ik - k0[ ~< 6/2 or Ik + k0l ~< 6/2 and X+k0 (k) = 0 if Ik - k0l ~> 6 and Ik + k01 ~> 6. Using the eigenfunction ~'pez,k of L(pe 2, ik) associated to the eigenvalue )~(pe 2, k) together with the eigenvector q~* p82, k ) = 1 we define Ec through p~2,k of the adjoint problem with (q~psZ,k q~* its Fourier transform
f~ [E~u](k)- X+ko(~:)(f~[u](~:), r
k)q'p~ k.
(6.12)
Here the functions q~ and q~* of k need only be defined in the two balls of radius 6 around k0 and - k 0 . This formula is certainly well-defined for functions u 6 L2(~2) and it is established in [123, Lemma 5] that the operator Ec is a bounded linear operator from L2ul(S-2) into This arises from the fact that all q~e,k are smooth functions of y 6 r and that the Fourier transform in x-direction of Ecu is localized. Additionally the construction is such that Ec commutes with Le, i.e., EcLe = LeEc. By this definition Ec depends on e, but for notational convenience we suppress the dependence. We can now write solutions u(t) of (6.1) in the form u(t) = Uc(t) + Us(t) with u~(t) X,~. However, we warn the reader that generally u~ =/: E~u since we define the two parts via
wk'2(S'2).ul
OtUc = Leuc + EcN(e, Uc + Us),
OtUs = Leus + EsN(e, Uc + Us).
(6.13)
The quantitative version of the quadratic interaction principle (6.11) takes the form
IIEcN(~,
uc>llx
-
O(ll,,c ..~).
To be more precise we make the following assumptions on the nonlinearity:
IIE~[N(~,Uc +u~>- N(e, ~c-Jr-~)] II~ CN ([ ttUc tRY+ I1~c [Iy]n~ + liU~ tt~ + II~ I1~)11Uc -- ~c 11~ (6.14) where nc - 2 and ns - 1. Of course, this is assumed to hold only for Ilu~ IIY, I1~ IIY ~ ~1. The linear operator Le generates a semigroup which, by construction, satisfies the estimates
Ile"~ [Ix _+~c ~< Ce c~:t,
Ile"~ I1~_~ ~ ~< Ct-~e-•
Ile"~ II~s_~, ~< Ce -• for t > 0, where C, y > 0 and/3 ~ [0, 1). The approximation result can now be formulated as follows.
(6.15)
The Ginzburg-Landau equation in its role as a modulation equation
817
THEOREM 6.1. Let the above assumptions be satisfied. Choose any rl > 0 and any solution A = A(r, ~e) :[0, rl] x F~d --+ C of the GLe (6.3). Assume that there are x > 1, so > 0 and C R, CA > 0 such that VA is an approximate solution of (6.1) with
IIv~(,, .)I1~ ~
.)II x <~ Cl~e~+2'
for all s 6 (0, s0) and all t ~ [0, rl/s2]. Then for all D > 0 there exist el ~ (0, s0) and C > 0 such that for all s ~ (0, s l) the following holds. If the initial datum uo ~ Y satisfies
I E c [ u o - VA(O, ")] [y ~< De z
and
IIE~[uo- va(o,.)]llY ~ o~max~"~-'~'
then the unique solution u(t) = S t ( u o ) of (6.1) exists for t ~ [0, rl/S 2] and satisfies the estimates
IE~[,(,, .)- va(,,-)] II~ ~
If x > 1 then the errors are relatively small compared to the solution u as well as compared to the approximation VA. PROOF. We insert the ansatz u = (Ec VA + rc) + (Es VA + rs) in (6.1) and obtain, by the variation of constant formula, r~(t)
=
e tLC
r~(O) +
f0 t e(t-s)L~Eu(Res(VA) -N(s,
+ N(e, VA(S) + r(s)) Va(s)))ds
ford 6 {c, s}. Using the abbreviation m e ( t ) = suPs~[0,t] ]]rcl]Y and m s ( t ) = I[rs(t)llY, (6.14) and (6.15) together with the temporary assumptions mc(t) ~< Cce K <~ CA8 ~ (~1,
ms(t) ~< Cse ~< 61,
(6.16)
we find the estimates
-) mc(t) <~ CeCe-tmc(O) +
/o' /o'
CeCe2(t-s)(cRe~c+2 + Cf[4C2ae2mc + 2E(Ca + Cs)ms])ds,
m~(t) <~ Ce-• +
C(t - s)-~e-•
+ Cx[2Caemc + ~lm~l) ds.
818
A. Mielke
The second estimate has the advantage of an exponentially decaying term in the integrand. With this, one can show that there exists a constant ~" which depends only on 81, fl, y, C and C N (but not on Cc and Cs) such that ms(t) ~<~'(e-•
4- CRe K+l 4- CNCAEK+I).
(6.17)
Clearly, the estimate for ms in (6.16) is satisfied when letting Cs = ~(D 4- CR 4- CNCA) and assuming el ~< min{1, 81/Cs}. Inserting (6.17) in the first inequality Me(t) --e-ceZtmc(t) is found to satisfy
Mc(t) ~< al 4- a2t 4- b
f0t Me(s) ds
with b = 4e2CNC 2, al - Cmc(0) + eC2ms(O), and a 2 - - CRe x+2 + 2CCN(CA 4- Cs) • ~'6(CRe x+l + CNCAeX+I). Using Gronwall's inequality we obtain the estimate me(t) ~< e ce2t Me(t) ~< eCE2t[ale bt + az(e bt -- 1)/b)] and for t 6 [0, r l / e 2] we obtain mc(t) ~< C3 [mc(O) 4- e ms(O) 4- eK],
(6.18)
where C3 is still independent of e ~< el and Cc. We now choose Cc = C3(2D + 1) and by making el smaller if necessary we are able to fulfill the assumptions for mc in (6.15). The estimates (6.17)and (6.18)establish the result. D
6.3. The attractivity of the set of modulated patterns Here we treat the question why initial conditions in the modulated form u0 UA are of such an importance. We say that such a u0 has a modulated structure. The GLe is formally derived to describe the evolution of the linearly unstable modes. In weakly nonlinear theory one expects that these modes dominate the dynamics of the full problem under arbitrary initial conditions. In the previous section the error estimates were shown under the assumption that the initial conditions already possess the modulated structure. Here we explain why, after the time tl = Tl/e 2, all small solutions of the full problem develop this mode structure and, henceforth, can be described by the solutions of the associated GLe. The first result in this direction was obtained in [48]. For our choice of function spaces the result is given in [124]. Since the Ginzburg-Landau formalism is only a local theory, this attractivity can only be expected for initial conditions in a small neighborhood of the trivial solution. Theorem 6.1 given above already gives a first result in that direction. If we start with u0 such that IlE~uollY <<,(CA 4- D)e and a suitable part Ecuo, then after time t = r l / e 2 we have IIE~u(t, ")llY ~< IIE~VAII + Ce ~+l and it is easy to see that EsVA is of order e 2. Since x > 1 we conclude that the stable part is automatically reduced from size e down to size e 2. However, the attractivity result needed for the application of the Ginzburg-Landau -
-
The Ginzburg-Landau equation in its role as a modulation equation
819
formalism is much stronger since we also have to show that the modes of the central part, considered in Fourier space, concentrate around the critical modes +k0. To formulate the results more precisely we have to define an operator which extracts the modulation function A from a modulated pattern. The key step for isolating the GinzburgLandau mode A 6 Z from a general function u 6 Y is to find an approximate inverse of the mapping A F-, UA. We introduce the linear operator
Z-+Y, Ke"
A
~
8 l-d
[3c[-~{xo(k-ko)[f;A](k-k~8
see also (2.10). Here Xo is a cut-off function with Ikl >/~. The approximate inverse is defined as
.k } + C .c.] ,
(6.19)
xo(k) -- 1 for Ikl ~< 6/2 and - - 0 for
Y---> Z, Me"
{X0(ex)(45;e2.ko+e K, [ f x u l ( k o + E~))}.
u ~-~ e d - , f - : ,
(6.20)
For sufficiently smooth A 9 Z we have A = Me[KeA] + O(e) for e --+ 0 and Ecu = Ke[MEu] + O(e), compare to (6.12) and see [124] for the proofs. In order to formulate the following results as simple as possible we restrict our attention to one particular equation for u, namely the one-dimensional version of (2.20):
OlU - - - - (1 -Jr- 02)2// -+- b(3 -Jr- 02)0xt/ Jr- 8 2 u - - u 3 + g u 2 Oxu, x
(6.21)
which has the associated one-dimensional cGL 0rA--
(4+ib)O2A
+
A
-
(3 -
ig)lAI2A
(6.22)
We formulate the following result with the notations of the general setting, since most of them carry over up to additional technicalities (e.g., due to Sobolev embeddings or different scaling factors depending on the number of unbounded directions). This might change the exponent rr and the underlying spaces but not the general structure of the theory. See Section 6.7 for an application of these principles to the Navier-Stokes equations. For the above equations the function spaces can be chosen as X -- Leul(IE, R), Y -H ul(IE, l 1 R) and Z -- Hul(R, C). In particular, Theorem 6.1 is applicable with fl -- 1/4 for these choices. The operators Ke and Me (which, in the present case, simplify a lot due to cb~2.k -- aS~2,k -- 1) satisfy
IIK~AIIy <~CsllAIIz,
IIz
C
IIM~ u IIz ~< ~_ IIu IIr,
C(r)
(6.23)
II.ll
where r > 0 and all constants C and C (r) are independent of e > 0. While the first estimate is trivial, the second estimate shows that rescaling yields an additional bad factor e - l .
820
A. Mielke
However, this factor disappears if we let act the linear semiflow for a suitable long time, see Section 2.2.1. Thus, in the linearized flow every solution u(t) = etL~uo is attracted to a modulated pattern, namely u ( r / e 2) -- K~A, + O(e 2)
with A, -- Meu(r/e2).
The following result (see [ 105,106]) is the nonlinear version of this basically linear effect. THEOREM 6.2. Consider Equations (6.21) and (6.22) with the given function spaces X, Y and Z. For all C1 > 0 there exist Co, eo, r0 > 0, such that the following is true. For all e (0, eo) and uo ~ Y with Ilu0llv <<.Cle the solution u(t) = S~(uo) exists for t ~ [0, ro/e 2] and A 1 = ME u (to / e 2) satisfies [[A, [[z <~ Co
and
Ilu(~0/e z) -
KeA,
C065/4"
(6.24)
We define the set of modulated patterns, MP-
{K~A" IIAllz ~ C0} C r,
which is a small, very flat ellipsoid in the phase space Y. Then MP attracts all solutions starting in the ball of radius C1E around 0 6 Y, at least up to an error of order e 5/4. This attractivity property of the modulated pattern gave rise to the title "The Ginzburg-Landau manifold is an attractor" for [48]. The development of the mode structure is essentially a linear effect, e.g., from the linear theory in Section 2.2.1 we see that the time scale O(1/e 2) is necessary for the solutions u(t) to develop the scaled mode structure. For the nonlinear result it is then necessary to show that the solutions u(t) = $~ (uo) exist over the desired time interval t E [0, rl/t~2]. For general problems with quadratic nonlinearity, solutions of order e exist only on the time scale 1/e, but again the fact can be used that quadratic interactions of critical modes generate damped modes. The time Z" 1 has to be chosen sufficiently small to avoid blow-up of solutions, which may occur in the case when Re~" ~< 0 in (6.3). Under suitable assumptions on A0 and ~" in (6.3) (e.g., Re~" > 0 and d ~< 2) the solution A(r) = ~r (A0) exists for all time (see Section 4.4) and rl can be chosen arbitrarily. Having Theorem 6.2 it is possible to apply the approximation theory of the previous section. In particular, the assumptions of Theorem 6.1 are fulfilled if we choose u(0) = S ~ r0/~2(u0) , and A(0) ~ A1 and take any time rl > 0 such that A(r) ~r(A1) exists for r e [0, rl].
6.4. Shadowing by pseudo-orbits and global existence Finally we consider the case where the GLe has a global semiflow with a global attractor ,AGE, see Sections 4.4 and 6.5. Hence, we may use the approximation property as well as the attractivity to control all small solutions u(t) = S~ (uo) of the original problem (6.1) for arbitrarily long time intervals. The idea is to use the attractivity property in a first step
The Ginzburg-Landau equation in its role as a modulation equation
821
to find the modulated structure u('~0/8 2) -- [K~A',] + 0 @ 2) where A'l = M~u(ro/s2). In a second step the solution can be described over a time interval ['roA/e2, (to + Zl)/e 2] by the Ginzburg-Landau approximation UA~ (t) with A1 (z') : ~r-r0 (A 1). At the end of this interval the error between the true solution and the approximation might be somewhat larger; however, from the attractivity we know that u((r0 § r l ) / e 2) is close to a different modulated pattern K~ A2 from which the approximation theory may start on the next time interval. The exponential attractivity of the absorbing ball for the GLe controls the size of the solution A j, j c I%1,and guarantees their uniform boundedness for all r >~ r0. In this way we can shadow the true orbit u(t) -- S t (uo) for all t > 0 by a sequence of solutions of the GLe. DEFINITION 6.3. Let r, > 0 and 6 > 0. A function A E L~((0, ~ ) , Z) is called a (r,, 6)pseudo-orbit in Z for (6.1) if for all n 6 N the relations A((n-
1)z, + r ) - - ~ , : ( A ( ( n -
1)T,))
for all r E [0, z,),
[ I A ( n r , - O ) - G r , ( A ( ( n - 1)r,)) 1[z ~<6 hold, where A (r - 0) -- l i m ? / r A (F'). The following result shows that each true orbit of (6.21) can be shadowed after a transient period by a pseudo-orbit of the associated cGL (6.22). The essential feature here is to show that the jumps in the pseudo-orbits are small in terms of s, viz. 8 - o(el/4). See [124, Theorem 3] for a proof. THEOREM 6.4. Under the assumption of Theorem 6.2 for all r, > 0 and C, > 0 there exist positive constants so and C such that for all s 6 (0, s0] the following is true: For all initial conditions uo with Ilu0llY ~< C,s the solution u(t) : St(uo) exists for all time and there is a (r,, csl/4)-pseudo-orbit A for(6.22)which satisfies IIA(0)IIz <~ C and approximates u (t) as follows:
This shadowing technique can be used to show that in fact all solutions starting in a small but s-independent neighborhood H of 0 in Y exist for all time and are finally absorbed into a ball of radius C s. THEOREM 6.5. Under the assumptions of Theorem 6.2 there exist positive constants so, ~o, and C such that for all s ~ (0, s0] all solutions u(t) - S t ( u o ) of (6.21) with Ilu01lr <~ 80 exist for all t > 0 and satisfy limsupt~o o Ilu(t)liY <~ Cs.
It is possible to prove global existence for the solutions u(t) = S~(uo) of (6.21) by using the methods in [ 105]. However, using energy estimates the best one can obtain is an absorbing ball of radius C s 1/2. The present result constitutes a new way to construct absorbing sets, namely by studying the dynamics rather than applying energy estimates.
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A. Mielke
6.5. Comparison of attractors According to Section 5 the GLe has a global attractor ~GL C Z. Similarly there exist local attractors A e for the semigroups S~. Local attractor means here that there is a neighborhood H e of 0 in Y, which is positively invariant with respect to the semiflow S t. According to Theorem 6.5 the sets/.4 e can be chosen such that they contain a ball of radius g0 > 0. Now every bounded B C H e is attracted to .Ate in the weighted norm in Y: distrw (S t (B), A e) --+ 0
for t --+ oo.
The question is now how well the attractors A e can be described by the attractor .AGE. The following result is given in [ 105] (for generalizations see [ 114,136]). THEOREM 6.6. Under the assumptions of Theorem 6.2, for every 6 > 0 there exist C > 0 and eo > 0 such that for all e 6 (0, e0] the estimates distz~, (MeA e, AGL) ~< ~r
and
distr,, (EsA e, {0}) ~< Ce 5/4
hold. This result means that A e is upper semicontinuous towards the attractor .AGL of the Ginzburg-Landau equation in the sense that 1 distzu, (MeA e, .AGL) + -distr., (EsA e, {0}) ~ 0
E
for e --+ 0.
Thus, the solutions u in the attractor A e have relatively small stable parts Us = Esu and the critical part Uc is given by a modulated pattern Ke A 6 Y with A 6 ~A,GL C Z. In this way we have obtained an upper bound on the richness of the attractor A e . In fact the upper semicontinuity result can be sharpened by combining it with the theory of pseudo-orbits as given in Section 6.4. Each orbit in A e can be shadowed by a (r*, Cel/4)-pseudo orbit A of cGL, which lies completely in AGE, see [ 136]. It is an unsolved problem to show the opposite direction, namely that A e is also as rich as the attractor AGE. This would provide a lower bound on the complexity of the attractor. In mathematical terms this amounts to lower semicontinuity in the sense distzu, (.AGL, Me A e) --+ 0 for e --+ 0. The definition of the attractors Me,Ate and AGL involves the limit of time r --+ oo. (Note that t --+ oo without e2t --+ oo is not interesting.) Thus, comparing the attractors means that we have chosen to consider first the limit r --~ oo and then we consider the limit e ~ 0. However, in practical cases one always deals with finite time r and finite e > 0, and it is not always clear whether we are closer to r ~ cx~ or to e ~ 0 in a given setting. Thus, it appears reasonable to study the double limit (r, e) -+ (oo, 0) where no specific order is supposed. We follow here the ideas of [65], which were already used in Theorem 5.3 to study the attractor on very large domains.
The Ginzburg-Landau equation in its role as a modulation equation
823
We define a generalized limit attractor .A* in the following way A* = [A E Z: Zle,,--~ 0 Zlr,,--+ cx~ 3u,, with sup Ilu,,llY < cx~ #tE~
~,,/~(u,,) -
such that IIM,,,S ~''
All z -->0 / .
Here the time sequence t,, = r,,/e 2 is chosen such that it tends to infinity in the GinzburgLandau scaling. In contrast to T h e o r e m 5.3 we can take here the uniformly local convergence (in Y - Hull(R, C)) rather than the localized convergence in Yu,. The following result relies on a slight generalization of the approximation estimate in Section 6.2 which follows by combining the approximation and attractivity results: For each R > 0 and r0 > 0 there exist e0 > 0 and C I > 0 such that for all A E Z with ][AI[z ~< R and all e E (0, e0) the following estimate holds:
[ [ M ~ S8 / ~ 2 ( K ~ A o ) - ~(Ao)llz <~c,(r
'/4
(6.25)
f o r t E [0, rO].
THEOREM 6.7. Under the above assumptions we have .A* = .A[GL.
PROOF. We first show .AGL C w4*. Take A ~ .AGL, Z'n ~-- n and, by the invariance of J~GL, choose B, E AGL with Gr,, ( B , ) = A. Defining u , - K~,,B, we obtain IlunllY <~ Ce,l~/4 ro. Here we still have the sequence (e,,) at our disposal. With (6.25) and t,, = r,,/e 2 we obtain
Il
s*" (u,, - All z ~< IIM,,,S ~'',,,(K,,.8,,)-
c,
),,,' /4.
Since the sequence r, = n was already fixed, we can now choose e,, such that the righthand side in the last estimate converges to 0 for n --+ oo, which implies A E .,4*. To show A* C AGL we take A E A* and A, = Mc,,S~,~'(u,,) with [[A - A,[lz ~ 0, e,, --+ 0, t,, -- r , / e 2 and [[u, [iv ~< r0. Choose ~" >~ 1 and restrict n E N such that r,, > ~. Let v,, -- ,.q~"~t,,_~)l~;,~(u,,) and B,,.~-- M~,, v,,. Then, by the attractivity theorem 6.2 we have [[B,,.~-[lz ~< C with C independent of n and T. Moreover, distz,,, ( a , AGL) ~< IIA - A,, [Iz + IIA,, - GT"(B,,.7-)IIz + distz,,, (GT(B,,.7"), AGL). For given ~ > 0, we can choose ~" so large that the third term is less than ~/3 for all n. Clearly the first term is less than 6/3 for sufficiently large n. While keeping ~" fixed we can estimate the second term as
IIM , + c.
824
A. Mielke
.-, 5/4
where we have used the approximation theorem 6.1 and IlVn - Ke,,Bn,~'IIY <, t e n for the first term and (6.25) for the second. By choosing n sufficiently large we have thus proved distz~, (A, AGE) ~< 6 where 6 > 0 was arbitrary. Hence we conclude A e ~4GL. [-] 6.6. Comparison of inertial manifolds Another way to compare the dynamics between the original system and the associated GLe is obtained by restricting the analysis to periodic boundary conditions, where the period of the original system has to be large in order to obtain the GLe on a fixed domain independent of e. At the moment the results are rather limited, and we only report on the work in [108] where a general method is given to compare the inertial manifolds of two different parabolic problems which may have a different number of components, a different order, or are even posed on different domains. We present the result only for the Swift-Hohenberg equation (SHe) ut -- - ( 1 + 02)2u + 62U -- U 3, which after the rescaling if(r, ~) = 6-lu"(62"t ", 6~) takes the form Orfi'-- - e - 2 (1 + e 2 0~)2~. + ft._ fi-3.
(6.26)
The associated modulation ansatz is ~"(r, se) = A(r, ~)e i~/e 4- c.c. 4- O(e)
(6.27)
which leads to the associated real GLe O r A - - 4 0 ~ A + A - 3IAI2A.
(6.28)
We complement (6.26) by the periodicity assumption fi'(r, ~ + g) = fi'(r, ~) for all r ~> 0 and ~ e R. The period g > 0 is considered to be fixed. In particular, we choose the appropriate phase space Xe,per -- {U" E H~oc(R)" ~'(~ + g) -- h'(~) for all se e R}. With the modulation ansatz (6.27) this gives the quasiperiodic boundary condition A (~ + g) = e iz A(~) with y = (e/e) mod2zr. Thus, the associated phase space for the GLe is Ze,•
{A e H~oc(IR, C)" A(se + g ) - ei•
all ~ EIR}.
Because of the new boundary conditions it is a slight variation of a classical result that the rGL (6.28) restricted to the phase space Ze,• has an inertial manifold, see, e.g., [44, 141]. Particularly, the eigenvalues of the linear part are given by )~n -- 1 - 4 ( y + 2zrn)2/g 2, n e Z, and thus there are sufficiently large spectral gaps. To compare the two quite different systems (6.26) and (6.28) we have to decompose both phase spaces into three parts: Xe,per = X1 (~ X2 (~ X3,
Ze,• = Zl 9 Z2 9 Z3.
The Ginzburg-Landau equation in its role as a modulation equation
825
These decompositions are associated with the spectral parts of the linear operators such that X l, Z l contain the largest eigenvalues, X2, Z2 the intermediate ones, and X3, Z3 the smallest ones (going to - c ~ ) . The important point is that there are isomorphisms Tj : Xj Zj for j = 1,2 which make the two systems comparable. However, we do not need to compare X3 with Z3, as the contributions of these subspaces on the flow reduced to the inertial manifold will be small. For the rGL and assuming V 9 (-zr, 7r] we define the subspaces
Zj = span{e i(2Jrk+z)~/e, ie i(2~k+z)-~/e" k 9 Jj(nGL, y, e) } where the index sets are given by Jl (nGL, y , e ) -
{k 9 Z: [k -+- •
I
-+- 3/4},
J2(nGL, Y, 6 ) = {k 9 Z: k r J, (r/GL, y, E), Ikl < e-a}. Here the parameters nGL 9 N and 6 9 (1/2, 1) are chosen later. In particular, nGL will be chosen in such a way that certain spectral gap conditions are satisfied: the spectral gap between the eigenvalues associated to Zl and Z2 is of approximate size 327rZnGL//~2. The space Z3 is then simply the closure of the span of all the remaining eigenfunctions (exponentials of the type in Zj). The spaces Xl and X2 are constructed by simply applying the modulation transformation (6.27) to the functions in Z l and Z2, respectively. For sufficiently small e, these mappings are isomorphisms by the construction of the index sets Jl and ,/2 and the assumption 6 < 1. We use the notation T j : X j ~ Zj to indicate these isomorphisms. It is shown in [108] that both systems have an inertial manifold which, after modifying the systems outside an absorbing ball, take the form A/gx(~) = graph(Rx) = {fi'l + Rx(~ff,) 9 Xe,per: fi', 9 Xl }, Y~z(Y) = graph(Rz) = {A1 + R z ( A l ) 9 Ze,y: Al 9 Zl }, where the reduction functions Rx : XI --+ X2 • X3 and Rz : Zl --+ Z2 9 Z3 are globally Lipschitz continuous mappings. The sets A//x and A//z are called inertial manifolds for (6.26) and (6.28), respectively, as these manifolds are invariant under the flow and they attract all solutions exponentially for positive times. Inserting the reduction functions in the full infinite dimensional problems the system can be reduced to the finite-dimensional flow in the inertial manifold itself. Using fi'l 6 X l and A l 6 Z l as coordinates the reduced flows take the form
d s
-
Fred
-
-
-2(I + e2O#)2fi", +fi",- P, [(fi"l + Rx(fi",))3],
d - - A l -- Gred(Y, Al) -- 40#Al + Al -- 3QI[IAI2A], where A - - Al + Rz(Al), dr where PI :X ~ X l and Q l :Z --+ Z l are the associated L2-orthogonal projections.
A. Mielke
826
It is now possible to compare the reduced flows of the two systems, since they are defined on finite dimensional spaces X1 and Z l which are isomorphic. The following result holds, see [ 108]. THEOREM 6.8. For given 6 ~ (1/2, 1) there exists C > 0 such that for each s ~ 1 there exists nGL 6 N with nGL ~< Cs 3 and a qo E N such that for all y ~ (-Jr, Jr) the following holds: The rGL (6.28)posed in Ze,• has an inertial manifold .A//z (y) with dimension less than 4nGL + 4 and the reduced vector field Greo(Y, ") : Zl --+ Zl. Moreover, for all q E N with q > qo the Swifi-Hohenberg equation (6.26) posed in Xe,per and with e = s - y) has an inertial manifold A/Ix(s) of the same dimension as .A//z(y). The reduced vector field Fred(e, "):Xl----> X! on A/Ix(e)satisfies
IlGred(y,A,) - TIFfed(E,
TU'A,)II
<~ c ( e l l A , I I z + e~).
It should be noted that the reduced vector field Fred(6, ") does not converge to a limit for e --+ 0 but it has to approach a family of vector fields parametrized by y E S 1. It is simple to check (e.g., by considering the steady states) that the vector fields Gred(Y, "), y E ~1, are genuinely different. We now point out some differences between the limit flow Gred(Y, ") and the reduced flow Fred(e, ") of SHe. Both, the SHe and the rGL are gradient systems, namely with respect to the functionals
E(A) =
f0 [
] d~.
210~AI 2 -~lA 1 12 +~lAI 3 4
Thus, the flows on the inertial manifolds .AAx and A//z are gradient-like systems in the sense that they have a strictly decreasing Liapunov function. In particular, this implies that all solutions in the global attractors have the or- and co-limit sets in the set of equilibria. Note, however, that there is a major difference between the flow of SHe and rGL. Both systems are translationally invariant but only rGL has an additional phase invariance A w-> e i~ A. In fact, the set of equilibria in rGL was studied explicitly in Section 4.2.2. The rotating plane waves A (x) = r e ik~ form a circle of equilibria, since the action of space translation and phase multiplication coincides. Moreover, there are steady states in Ze,• for which [A (.)1 is periodic but nonconstant. Such solutions occur always in a two-dimensional continuum of equilibria which has the topological type of a toms, see, e.g., [21,39]. If we now consider generic s and y then the set of equilibria for Gred(Y, ") consists of finitely many circles and toil as described above. Moreover, we can assume that all these manifolds of equilibria are normally hyperbolic. If we now consider the flow of Fred(e, ") with s = y mod27r then the flow is a small perturbation of Gred(Y, ") and we conclude that all the circles and tori persist as normally hyperbolic invariant manifolds. The flow on the circles will still be trivial as the translation symmetry of SHe forces equilibria to occur in circles. However, the flow on the tori will no longer be trivial: generically one
The Ginzburg-Landau equation in its role as a modulation equation
827
expects that only a finite number of circles of equilibria will persist which are connected by heteroclinic orbits inside the torus. Thus, we see that the inertial manifold provides a tool to study the breakup of the normal form symmetry which is present in rGL, cf. [94].
6.7. Results on the Couette-Taylor problem We finally return to hydrodynamical problems to emphasize that the whole theory developed above is applicable to general partial differential equations, in particular to the three-dimensional Navier-Stokes equations. The Rayleigh-B6nard problem was studied in [ 123] and the Poiseuille problem in [ 18]. Here we restrict ourselves to the Couette-Taylor problem, give a rough sketch of the relevant results and refer to [135,114,136] for the details. We study the Navier-Stokes equation exactly in the setting described in Section 3.2 and write the velocity field v(t,-) :s - R • Z" --+ I~3 as Vcou + u, where u is the perturbation of the Couette flow Vcou. The equations for u are
Otu -- A u - V p -
|
~[(Vcou.V)u + (u.V)Vcou + (u.V)u],
in s
(6.29)
0 -- divu, u
=
0
f r u l (t, x, Y) dy - 0
on0F2, for x 61K.
The Reynolds number 7~ is assumed to be slightly above the threshold, namely ~ = 74o + e 2. Moreover, we have incorporated the condition that the mass flux through each cross section {x } x r is 0 (here u l 6 R denotes the axial component of u 6 R 3). We assume that the other parameters are chosen such that the Ginzburg-Landau formalism is applicable. Then, the associated GLe is automatically one-dimensional and real:
O v A - y~ O~A + v2A - y3IAI2A.
(6.30)
By numerical calculations it can be shown that the coefficients yj are all positive. The mathematical setup of (6.29) can be formulated in the spaces X--[ueH2ul(~,~3) 9 divu=0, ulas2-0, fruldy--O}. We also use the space X~, when the uniformly local norm is replaced by a weighted norm, where the weight w is chosen to be independent of y 6 2', e.g., w(x, y) = 1/cosh(x) or 1/(1 -+- X2). Problem (6.29) defines a local semiflow ($~)t 3o on the space X. In general the question of global existence for the three-dimensional Navier-Stokes equation is not yet solved. However, using the Ginzburg-Landau formalism it is possible to obtain new results in
828
A. Mielke
that direction. Denote by (~r)r~>0 the global semiflow of the rGL (6.30) on the space Z = H 2 (R, C). For a solution A:[0, cx~) --+ Z the modulation ansatz is given by UA (t, X, y) -- e A(e2t, ex)e ik~ ~0,k0 (Y) 4- c.c.
(6.31)
The operators Ke :Z --+ X and their approximate inverse Me :X --+ Z are defined exactly as in (6.19) and (6.20), respectively. Using these specifications, the approximation theorem 6.1, the attractivity result in Theorem 6.2 as well as the shadowing by pseudo-orbits (Theorem 6.4) can be established. We will not repeat these results as stated theorems once again. However, we formulate two results which are derived by using the Ginzburg-Landau formalism but can be formulated independently of that theory. As a parallel to Theorem 6.5 we obtain the following result (see [135, Theorem 1.4]) which proves global existence for certain interesting solutions of the three-dimensional Navier-Stokes equations.
THEOREM 6.9. There exist small positive numbers ro and Eo and a large positive C such that for all e ~ (0, eo) the following holds, l f uo ~ Z satisfies [luollx ~ ro, then the solution u(t) -- S~ (uo) of the Navier-Stokes equation (6.29) exists for all t ~ 0 and satisfies
Ilu(t>ll
~ Cro
for allt ~ 0
and
limsupllu(t)[[ x ~ c c . t----~oo
In particular, the set Be = [,.Jt>>.oS~({u: Ilullx ~ r0}) is positively invariant and contains the ball of radius ro in X. Moreover, this set is absorbed into the much smaller ball, namely that of radius 2CE in finite time. Note that the small solutions u 6 Z might have arbitrary large or even infinite kinetic energy fs~ lu(t, x, y)l 2 d(x, y) in the classical sense of the Navier-Stokes equation. However, this energy must be spread over the cylinder so that in any finite subregion (xl, x2) x Z' the kinetic energy is bounded by C (x2 - x l). In analogy to Section 5 it is possible to define a local Couette-Taylor attractor ,A~T with respect to the positively invariant set Be, see [ 114,136]. Clearly, A~T will lie in the smaller ball of radius Ce. It is an (X, Xw)-attractor with the following properties: (a) A~T is nonempty, closed and bounded in X and compact in Xu," (b) A~T is invariant under the flow of (S~)t~>0; (c) ,A~T attracts all subsets B of Be C X in the Xw-metric, viz., E
distx., (Sf (B), ACT ) ~ 0 (d)
for t ---> oo;
.A~T is invariant under spatial translations x ~ x 4- x0 and under azimuthal rotations
~ ~ + ~b0 (cylinder coordinates, see Section 3.2). In fact in [ 136, Theorem 6.1 ] it is shown more" The Couette-Taylor attractor ,A~T is also upper semicontinuous towards the Ginzburg-Landau attractor AGL in the sense of Theorem 6.6. Moreover, the whole attractor lies in the subspace Xri C X of rotationally invariant functions. This can be explained by the fact that in the regime under consideration the instability occurs in the rotationally invariant subspace. Perturbations which have a nonzero
The Ginzburg-Landau equation in its role as a modulation equation
829
azimuthal wavenumber are still uniformly damped. (If the first instability arises by functions which are not rotationally invariant then the associated eigenvalues cannot be simple and our basic spectral assumptions (2.3) are not satisfied.) We decompose u ~ X into u = Uri 4- W where Uri E Xri is rotationally invariant whereas w has mean 0 when integrated over dp" fs~ w(t, x, r, 4~) dO - 0. Denote by P" X --+ Xri the associated projection, then we have distx((I - P ) S t ( B c ) , {0}) ~< Ce -ut
fort >~ 0,
where C, ot > 0 are independent of ~ ~ (0, e0). Finally we mention that the existence of modulated front solutions in the Couette-Taylor problem is established in [67]. These solutions correspond via the ansatz (6.31) to simple front solutions A(r, ~) = A(~ - Cfront'g) of (6.30) where Cfront : (..O(e). However, the existence proof does not use the Ginzburg-Landau formalism; it is rather based on a singular version of the Kirchg~issner reduction for time-periodic parabolic problems as described in Section 6.1.
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CHAPTER
16
Parabolic Equations: Asymptotic Behavior and Dynamics on Invariant Manifolds
R Polfi~ik* Institute of Applied Mathematics, Comenius UniversiO', Mlynskd dolina, 842 48 Bratislava, Slovakia
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. State space, smoothness and compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The comparison principle and monotone dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . 4. One space dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Symmetric domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Gradient-like systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Dynamics on invariant manifolds and realization of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
*Supported in part by V E G A grant 1/1492/94. H A N D B O O K OF D Y N A M I C A L SYSTEMS, VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 835
837 839 843 852 859 864 867 876 876
Parabolic equations
837
1. Introduction
This survey is devoted primarily to second order parabolic equations of the form ut = L u + Bu
f(t,x,u,
Vu),
= 0,
xEY2, t > 0 ,
(1.1)
x E 0s
(1.2)
t > 0.
Here Y2 is a domain in ]l~N, N >~ 1, L is a second order elliptic operator, f is a real-valued function, and B is a boundary operator of the form OU
B u - - ~ ( x ) - x - - + V(x)u,
OV
(1.3)
v being the unit outward normal vector field on 0Y2. Boundary conditions of Dirichlet (/3 = 0 with g :/= 0 everywhere), Neumann (g = 0 with fl --/- 0 everywhere) and Robin (/3 --/: 0 everywhere) type are considered. We impose appropriate regularity hypotheses on the above functions and domain and always assume that f is periodic in t (this in particular includes autonomous equations). We study qualitative properties of solutions that are global and stay bounded in a suitable norm. For the sake of simplicity of exposition we do not attempt to present the results in the forthcoming sections in their most general form. This is the main reason why our discussion usually refers to the above scalar semilinear equations on bounded domains. Often extensions to a broader class are possible and will occasionally be mentioned. In particular, quasilinear or fully nonlinear equations, nonlinear boundary conditions, unbounded domains and special systems of equations will be commented upon. Boundary-value problems of the form (1.1) and (1.2) are frequently met in research publications. On the applied side, interest in these equations stems from their relevancy in modeling reaction and diffusion phenomena in, among others, population dynamics and genetics, heat conduction, chemical reactions and combustion theory, cf. [72,146,150,12, 30,18]. For example, as a population dynamics model, (1.1) can be interpreted as follows. Assume a population of organisms inhabits a region s Denote by u -- u (x, t), x E s the density of this population at time t. The growth rate of u, u t, depends on several factors. Firstly, it is the birth and mortality rate of the population, that is typically a nonlinear function of u. In heterogeneous environment, this function can also depend on x, and if the environment changes with time, due to seasonal variations, for example, it is also natural to assume that the nonlinearity depends on t periodically. Secondly, the random dispersal (migration) of the population contributes the term L u = Au in the equation. This is obtained via application of Fick's law, or alternatively, using the equations for the Brownian motion. More general second order elliptic operators may be more appropriate here, see [150]. Finally, other forms of dispersal of organisms may be present which are not random, hence not taken account for by the Laplacian. One can think of advection due to physical forces, such as currents in the environment, or the ability of the organisms to sense the environment and thus find a preferred direction of movement. In both cases, an additional term of the form b ( x , t) 9 V u should be included in the equation.
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P. Pold(ik
Combining the above, one arrives at an equation of the form (1.1). Condition (1.2) characterizes the boundary of the environment. For example, the boundary can be lethal to the individuals, thus Dirichlet condition is assumed on ~ S-2, or the environment is isolated from its surroundings in which case Neumann (nonflux) condition is imposed. Equations (1.1) are also attractive for purely theoretical reasons. They are relatively simple and often tractable with several analytical tools (maximum principle, Lyapunov functions, invariant manifolds), yet their solutions exhibit a vast variety of interesting phenomena. The qualitative theory of parabolic equations underwent a rapid development in the last two decades and became wide in scope. Obviously, only some topics can be included in this survey. We have made the selection focusing almost exclusively on asymptotic behavior of bounded global solutions. Our aim is to provide the reader with a picture of the present understanding of the large time behavior of such solutions under various structural assumptions. More specifically, the following material is reviewed. We start with the most general case, assuming only minor regularity on the data in (1.1) and (1.2). Invoking the maximum principle, we put the equations in the context of monotone dynamical systems. Several theorems are then available describing the behavior of typical trajectories, that is, trajectories whose orbits fill an open and dense set in the state space. In particular, we note that bounded solutions of autonomous equations typically converge to an equilibrium, whereas solutions of time-periodic equations approach periodic trajectories of possibly high minimal period. In this context, the existence of asymptotically stable subharmonic solutions is a relevant problem and will be discussed. These results are presented in Section 3. In Section 4, problems on one-dimensional domains are considered. The behavior of all bounded trajectories is described for boundary conditions of type (1.2), as well as for periodic boundary conditions. We devote some space to the study of linear variational equations and their so-called Floquet bundles. These play an important role in the study of perturbations (regular or singular) of one-dimensional problems. All these results depend in a crucial way on properties of the zero number functional that are also recalled in the section. In their simplest form, the zero number properties follow from the maximum principle combined with the Jordan curve theorem in the (x, t) plane. This is of course lost when we pass to higher-dimensional domains, and with it is lost a chance for a universal description of the asymptotics of all bounded trajectories. Indeed, as we show in the last section, there are essentially no restrictions on possible behavior of general solutions. This may not apply, however, if an additional structure is imposed on (1.1). Some specific examples are discussed in Sections 5 and 6. In Section 5, we consider equations on a ball. Assuming spatially homogeneous nonlinearity, the equation becomes equivariant under the natural action of the group of rotations. For Dirichlet boundary condition, a complete space-time asymptotics of positive solutions is found. First, employing the moving hyperplane method, their asymptotic spatial symmetrization is shown. This implies that the large time behavior of positive solutions is "controlled" by a problem with a 1D spatial variable (and its zero number functional). This eventually leads to the asymptotic periodicity of positive solutions. Gradient-like equations, that is, equations admitting a Lyapunov functional are discussed in Section 6. By LaSalle's invariance principle, any bounded solution of such an equation
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approaches a connected set of equilibria. We present two theorems giving sufficient conditions for the solution to actually converge to a single equilibrium. The first one requires the (local) normal hyperbolicity of a set of equilibria- potential elements of the co-limit set. The other one applies to analytic equations with the gradient (not merely gradient-like) structure. We list several application of these theorems. Note that even for the semilinear heat equation ut = A u + f ( x , u ) ,
u = 0,
xES2, t>0,
(1.4)
x E OS2, t > 0,
(1.5)
with S2 being a two-dimensional disk, bounded solutions are not necessarily convergent. We give a more precise statement about this in Section 6. The entire Section 6 is devoted to the discussion of possible dynamics of semilinear parabolic equations. The problems of existence of chaotic dynamics and of trajectories having high-dimensional limit sets is addressed. Also the effect of the dimension of the spatial domain Y2 on the dynamics is examined. There is a method with rather general applicability that has been designed to deal with similar problems. It is based on realization of vector fields on invariant manifolds and works roughly as follows. One considers vector fields generated by equations in question on invariant finite dimensional manifolds. Varying data in the investigated PDE, one tries to discover if there are any restrictions on the class of vector fields one can obtain this way. With no, or not too significant restrictions, one accounts for an "arbitrary" dynamics. We indicate basic ingredients of the method and survey recent results. Preliminary material is collected in Section 2. Here the dynamical systems generated by Equations (1.1) and ( 1.2) are introduced and their compactness and smoothness properties are discussed. We would like to bring the readers attention to Hale's survey [87] that is also devoted to scalar parabolic equations. The focus of that paper is different from ours and it is a good source of additional results and references.
2. State space, smoothness and compactness In this section we introduce dynamical systems defined by problem (1. l) and (1.2). Let s be a bounded domain in ]~U with boundary 012 of class C 2+~ for some 0 > 0. Let L be a second order differential operator N Lu -- ~
(2.1)
aijUxixj,
i.j=l aij satisfy the following requirements. (H 1) (Regularity) aij are H61der continuous on I2.
where
(H2) (Ellipticity) Zi,j=l U ai.j(x)~i~j positive constant c t.
~ Cl 1~]2 (x E ~"2, ~ -- (~1 . . . . . ~N) E R N) for some
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Functions a, b defining the boundary operator B, see (1.3) are assumed to satisfy (H3) /3, y are of class C 1 on Os and either/~ _= 0 with y :/: 0 everywhere or/3 -r 0 everywhere. Our assumptions on the nonlinearity f : (t, x, u, ~) w-~ f (t, x, u, ~) are as follows. (N 1) For some integer m ~> 0, f is continuous on It~ x a"2 x IR x ]t~ u together with all its partial derivatives with respect to (u, ~) up to order m. (N2) f is locally H61der continuous in t and (if m - - 0 ) locally Lipschitz continuous in (u, ~). More specifically, there is a 0 > 0 such that for any bounded set B C I~ X ~ X ~ X I~ x one has
< c(I,-il
~
with some constant c -- c ( B ) > O. (N3) For some r > 0, f is r-periodic in t. Under these assumptions, one can study (1.1) and (1.2) in the context of analytic semigroups and abstract parabolic equations. A standard approach to the existence and regularity of solutions, as found in [6,52,76,94,122,153] and other texts, is outlined by the following scheme. One chooses a basic space X, on which L (under a given boundary condition) defines a sectorial operator, that is, it generates an analytic semigroup. Then, one considers an appropriate scale of Banach spaces X ~, 0 < ot < 1, between the domain of L, X l , and the basic space X. Among these, one chooses a space X ~ such that the Nemitskii operator defined on X ~ by the nonlinearity f is locally Lipschitz continuous. Complementing (1.1) and (1.2) with an initial condition u(., 0) - - u 0 with u0 e X ~, one obtains a well posed problem. For a comprehensive discussion of the spaces on which elliptic operators generate analytic semigroups see [ 122]. One also has several possibilities for the constructions of the intermediate spaces X ~. Typically, fractional power spaces [76,94,122,153], or spaces obtained by an interpolation method [6,52,122] are used. To make a specific choice, we set X = L P (~C2)
with N < p < cx~. Let A be the X-realization of L under boundary condition (1.2); that is, A is the operator u ~-, Lu with domain D(A)-
X l -- {u ~ W2'p(s2) 9u satisfies (1.2)}.
Then A is the generator of an analytic semigroup on X. Let X ~, 0 ~< c~ < l, be the fractional power spaces associated with A. We choose (p 4- N ) / 2 p < ot < 1; the space X ~ is then continuously imbedded in C1 (~-). Thus the Nemitskii operator / /(t, u)(x)-
f(t,x,
u ( x ) , Vu(x))
(u E X ~)
is well defined on R x X ~ with values in X, it is locally H61der continuous in t, locally Lipschitz in u and of class C m in u.
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Write (1.1) and (1.2) in the abstract form:
u, = Au + f (t, u).
(2.2)
For the following existence and regularity results we refer the reader to [94,122]. Given u0 ~ X ~ there is a unique solution u(., t, u0) of (2.2) satisfying the initial condition u(., t) = u0(.). This solution is defined on a maximal interval [0, T), T = T (u0) ~< ec and if T < ec then Ilu(-, t, u0)llx- is unbounded. The function t ~ u(., t, u0) is contained in
c([o, T), x
n C ((0,
X
n C((O,
X').
For any fixed t > 0, the time-t map u0 ~ u(., t, u0) is defined on an open set in X ~ and, when viewed as an X~-valued function, it is of class C m (see [94, Section 3.6]) and compact (see [97, Section III.21 ]). Under the additional assumption that f is H61der continuous in x it follows that u(., t, u0) is a classical solution of (1.1) and (1.2) (cf. [94, Section 3.6], [ 122, Section 7.3]). If Equation (2.2) is autonomous and all solutions are global (that is, their maximal interval of existence is [0, ec)) then the map (uo, t ) ~ S(t)uo = u ( . , t , uo) defines a continuous-time dynamical system (semiflow) on X ~ . This is to say that the map is continuous, S(0) is the identity on X ~ and the semigroup condition S(t + s) = S ( t ) S ( s ) (t, s >~O) holds. If (2.2) is time periodic and the solutions are global, the period map S ( r ) generates a discrete-time dynamical system on X ~ . In this case, S ( n r ) = S ( r ) " (n -- 0, 1 . . . . ). The following standard notation and terminology is used throughout the paper. Let S be a semiflow on a Banach space Y. The trajectory of a point x E Y is the set {(t, S(t)x): t >~ 0}; its orbit is the set O ( x ) = {S(t)x: t >~ 0}. The w-limit set of x is defined by
N
s~>0
so og(x) consists of the limit points y = limS(t,,)x where t,, ~
O(G) = U
cx~. For G C Y, we denote
O(x).
xEG
If S(t)z = z (t >~ 0), z is called an equilibrium. We say of a point x (or of its trajectory or orbit) that it is convergent if S ( t ) x converges as t ~ cx~ to a point z, necessarily an equilibrium. A semiflow is compact if S(t) is compact for each t > 0. For discrete-time dynamical systems, the above notions are defined in an analogous way (S(t) is replaced by S t where S is a map and t a positive integer). We next state explicit conditions that guarantee that all solutions of (1.1) and (1.2) are global and thus the problem generates a semiflow or a discrete-time dynamical system
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842
on X '~. Although these are not the most general assumptions, they are suitable for our purposes. (G 1) There exist an e > 0 and a continuous function c" IR+ --+ IR+ such that
[f(t,x,u,~)[ ~ c(p)(1
+
I~12-~)
(p > 0, (t, x, u, ~) ~ [0, r] x s x [ - p , p] x ]I~N). (G2) There is atc > 0 such that
u f ( t , x, u, O) <~0
((t, x) ~ [0, r] x ~ , lul ~ K).
(G3) If/3 ~ 0 (so that/3 -r 0 everywhere, by (H3)), then y ~> 0. The last condition refers to the functions defining the boundary operator. Note that we only need to assume that y does not change sign on 0s (iii) is achieved by multiplying (1.2) by - 1 if necessary. One shows that any solution u of (1.1) and (1.2) is global in two steps. First, one uses (G2) and (G3) to find an L ~ bound on u. In fact, if k ~> x is any constant such that the estimate
k lu(x,t)l holds for t = 0 then the same estimate hold for any positive t as long as the solution exists. This follows from the fact that k is a supersolution and - k is a subsolution of (1.1) and (1.2) (see [97, Section III.21] for details). Once an L ~ bound is established, the growth condition (G1) implies that u(., t) is also bounded in X ~ (cf. [4], [122, Section 7.3.2]) which implies that it is global. An even stronger conclusion can be drawn from (G 1)-(G3). Set
D-
{~ ~ x ~. - ~ ~< ~(x) ~< ~ (x ~ s2)}.
Then the following statements hold true: 9 for any u0 ~ X ~ there is a 0 -- O(uo) such that for any t > 0 one has u(., t, u0) 6 D, 9 D is positively invariant: u0 ~ D implies u(., t, u0) E D (t ~> 0), 9 S(t)D is relatively compact in X ~ for any t > 0. This strong compactness property implies that (1.1), (1.2) has a compact global attractor (see [86], for example). The first two properties follow by comparison arguments as above, the last property is a consequence of (G1) and the compactness of S(t) (see [97, Section III.21 ]). Another condition under which all solutions are global is that of f being (globally) Lipschitz. In this case S(t) is also defined everywhere and compact for t > 0. This remark is useful when individual bounded solutions or a set of solutions Contained in a bounded set in X ~ are considered. For such solutions all vectors (u(x, t), Vu(x, t)) are contained in a bounded set B0 in R x ]~U and the values of f at the points out of IR x ~- x B0 are irrelevant. We can modify f , without effecting the solutions in question and the standing
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hypotheses, so that f becomes globally Lipschitz. This allows us to assume, without loss of generality, that (1.1) and (1.2) generates a compact dynamical system on X ~ . Existence and regularity theorems similar to those mentioned above are also available, under appropriately modified assumptions, if Y2 is not bounded (cf. [52,94,122]). However, the time-t map is no longer compact in general.
3. The comparison principle and monotone dynamical systems The maximum principle has long been used for various purposes in the study of parabolic equations. For nonlinear equations one usually needs it in the form of a comparison principle which we now state for (1.1) and (1.2). We use the notation from Section 2 and assume (H1)-(H3) and (N 1) and (N2) of that section to be satisfied. COMPARISON PRINCIPLE. Let uo,-~o E X ~ satisfy uo(x) <~-fro(x) (x E ~ )
and
uo ~ -uo.
Then the following inequalities hold as long the indicated solutions exist: (ci) u(x, t, uo) < u(x, t,-fro) (x E ~ , t > 0), (cii) f o r any x E 0 s t > 0 one has either u ( x , t, uo) < u ( x , t,-ao)
or else u(x, t, uo) - u(x, t,-uo)
and
Ou Ou -Ov - ( x , t, uo) > ~ (x ' 't -uo) "
See [75,170,189]. Note that a slight modification of the standard proofs is needed to deal with solutions that may not be classical (see [52]). Thus the dynamical system of (1.1) and (1.2) preserves the pointwise ordering the of the state space X ~ . In this section, we study consequences of this property. One can find several other classed of differential equations (ODEs, FDEs and some parabolic systems) whose semiflows respect an order structure. This naturally calls for an abstraction covering all these equations. In a series of papers from early 80s, Hirsch introduced the notion of monotone dynamical systems and initiated their systematic study (see [100-102] and references therein). Independently, Matano considered a similar class of semiflows that he called order preserving (see [ 132-134]). Triggered by their work, many other researchers contributed to the theory. For a comprehensive treatise of continuoustime systems the reader can consult the monograph of Smith [186]. The monograph of Hess [97] contains several early results dealing with discrete-time systems with special attention to time-periodic parabolic problems. In accordance with the general focus of this survey, we confine our account to results on asymptotic behavior of solutions. In particular, we discuss a very successful area of the theory of monotone dynamical systems - a description of the behavior of "most" bounded
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trajectories. We present relevant results for both discrete and continuous time systems. Abstract results are given without proofs, sometimes short sketches are included to help the reader's intuition. Applications to (1.1) and (1.2) are given with more details. We first introduce basic notation and definitions. In the whole section Y is an ordered Banach space with norm I1" II and order cone Y+. Recall that an order cone is a closed convex cone such that Y+ M ( - Y + ) = {0}. We assume that Y is strongly ordered which means that int Y+, the interior of Y+, is nonempty. For x, y E Y we write
x<<, y x < y x<
ify--xEY+, if x ~< y and x ~- y, if y - - x E int Y+.
The reversed signs are used in the usual way. Two points are said to be related (or ordered) if they are related by ~< or ~>. The notation A ~< B (similarly for < and <<) between two sets means that x ~< y whenever x 6 A and y E B. A mapping F : D ( F ) C Y --+ Y is said to be monotone if x, y E D ( F ) and x ~< y imply F(x) <<,F(y). It is called strongly monotone if x, y E D ( F ) and x < y imply F(x) << F(y). A linear strongly monotone map T is also called strongly positive; it is equivalently characterized by the property T(Y+ \ {0}) C int Y+. A semiflow (x, t) ~ S(t)x on Y is strongly monotone if S(t) is strongly monotone for each t > 0. Let us show that (1.1) and (1.2) belongs to this class of dynamical systems. Recall that the fractional power space Y = X ~ is continuously imbedded in C l ($2). Define the positive cone by
and observe that it has nonempty interior given by intY+ := {u E Y: u(x) > 0 (x E ~-)} in the case of Neumann or Robin boundary condition, and int Y+ "--
u E Y" u(x) > 0 (x E $2) and ~v(X) < 0 (x E 0I2)
in the case of Dirichlet boundary condition. The comparison principle implies that the time-t map of (1.1) and (1.2) is strongly monotone for any t > 0 (note that if fl -fi 0, the boundary condition (1.2) excludes the second possibility in (cii)). We next present a theorem on the behavior of typical trajectories for semiflows. We say that a point x E Y (or its trajectory or orbit) is quasiconvergent (relative to a semiflow) if O(x) in relatively compact and o~(x) consists of equilibria. We say that a set D C Y has the monotone approximation property if for any x E D there is a monotone (either increasing or decreasing) sequence xn 6 D converging to x. Any nonempty open set has this property due to strong orderedness.
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Parabolic equations
THEOREM 3.1. Let S be a compact strongly monotone semiflow on Y. Let D C Y be an open set (or more generally, a set with the monotone approximation property) such that f o r some ~ > 0 O ( S ( 6 ) ( D ) ) - {S(t)D: t ~ 6] is bounded. Then the set o f quasiconvergentpoints contains an open and dense subset o f D.
This theorem, in a slightly weaker form, is due to Hirsch [ 102]. The proof of a stronger version, similar to the one above, was given by Smith and Thieme (see [186,187]). It is noteworthy that Theorem 3.1 contains no smoothness assumption on S(t). We apply the theorem to autonomous problems (1.1) and (1.2): COROLLARY 3.2. Let hypotheses (H1)-(H3), (N1), (N2) be satisfied (with m possibly equal to O) and let f be independent o f t. Let D C X ~ be an open set (or a set with the monotone approximation property) such that
(B)
O ( D ) ---- {u(., t, uo): u 0 6 D ,
t>~0}
is bounded in X ~
(in particular, f o r any uo ~ D the solution u(., t, u0) is global). Then there is an open and dense subset G o f D such that f o r any uo ~ G the solution u (., t, uo) is quasiconvergent. If hypotheses (G1)-(G3)are satisfied then (B) can be omitted (and one can take D ---- X~).
PROOF. As remarked at the end of Section 2, modifying f outside O(D), we may assume that (1.1) and (1.2) generates a compact strongly monotone semiflow on X ~. The first conclusion then follows directly from Theorem 3.1. If (G1)-(G3) are satisfied then all solutions are global (in this case no modification of f is needed) and O ( S ( 1 ) ( D ) ) is bounded in X ~ for any bounded set D. We thus obtain the conclusion for any D bounded and consequently for D -- X ~ . D We list three fundamental properties of strongly monotone compact semiflows that lead to the conclusion of Theorem 3.1. We assume here that x, y are points in Y with relatively compact trajectories. CONVERGENCE CRITERION. If the orbit O ( x ) contains two distinct related points then co(x) is a singleton; that is, x is convergent. NONORDEREDNESS PRINCIPLE. co(x) does not contain any two distinct related points.
LIMIT SET DICHOTOMY. If y > x then either co(y) >> co(x) or else co(y) = co(x) and this joint limit set consists of equilibria (that is, x, y are quasiconvergent). While the proof of the first two properties is an easy exercise, the limit set dichotomy is nontrivial (see [ 102] for the original proof and [ 186] for an improved version). To give the reader some intuition about Theorem 3.1, we sketch the proof of the density of the set of
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846
quasiconvergent points. Denote this set by Q. We fix any x 6 D \ Q and show that it can be approximated by a quasiconvergent (even convergent) point y. By the limit set dichotomy, for any y E D with y > x one has co(y) >> co(x). Moreover, using compactness arguments and nonorderedness of co (x), one can show that the Hausdorff distance between these limit sets is bounded below by a positive number e0 independent of y > x. Now, if y > x is close to x then, by continuity, its trajectory stays near the trajectory of x for a long time. In particular, S(t)y gets close to co(x) for some t = to. Of course, S(t)y has to eventually get close to its limit set co(y) which is above co(x)" co(y) >> co(x), and of distance at least e0 from co(x). It follows that the orbit of y contains two points S(to)y and S(tl)y, near co(x) and co(y), respectively, such that S(to)y << S(tl)y. By the convergence criterion, y is convergent. If there is no point y 6 D with y > x that is sufficiently close to x, one uses analogous arguments for y < x (referring to the monotone approximation property of D). This shows the density of Q. We remark that no result analogous to Theorem 3.1 is available for compact strongly monotone discrete-time dynamical systems if they are merely continuous (or even Lipschitz). Whether any characterization of typical dynamics is possible in this case or not is still an open problem. It is a natural question to ask, under what conditions can quasiconvergence in Theorem 3.1 be replaced by convergence (it is a folklore knowledge that one does need additional assumptions although this has not been documented by explicit examples). Here are two easy sufficient conditions involving the set of all equilibria (further this set is denoted E). If E is either totally disconnected (that is, all its connected components are singletons) or totally ordered (that is, any two equilibria are related) then quasiconvergence is the same as convergence. This follows from the connectedness of the limit sets in the former case and by the nonorderedness principle in the latter. It is more interesting that typical convergence occurs, without any a priori assumptions on the set of equilibria, if the semiflow is of class C l . We now present results of this type for differentiable monotone dynamical systems. We start from maps and then obtain the corresponding theorem for semiflows as a corollary, although, historically speaking, this is not the chronological order (see the bibliographical notes below). Consider a compact C l map F" Y ~ Y. We denote by Fix(F) the set of all fixed points of F. For a z 6 Fix(F) let Q(z) denote the spectral radius of F'(z). We say that z is
linearly stable linearly unstable neutrally stable
if Q(z) < 1, if Q(z) > 1, if Q(z) - 1.
By standard stability results, a linearly stable fixed point z is stable in the nonlinear sense: given any neighborhood U of z there is another neighborhood V such that F n (V) C U, n = 1,2 . . . . . Also z attracts all orbits starting in a neighborhood of itself. On the other hand, for z to be stable it is necessary that it be at least neutrally stable (that is, neutrally stable or linearly stable). A point z 6 Y is periodic (relative to F) of period p if F p (z) -- z. A periodic point is said to be linearly stable (and similarly for instability and neutral stability) if it is such for the map FP.
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THEOREM 3.3. Let F : Y ~ Y be a compact C 1 map such that F ' ( x ) is strongly positive f o r any x E Y. Let D C Y be an open subset such that O ( D ) = U,=0.1 .... F " ( D ) is bounded. Then the following two assertions hold: (i) There is an open and dense subset G C D such f o r any x E G there is a periodic point z that is at least neutrally stable and such that co(x) = O(z). (ii) There is a Po -- po(D) such that if z E clO(D) is a periodic point that is at least neutrally stable then its minimal period is bounded above by Po. The first result, assertion (i), is due to Polfi6ik and Tereg6fik (see [164,165]). Assertion (ii) was proved by Hess and Polfi6ik [98]. In these papers, a slightly higher regularity, F E C l'~ for some 0 > 0, was assumed. Later TeregSfik [ 197] gave different proofs with FEC l Statements (i) and (ii) combined imply that if F is replaced by F k with a sufficiently large k then a typical trajectory with initial condition in D is convergent. As we discuss bellow, this may not be true for the original map F. However, typical trajectories are convergent if F is a time-t map of a semiflow. COROLLARY 3.4. Let S and D be as in Theorem 3.1. Assume in addition that f o r some to > c~the map S(to) is of class C l and its derivative DS(to)(x) is strongly positive f o r an), x. Then the set of convergent points contains an open and dense subset of D.
PROOF. Let F = S(to). By Theorems 3.1 and 3.3, there is an open and dense subset G C D such that for any x e G the limit set co(x) of x relative to the semiflow consists of equilibria and, at the same time, the discrete orbit of x relative to F approaches a single periodic orbit. This readily implies that co(x) is a single equilibrium. D We remark that in applications the assumption that DS(to)x be strongly positive is hardly a significant additional restriction. For example, if S(t) is the time-t map of (1.1) and (1.2), then DS(to)x is the time-t map of a linear variational equation. One shows its strong positivity using the maximum principle in much the same way as in the verification of strong monotonicity of S(t) (which is the comparison principle formulated above). We have the following corollary on (1.1) and (1.2). COROLLARY 3.5. Let hypotheses (H1)-(H3), (N1)-(N3) be satisfied with m >~ 1. Let D be an open set in X ~ such that (B)
O ( D ) -- {u(., t, uo)" uo E D, t >l O}
is bounded in X ~.
Then there exist an open and dense subset G C D and a positive integer qo such that f o r any uo E G there is a solution c~(x, t) of(1.1) and (1.2)with the following properties: (i) ~ ( x , t) is qr-periodic in t f o r some q <~qo, (ii) ~b(., 0) is at least neutrally stable as a periodic point o f the period map uo w-~
u(., r, u0), (iii) I[u(', t, u0) - ~b(., t)llx- --~ 0 as t --+ ec, (iv) if f is independent of t then so is ~ (hence it is an equilibrium).
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If hypotheses (G1)-(G3)are satisfied then (B) can be omitted. PROOF. As in the proof of Corollary 3.2, we may assume that all solutions of (1.1) and (1.2) are global. Then the period map F :u0 w+ u(-, r, u0) satisfies the hypotheses of Theorem 3.3. Let G be the open and dense set given by that theorem and let u0 6 G. Then F n ( u o ) = u ( . , n r , uo) approaches the orbit {~b0,F(~b0). . . . . Fq(~b0)} of some periodic point of F which is at least neutrally stable. Here q is the minimal period of F n (~b0); it may be assumed bounded above by a constant q0 independent of u0 (by statement (ii) of Theorem 3.3). Taking the solution 4) of (1.1) and (1.2) with the initial condition 4) (., 0) = 4~0, we see that it satisfies assertions (i) and (ii) of Corollary 3.5. The convergence of the discrete orbit and the continuity with respect to initial conditions gives (iii). The fact that 4) is an equilibrium in case (1.1) is autonomous follows from Corollary 3 . 4 . 5 Let us now return to the abstract results in Theorem 3.3 and Corollary 3.4. Although they are in a sense analogous to one another, the result for continuous-time systems is much easier to prove and was in fact proved much earlier. See [ 186,188] for the proof of Corollary 3.4 (not using Theorem 3.3); earlier less general results were proved in [117, 147,154,206]. In the following remarks we want to illuminate a few major differences between discrete and continuous time strongly monotone systems. Doing that, we also want to mention a theorem that is crucial for the proof of typical convergence and is of independent interest. We start by recalling the Krein-Rutman theorem; the proof can be found in [3,54,113, 196] and many other texts. THEOREM (Krein-Rutman). Let T be a compact strongly positive operator on Y. Then its spectral radius p ( T ) is positive and it is a simple eigenvalue of T. Furthermore, there is a decomposition Y = Y I OY2 of Y into closed subspaces such that the following statements hold true: (i) Y1 is the one-dimensional eigenspace of T corresponding to the eigenvalue p(T); it is spanned by an eigenvector v >> 0. Y2 does not contain any positive vector: Y2 n Y+ = {o}.
(ii) Y2 (and of course also YI ) is invariant under T: T Y2 C Y2 9 (iii) The spectrum of the restriction TlY 2 is a compact subset of {)~ ~ C: I)~l < p(T)}. This theorem is very useful in analysis of C 1 maps whose derivatives at any point are strongly positive. More precisely, it is useful for local analysis, or investigation of orbits near fixed points. In particular, one can show the existence of invariant manifolds modeled on the invariant subspaces Y1, Y2 and other useful properties (cf. [141,155,185]). Now, when the map in question is a time-t map of a strongly monotone semiflow, one knows, by Theorem 3.1, that after a transient period of time, a typical orbit stays close to a set of
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equilibria. Local analysis gives a crucial information on the behavior of such orbits which eventually leads to the conclusion of Corollary 3.4. The situation is quite different with discrete-time systems as one has no a priori information on the structure of limit sets of typical trajectories. Therefore a generalization of the Krein-Rutman theorem is needed that could be used for the linearization along a general compact orbit, not just along fixed points or periodic orbits. The following theorem on the existence of exponential separation is such a generalization. By Y* we denote the dual Banach space of Y. We say a functional ~ E C* is strictly positive if g(v) > 0 for any v > 0. The set of strictly positive functionals is denoted by C*THEOREM 3.6. Let K C Y be a compact set and F : K --+ K a homeomorphism. Let {Tx }xEK be a f a m i l y ofstrongly positive compact operators on Y such that the map x w-~ T~is continuous in the operator norm. Then there are subspaces Ylx, Y2x (x E K ) o f Y with the following properties: (i) Ylx = span{vx } where vx >> 0 (x E K ) and x ~-+ vx : K --~ Y is continuous. (ii) Y2x -- ker gx where gx E C* and x w+ ~.~. " K --+ Y* is continuous. In particular, Y2x A Y+ = O, and, combining this with (i), Y = Ylx 9 Y2x (x E K). (iii) (lnvariance) T~-(YI.r) = YIF(x)and Tx(Y2.~-) C Y2F(x) (x E K). (iv) (Exponential Separation) There are constants M > O, 0 < g < 1, such that f o r any x E K and any w E Y2x with 11w l! = 1, one has
II
II
Mg/"
II
II
where T ~ ' = TF,,-,c~)TF,z-2(x)... TF(x)T~-.
In a particular case, when K consists of just a single element, the above theorem is equivalent to the Krein-Rutman theorem. Theorem 3.6 was first proved by Polfi~ik and Tereg~fik [165] under the additional assumption of F being one-to-one on K. The extension to maps that are not necessarily one-to-one is due to Tereg~fik [197]. Note that if F is the period map of (1.1) and (1.2) then it is one-to-one by backward uniqueness. However, some functional differential equations with monotone semiflows do not enjoy this property and the extension is useful there. In finite dimensions, Theorem 3.6 is due to Ruelle [176]. Mierczyfiski [ 139] proved an early result in infinite dimensions under severe restrictions on the structure of the order cone. For recent applications of the theorem see [ 142,143,183]. Theorem 3.6 is typically applied to linearizations of C l maps: if F is as in Theorem 3.3 and K is a compact invariant set of F, then T,- = D F ( x ) is a family satisfying the hypotheses. This allows one to employ the linearization in the study of the behavior of trajectories of F near a general compact invariant set K. For example, one can introduce the first Lyapunov exponent along an orbit {F" (z)} C K by X 1 (z) - lim sup
log IlD(F")(z)v:ll
II-----> ~X2
where v: is the positive vector as in Theorem 3.6, and use it to define the linear stability and instability. Similarly as in Osceledec's theorem in the ergodic theory, one can show
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that for "most" points z e K, ~l(z) is actually the limit not just the superior limit. Such points are called regular. If K -- co(x) for some point x, one can further show that if co(x) contains a regular point z which is at least neutrally stable in the sense that )~1(z) ~< 0, then z is a periodic point and co(x) is equal to the orbit of z. On the other hand, if each regular point z e co(x) is linearly unstable; that is, )~l (z) > 0, then co(x) cannot attract an open set (and therefore cannot attract two related points). These results are proved in [ 164] as crucial ingredients of the proof of Theorem 3.3. In [ 197], Theorem 3.3 is proved in a different way, the key step being an application of Theorem 3.6 to an auxiliary family of compact strongly positive operators. Smooth strongly monotone maps have another interesting property closely related to Theorem 3.3(ii). It says that if F and D are as in that theorem and p0 satisfies (ii), then (ii) holds (with the same p0) for any map F that is C 1 close to F (see [98]). Thus the bound on the minimal periods of stable periodic orbits is not increased under small perturbations. This is especially interesting if the unperturbed map has no stable periodic orbits, except for fixed points, so that most of its orbits are convergent. The same is then true for the perturbations. By Corollary 3.4, this applies in particular to small perturbations of autonomous equations. See [98,165] for other applications. Let us now discuss the opposite situation, when F has stable periodic points with high minimal periods. In the context of r-periodic differential equations, the situation occurs when there exist periodic solutions of period nr, where n > 1 is an integer (these are called s u b h a r m o n i c solutions). Examples of parabolic equations that admit such solutions where found independently by Tak~ie [194,195] and Dancer and Hess [49]. Their results in particular imply that for some time-periodic nonlinearity f -- f ( t , x , u) the equation ut -- A u + f ( t , x , u ) ,
x E S-2, t > O
(3.1)
(under Neumann or Dirichlet boundary conditions) has linearly stable subharmonic solutions. Although they work with specially chosen domains $2, the result appears to be true for any domain in R N, N ~> 2. The situation is quite different when Equations (3.1) with spatially homogeneous nonlinearities, f = f (t, u), are considered. The existence of stable subharmonics depends on the domain then. As noted in [ 165], the Neumann problem cannot have stable subharmonic solutions if $2 is convex. This follows from a result of Hess [96] saying that any stable periodic solution is necessarily spatially constant and thus satisfies ut = f ( t , u) (see [31, 130] for earlier results of this type for equilibria of autonomous equations). As there are no subharmonic solutions of the scalar ODE, there are no stable subharmonic solution of the reaction-diffusion equation. On the other hand, for some nonconvex domains, spatially homogeneous equations with linearly stable subharmonic solutions have been constructed (see [ 166]). Generally speaking, the strong monotonicity structure does not allow one to obtain information on the asymptotic behavior of every trajectory (as opposed to a typical trajectory). This is well demonstrated by the realization results discussed in the last section. There are interesting additional assumptions, however, which imply that all trajectories are convergent. For example, if for a compact strongly monotone map one is able to show that all fixed points are stable (in the nonlinear sense) then all bounded trajectories are convergent.
Parabolic equations
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See [1,2,97,191,193] and references therein for theorems of these type and applications. No smoothness hypotheses are needed for such theorems. Often, in monotone dynamical systems, the explicit knowledge of a distinguished class of solutions (equilibria or periodic solutions) allows one to describe the global dynamics in detail; see [22,55,97,208] for examples and further references. Let us briefly discuss a few other results on monotone dynamical systems pertinent to the asymptotic behavior. Revisiting the convergence result of Corollary 3.4, we mention that a more specific description is available on the meager set D \ G containing nonconvergent trajectories. One can show that it is contained in the union of countably many manifolds of codimension 1 that form boundaries of the domains of attraction of stable fixed points or discrete periodic orbits (see [ 144,155,191,193,197]). Our next remark is concerned with the manner typical trajectories of C l strongly monotone semiflows converge to their limit equilibria. It has been proved that if there is no simple ordered arc of equilibria then typical trajectories are eventually monotone in time (see [140,141,155]). The reason is that most trajectories approach their limit equilibrium tangent to a positive or negative eigenvector given by the Krein-Rutman theorem. Absence of simply ordered arcs of equilibria is a generic property of many classes of monotone dynamical systems (cf. [156]). Moreover, what is perhaps more interesting, one can ensure it assuming that the nonlinearity in the equation is analytic and the set of fixed points is bounded (see [107,108,187,207,213] for results of this type). Several authors have studied the asymptotic behavior of strongly monotone dynamical systems with an additional structure. Systems equivariant under a monotone action of a continuous symmetry group were studied in [145,192] and more recently in [136, 137]. A representative theorem says that a typical trajectory approaches a set of symmetric points. This result holds for systems with continuous as well as discrete time and does not depend on any smoothness assumptions. Applications include reaction-diffusion equations (3.1) where f -- f ( t , u) is independent of x and locally Lipschitz, and ~ is a rotationally symmetric bounded domain (a ball or an annulus). One obtains that a typical solution of such an equation under Dirichlet or Neumann boundary condition is asymptotically radially symmetric. To conclude the section we mention other types of equations that define strongly monotone dynamical systems on an appropriate Banach space. Scalar parabolic equations of much more general form than (1.1) have been studied from this point of view. For example, see [ 102] for a discussion of compact strongly monotone semiflows generated by quasilinear equations. One can also allow for nonlinear Robin boundary conditions, however, in that case the above abstract results do not apply directly. One has two possibilities to treat nonlinear boundary conditions: either consider a semiflow on a Banach submanifold (given by the nonlinear boundary condition) of a strongly order space, or, as in [5], consider the semiflow on a larger Banach space in a scale of interpolation spaces where the boundary conditions do not take place. In the first case, one needs to adjust the "linear" setting of the above results and in the second case one has to allow for a more general notion of strong monotonicity (Matano's definition of strongly order-preserving systems is more appropriate here). In either case modifications are straightforward. Strongly monotone semiflows or maps are also generated by certain types of systems of reaction diffusions. Cooperative systems of any number of equations or competitive
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systems of two equations are such examples (cf. [97,101,186]). For a discussion of strongly monotone maps generated by delay equations or reaction-diffusion equations with delays we refer the reader to [ 186] and [ 127,211 ].
4. One space dimension
The results stated in the previous section show that typical bounded trajectories of (1.1) and (1.2) exhibit a relatively simple asymptotic behavior- convergence to equilibria or periodic solutions. Unless additional structure of the equation is assumed, one cannot expect such a simple description to be valid for all bounded solutions (cf. Section 7). In this section, we discuss one-dimensional problems; that is, we assume N -- dim s to be equal to 1. In this case, the dynamics of bounded solutions is much better understood. A crucial property that distinguishes 1D equations among all problems (1.1) and (1.2) is that the Jordan curve theorem applies in the (x, t) space. In conjunction with the maximum principle it implies an interesting p r o p e r t y - the nonincrease of the zero number. In this section we discuss its consequences on asymptotics of bounded solutions. For a continuous function w:[0, 1] --+ IR let z(w) denote the number (possibly infinite) of zeros of w in (0, 1). We say that a differentiable function w has a multiple zero at x0 E [0, 1] if w(xo) = w'(xo) = O. Consider a linear parabolic equation of the following form Vt = V x x -+- b(x,
t)Vx + c(x, t)v,
x 6 (0, 1), t 6 (0, T),
(4.1)
where b, c are functions in L ~ ( ( 0 , 1) x (0, T)) (T ~< ~ ) . We couple (4.1) with separated boundary conditions fl(X)Vx + y ( x ) v = O ,
x=0,1,
(4.2)
where (fl(x)) 2 + (y(x)) 2 > 0 (x = O, 1) or with periodic boundary conditions v(0, t) = v(1, t),
vx(O, t) = Vx(1, t).
(4.3)
LEMMA 4.1. Let v(x, t) be a solution of(4.1) and (4.2) or of(4.1) and (4.3). If v ~ 0 then the following statements hold true: (Zl) z(v(., t)) is finite f o r any t > 0; (Z2) z(v(., t)) is nonincreasing in t; (Z3) (Diminishing Property). If v(., to) has a multiple zero in [0, 1]for some to ~ (0, T), then z(v(., t)) drops strictly at t = to, that is, f o r any 0 < tl < to < t2 < T one has z(v(., tl)) > z(v(., t2)). Statement (Z2) is relatively simple, it follows by an application of the maximum principle in two-dimensional domains. Results of this kind can already be found in a paper of Nickel from 1962 (see [149]), but they became widely used in the study of nonlinear equations mainly due to Matano's work [ 131 ]. The other two statements were first proved
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by Angenent [8] under stronger regularity requirements on the coefficients. A more recent theorem of Chen (see [36]) only needs b, c ~ L ~ ( ( 0 , 1) x (0, T)) and in addition it gives a detailed description of the nodal set of solutions (see [ 11,40,61 ] for related results on the zero number). Let us now consider nonlinear equations ut=uxx+
f(t,x,u,
ux),
xr
t>0,
(4.4)
assuming first the separated boundary conditions fl(x)vx + y ( x ) v = O,
x = O, 1
(4.5)
((fl(x)) 2 + (V(x)) 2 > 0 (x = 0, 1)). The nonlinearity f is assumed to satisfy hypotheses (N 1)-(N3) of Section 2. Applications of Lemma 4.1 in the study of the nonlinear problem are facilitated by the observation that the difference v of any two solutions of (4.4) and (4.5) is a solution of a linear problem (4.1) and (4.2). Obviously, since z(v(., t)) is finite, it cannot drop infinitely many times. This implies, in view of the zero number diminishing property, that beginning with some t > 0, v(., t) has only simple zeros. We show a typical way this property is used when giving the proof of the convergence result of Theorem 4.2. We use the notation of Section 2: X ~ is the state space of (4.4) and (4.5), and it is continuously imbedded in C~[0, 11. We stress that in Theorems 4.2, 4.3 we assume that the nonlinearity f is merely locally Lipschitz in u, ux. This assumption is sufficient for the equation for differences of any two solutions to have L ~ coefficients, hence Chen's zero number results [36] can be used. Originally, the theorems were published with the assumption of f being of class C l (at least). THEOREM 4.2. Let hypotheses (N 1)-(N3) be satisfied (with N -- 1). Let u(-, t), t > O, be any solution o f (4.4) and (4.5) that is global and bounded in X ~. Then there is a solution cp(x,t) of(1.1) and (1.2) that is r-periodic in t and such that ]u(.,t, u o ) - c P ( . , t ) lx~ --+ O
ast--+ ec.
If f is independent of t then so is 4) (it is thus an equilibrium).
This theorem has been proved in a more general context (allowing separated nonlinear boundary conditions and, with some restrictions, fully nonlinear equations) by Brunovsk~ et al. [29]. For spatially homogeneous equations, the result was obtained earlier by Chen and Matano [37] together with an additional symmetry information on the limit periodic solution. For autonomous equations, Zelenyak [212] proved the convergence, as early as in 1968, using his construction of a Lyapunov functional. Later Matano [ 129] gave a different proof (see also [89]). We include the proof of Theorem 4.2 for Neumann boundary condition as an illustration of the use of the zero number.
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854
PROOF OF THEOREM 4.2 FOR f l ( x ) ~- 1, F(x) = 0. The assertion is equivalent to the convergence of the discrete trajectory {u(., nr)}n to a fixed point of the period map. We first prove that the real sequence u (0, n r) is convergent. As X c~ ~-~ C l [0, 1], this sequence is bounded. Take the following difference of two solutions of (4.4) and (4.5): v(., t) := u(., t + r) - u(., t). As remarked above, Lemma 4.1 implies that v(., t) has only simple zeros for all sufficiently large t. In particular, by Neumann boundary condition, v(0, t) = u(0, t 4- r) - u(0, t) is nonzero for all sufficiently large t. This implies that the bounded sequence u(., n r) is eventually monotone, therefore convergent. Let s~ denote the limit of u (0, n r). Let COd(u) C X ~ denote the CO-limit set of the discrete trajectory {u(., nr)},z. Clearly, if 4) ~ COd(U) then 4~(0) is a limit point of u(0, n r ) thus
r
=~.
Let r q~2 E COd(t/) be arbitrary. We show they are equal. We take these functions as initial conditions for (4.4) and (4.5), denoting the corresponding solutions by Ul (x, t), u z(x, t), respectively. By the invariance of CO-limit sets under the period map, we have ui (., n r ) 6 COd(t~) for any positive integer n. Hence, U l (0, n r ) = u2(0, n r ) = ~ for any n. Now, take the difference of these two solutions:
v(x, t) = Ul (x, t) - u2(x, t). We have shown that v(0, n r ) = 0 hence, by Neumann condition, v(x, n r ) has a double zero at x = 0 for any n = 0, 1 . . . . . By the zero number diminishing property this is only possible if v = 0, in particular, t~l = ~2. This shows that COd(U) is a singleton thus u(., n r ) is convergent as asserted. [2 With periodic boundary conditions the zero number can still be applied. It does not yield the convergence of bounded solutions, however. To appreciate the difference in applications of the zero number to separated boundary conditions and periodic boundary conditions, consider the autonomous case f = f ( x , u, Ux). Let u be a solution of (4.4) with separated boundary conditions (4.5) or with periodic boundary conditions u(0, t) = u(1, t),
Ux(0, t) = Ux(1, t).
(4.6)
Then v ( x , t ) : = u t ( x , t ) solves a linear equation (4.1). By Lemma 4.1, the vector (v(O, t), Vx (0, t)) is nonzero for all sufficiently large t. In the case of separated boundary condition this implies that for a sufficiently large T the orbit {u(., t): t ~> T} continuously imbeds in R. For example, in the Neumann case, we have v(0, t) = ut (0, t) :~ 0 for large t (cf. the above proof), thus the map u w-~ u Ix=0 gives an imbedding of the orbit in •. In the case of periodic boundary conditions, no relation between v(O, t), vx(O, t) is postulated. Here, the vector map u ~ (u, Ux)lx=0 is to be considered and the imbedding of the orbit in the plane appears an optimal property one can try to prove. Verifying this property for orbits in the w-limit set, Fiedler and Mallet-Paret consequently obtained the following Poincar6-Bendixson-type theorem (see [67]).
Parabolic equations
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THEOREM 4.3. Let hypotheses (N1)-(N3) be satisfied and let f be independent o f t . Let u(., t), t > O, be any solution of(4.4) and (4.6) that is global and bounded in X ~. Then the co-limit set o f u is either a single periodic orbit or it consists o f equilibria and connecting (homoclinic or heteroclinic) orbits. For time-periodic equations with periodic boundary conditions one does not expect a simple asymptotics of solutions in general. In fact, Fiedler and Sandstede [71,181 ] have shown that chaotic behavior exhibited by some time-periodic planar vector fields can also be found in time-periodic equations with periodic boundary conditions. One can still prove that each co-limit set is imbedded in the plane, however. THEOREM 4.4. Let hypotheses (N1)-(N3) be satisfied with m >~ 1. Let u(., t), t > O, be any solution of(4.4) and (4.6) that is global and bounded in X ~. Then the co-limit set co~l(u) o f the discrete trajectory {u(-, nr)}, imbeds in ]K2; that is, there is a one-to-one continuous map M from co~t(u) into ~2. This theorem is due to Tereg~fik (see [198]). The proof is not as simple, as our earlier remarks could suggest (the proof is much easier in the autonomous case). Of course, for discrete trajectories, there is no consequence like Poincar6-Bendixson theorem. If f does not depend on x, f = f (t, u, ux), the equation is equivariant with respect to the natural action of the symmetry group S l: O ~ u(. - O) : S ! --+ X ~.
In this case, a more specific description of co~/(u) has been found by Fiedler and Sandstede: coj (u) is either a periodic orbit or a circle. In either case, cocl(u) is a subset of the group orbit 4~(" - 0): 0 ~ S l of a point 4~ c X ~ (see [71]). In particular, in the autonomous case, any periodic solution is a rotating wave; that is, it equals ~o(x - ct) for some 1-periodic function ~0 and constant c. The latter result was proved earlier by Massatt [128] and Matano [135] (see also [ 11,35,37]). It is an interesting question whether some of the above results are valid if the equation is perturbed so that the new equation does not have the form (4.4). As a typical example, one can consider a perturbation obtained by adding a nonlocal function eg(x, t, u, Ft) to the right-hand side of (4.4). Here e is a number, g is a smooth function r-periodic in t, and t~ is the spatial average of u" ft - f~ u d x . The zero number functional, considered along differences of solutions of such a perturbed equation, has no longer the nonincrease property in general. In fact, the dynamics of the nonlocal equations appears to be rather complicated (see [68,74,163]). However, if e is sufficiently small one can establish convergence properties similar to those formulated above for the local equations. There two different approaches to such perturbation problems. The method proposed by Chen and Polfi~ik [39] requires separated boundary conditions and is based on a study of Morse decompositions for the period map of (4.4) and (4.5). As shown in [39], there exist Morse decompositions with invariant normally hyperbolic curves as Morse sets. This robust structure can be effectively used to study perturbations of (4.4) and (4.5).
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A different approach was used by Tere~6~ik in his thesis [198]. There he was able to prove that any perturbation of (4.4) and (4.5) or of (4.4) and (4.6), which is small in an appropriate C l sense, inherits the properties of the unperturbed problem as stated in the above theorems. In particular, in the case of separated boundary conditions he obtains the convergence of discrete trajectories to fixed points, and in the case of autonomous equations with periodic boundary conditions he proves the PoincarT-Bendixson theorem. The method of [ 198] is based on another robust structure of the one-dimensional problems, the so called Floquet bundles and exponential separation for linearization of (4.4). We give some details on this interesting structure. It is an extension of the classical SturmLiouville theory which goes on similar lines as the extension of the Krein-Rutman theorem presented in the previous section. We formulate a theorem on Floquet bundles for linear equations (4.1) with b(x, t) = 0 (in the case of Dirichlet and periodic boundary conditions this can be achieved by means of a suitable substitution, cf. [44]). In the theorem 13 denotes the set of all functions c 9 L ~ ( ( 0 , 1) • IR) whose L ~ norms are not greater than a fixed constant R. The set/3 is endowed with the weak* topology of L ~ ( ( 0 , 1) • R). By a direct sum of an infinite set of subspaces of X ~ we mean the closure of the union of all finite sums of those spaces. THEOREM 4.5. Consider a linear problem (4.1) and (4.2) with c 9 13 and b = O. There are subspaces Yi (t; c), c 9 t 9 R, i = 0, 1 . . . . . of X ~ with the following properties: (i) Yi(t; c) = span{vi(.,t; c)}, where vi(.,t; c) is a solution of (4.1) and (4.2) on R with z(vi(., t; c)) =_ i; the map c ~-+ v i (., t" c ) "]3--+ X ~ is continuous (t 9 IR). (ii) One has oo
X ~ -- @
Yi (t; c)
( t 611~, c 6 1 3 ) .
i=0
(iii) (Invariance) Let k
Z~-(t;c)--(~Yi(t'c) i:0
oo
and
Z + (t; c) = @
Yi (t; c).
i:k+l
Then the space Z-~(t; c) is invariant in the sense that f o r any vr e Z~-(r; c) there is a solution v(., t), t e R, of (4.1) and (4.2) with v(., r) - vr such that v(., t) e Z~- (t; c) (namely, v is an appropriate finite linear combination of the solutions vi(., t; c)). This solution satisfies z(v(., t; c)) <~ k (t e R). The space Z+(t; c) is positively invariant: f o r any vr 9 Z + ( r 9 c) the solution v o f ( 4 . 1 ) a n d (4.2)with v(. , r) - vr satisfies v(. , t) 9 Z + ( r 9' c) (t ~> r). Moreover, z(v(. , t; c)) ~ k 4- 1 k (t ~> r).
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Parabolic equations
(iv) (Exponential Separation) There are constants M > 0, ~ > 0 such that if v(., t) Z k (t" c) and w(., t) ~ Z k+(t; c) are nontrivial solutions of(4.1) and (4.2) (t ~> r) then Ilw(',t)ll ~ Me_Z(,_s)Ilw(',s)ll Ilv(', t)ll IIv(', s)ll
(t ~> s ~> r)
where I1" II is the norm in X ~.
A similar result holds for periodic boundary conditions in which case the spaces Yi (t; c), i = 1, 2 . . . . . are two-dimensional and Y0(t; c) is one-dimensional. The standard Sturm-Liouville spectral theorem is a consequence of Theorem 4.4 where we take an autonomous equation (4.1). Theorem 3.6 presented in the previous section can be used to obtain the first of these spaces, Y0(t; c), and its invariant complement Z~-(t" c) (see [142,143,183]). This decomposition with exponential separation is available in any dimension. The complete decomposition into one or two-dimensional invariant bundles is only possible in one space dimension. Theorem 4.5 is due to Chow, Lu and Mallet-Paret [44]. A similar theorem was independently proved by Tereg6fik [ 198]. Earlier the Sturm-Liouville type results were proved for linear time-periodic problems (4.1) and (4.2), and (4.1) and (4.3) (see [ 11,33,43]). The latter results are very useful for analysis of the trajectories of nonlinear equations near equilibria or fixed points. Theorem 4.5 can be used for linearization along a general trajectory or for a difference of two bounded trajectories. This and a robustness result on invariant bundles with exponential separation are main components of the perturbation theorems of [ 198]. There are other interesting classes of evolution problems that can be viewed as small perturbations of one-dimensional parabolic equations (see [86] and references therein for a discussion of such perturbations from the point of view of semicontinuity of attractors). We mention here only a simple thin domain problem, ut Ou Ov
uxx + u~.~. + f (t, x, u),
= 0,
(x, y) e S2e, t > 0 ,
(4.7)
(x, y) E OI2~, t > 0.
(4.8)
Here S2e - {(x, y)" x ~ (0, 1), 0 < y < ge(x)} and it is thin in the sense that ge(x) = ego(x) + o(e),
as e --+ 0,
in an appropriate C l sense (go, e are positive). The nonlinearity f is assumed to be sufficiently regular and r-periodic in t. Consider the following one-dimensional problem u, =
1
go(x)
ux - 0,
(go(x)Ux)x + f ( t , x , u ) ,
x ~ (0, 1), t > 0, x-0,
1, t > 0.
(4.9) (4.10)
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P. Pold(ik
Assuming natural dissipativity conditions, one can show that (4.9) and (4.10) serves as a limit problem of (4.7) and (4.8) when e ~ 0. A more precise meaning of this statement is that for sufficiently small E > 0 both equations have inertial manifolds (that is, globally attracting invariant manifolds) of equal finite dimension, the flows on these manifolds can be represented by two vector fields on a Euclidean space R" and they are C l-close to one another of order o(e). This property for autonomous equations has been established by Hale and Raugel (see [89,90,171]). In [89] they used it, in conjunction with the gradientlike structure of the problem, in the proof of convergence to equilibrium for solutions of (4.8) and (4.7) with small e (and f = f ( x , u)). Although a similar study of thin domain problems with periodic time dependence has not been carried out, it is not difficult to extend the result on closedness of the inertial manifolds to this case. It then seems feasible that the methods proposed in [39] or [ 198], as discussed above for the nonlocal perturbations, can also be used to prove convergence to rperiodic solutions for (4.8) and (4.7). Note that the latter problem is no longer gradient-like in general, so the method of [89] does not apply. The properties of the zero number functional have many other interesting consequences, notably those in the global dynamics of (4.4) and (4.5). Although a complete account of such results is beyond the scope of this survey, we cannot resist touching upon some of them (see [87] for a more extensive discussion). Based on the zero number techniques are theorems on the Morse-Smale structure of (4.4) and (4.5) (see [7,33,95]), the global "linearization" and structural stability theorems [112,120,121] and theorems describing the attractor of 1D equations as a graph [25,109,173]. Many authors contributed to the investigation of the structure of the connecting orbits of dissipative autonomous equations (see in particular [26,66,69,70,78,95,189]). At present, a relatively complete description is available for separated boundary conditions. As proved by Fiedler and Rocha [69,70], in a generic situation, the structure of the connections is completely determined by an object as simple as a permutation of (1, 2 . . . . . n) (n is the number of equilibria); the permutation was introduced earlier in [78]. Fiedler and Rocha [69] further proved a very nice result to the effect that the permutation determines the flow on the attractor up to flow equivalence: if two generic dissipative equations have the same permutations then their attractors are homeomorphic and the corresponding flows on them are equivalent. Connecting orbits for periodic boundary conditions have not been fully understood so far (see [ 11 ] for partial results). Time-periodic equations do not seem to have been studied from this point of view at all. To conclude the section, we mention a few other equations where zero number techniques have been employed in the study of large time behavior of solutions. These include reaction-diffusion equations on R [61], some degenerate parabolic equations [63,62], elliptic equations on infinite stripes [27,38], curve shortening problems [10] and some types of ordinary and delay differential equations [77,65,123-125]. For applications to blowup problems see [37,73,80]. See also [34,115] for a study of autonomous nondissipative equations (4.4) and (4.5). Some of the results presented in this section have (nontrivial) extensions to equations that are almost periodic in t. An interested reader is referred to [ 183].
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5. Symmetric domains The results from the previous section do not extend to higher dimensional problems. In general, there is no discrete Lyapunov functional to replace the zero number (see [79] for a more specific statement to that effect). Yet, discrete Lyapunov functionals play an important role in the study of special classes of equations. We discussed above thin domain problems that can be considered as small perturbations of a 1D equation and thus inherit several properties of the latter (convergence of solutions in particular). In this section we examine problems that are not such small perturbations in any sense. Rather, their solutions, at least the positive ones, are asymptotic to solutions of a 1D problem. This is true, for example, of spatially homogeneous equations on a ball. The positive bounded solutions are asymptotically radially symmetric, hence in the limit they only depend on the one-dimensional radial variable. We start by discussing the symmetrization of positive solutions in a more general setup, considering equations that are equivariant under a reflectional symmetry. We choose the coordinates such that the symmetry hyperplane is given by x l = 0, thus assume the following hypotheses on the domain (in addition to the earlier requirement that the boundary 0 S2 be of class C 2+~ for some 0 > 0). Let e "--sup{xl" x - - ( X l , X 2 . . . . . XN) E n } , and set .-
{x e s2-
>
Let 79z denote the reflection in the hyperplane Hz := {x E R N" Xl = ~}. We assume (D.1) N = dims ~> 2, and S-2 is symmetric with respect to the hyperplane H0:790Y2 = Y2. (D.2) 79z (S2~) C ,f2 for all )~ E [0, ~). (D.3) For all x E 0S-2 with xl > 0, the exterior normal vector v(x) has the first component v~ > 0. The study of parabolic equations on symmetric domains was motivated by earlier symmetry results for elliptic equations, prominently by the celebrated paper [81 ] of Gidas, Ni and Nirenberg. One of their theorems says that if u is a positive solution of the Dirichlet problem A u + f (u) = O,
u = 0,
x E S-2,
(5.1)
xEOS2,
(5.2)
where f is locally Lipschitz, then u is symmetric with respect to H0: u ( 7 ) o x ) = u(x) (x ~ S-2). Moreover, Ox~u(x) < 0 for x 6 S-20. Several extensions of this result have been obtained. In particular, the equation can be made more general [ 116], the domain can have no smoothness [20,48] and special systems of equations can be considered [ 199]. Gidas, Ni and Nirenberg also proved an analogous theorem for equations on unbounded domains (see [82]). These, too, have been improved and extended in many ways (see [19]
P. Polgt(ik
860
for a survey and additional references). A different type of symmetry results for elliptic equations can be found in [111,118,119]. For future purposes, we sketch the proof of the symmetry of u. For any ~ 6 [0, ~) consider the following statement ,.q(~.): u ( ? z x ) >1u(x) and Ox~u(x) <<,0 for any x 6 ;2~. Using positivity of u, one shows that ,_q()~) holds true if ~ < g is sufficiently close to g. A modern argument for this, not using any regularity of 0 ~2, is based on the maximum principle for equations on domains of small measure, see [21,48]; originally this was proved using a version of the Hopf boundary lemma [81 ]. Now, we start moving the hyperplane H~ to the left, and keep moving it as long as S(~) remains valid. One shows, using the maximum principle, that H~ can be actually moved all the way through H0, which shows that u(?ox) >~u(x) and Ox~u(x) <<,0 for any x ~ ~0. Using a similar procedure from the left-hand side, one obtains the symmetry and monotonicity properties of u. The above method of moving hyperplanes has its origins in the work of Alexandrov; it was applied in the study of partial differential equations by Serrin [ 182] and by many other authors later on. Let us now turn to parabolic equations, consider
ut = Au + f (t, u), u = 0,
xe~,
t>0,
xeOS2, t > 0 .
(5.3) (5.4)
Assume first that the equation is autonomous: f = f ( u ) . Let u(., t) be a bounded solution that stays positive for all t. Since the equation has a Lyapunov functional, the co-limit set of the solution consists of equilibria. All these are nonnegative, as u is, therefore radially symmetric by the results on elliptic equations. We thus obtain an asymptotic symmetrization property (all limit functions are radially symmetric) for any bounded positive solution. This simple observation gives us a prototype of results we seek under more general hypotheses. For nonautonomous equations (5.3) the symmetrization is much more interesting, as no obvious Lyapunov functional exists to provide for an a priori information on the structure of the limit set. We assume the following on the nonlinearity f : I ~ x ~ --+ ItS: (M1) There is a 0 > 0 such that for each M > 0 (t, u) ~ f ( t , u) is H61der continuous of exponent (~-, O) ( ~ in t, 0 in u)in the region [T, T + 1] x [ - M , M], (T ~> 0) with the H61der norm bounded by a constant independent of T; (M2) For each M > 0, f ( t , u) is uniformly Lipschitz continuous in u in the region [0, oo) x [ - M , M]. For the time being it is not necessary to assume that f is periodic is t. When considering solutions of the above nonautonomous equation, we denote by ~(u) the set of all limit points in X ~ of u(., t) as t --~ oo. If [[u(., t)llL~ is uniformly bounded then ~(u) is a nonempty compact subset of X ~ and the distance of u(., t) to ~(u) approaches 0 as t -+ oo. THEOREM 5.1. Assume the above hypotheses on the domain and f are satisfied. Let u be a solution of(5.3) and (5.4) such that ][u(., t)ilL~ is uniformly boundedand u(., t) ~ Ofor
Parabolic equations
861
any t sufficiently large. Then each z ~ w(u0) is symmetric with respect to Ho. Moreover, either z =-- 0 on ~ , or z ( x ) > Of o r all x E ~2 and Olz(x) < Of o r x ~ ~o.
Thus bounded positive solutions have asymptotically similar symmetry properties as positive solutions of elliptic equations discussed above. This theorem is due to Hess and Pol~6ik [99]. In an independent work, [14,15], Babin proved similar results for a general class of autonomous parabolic equations with reflectional symmetry. More recently, Babin and Sell [ 16] extended the symmetrization results to fully nonlinear nonautonomous equations on nonsmooth symmetric domains. In an earlier paper, Dancer and Hess [50] established symmetry of positive time-periodic solutions. Although no sign hypotheses on the nonlinearity are necessary in the theorem, let us mention two assumptions that guarantee positivity of solutions. The condition f (t, 0) ~> 0 (t >~ 0) implies that a solution that is positive initially remains positive as a long as it exists. Another condition, f ( t , u) > 0 (t ~> 0, u ~< 0), implies that any bounded solution is positive for all sufficiently large t. In both cases, one uses the comparison principle and an appropriate subsolution. Conditions that imply boundedness of all solutions can be found in Section 2. The proof of Theorem 5.1 uses the method of moving hyperplanes, adapted to parabolic equations. For a fixed t, consider the above statement S(~) with u = u(., t) (u(., t) is a positive solution of (5.3) and (5.4)). One first shows that if t > 0 then S(~.) holds for ~ < sufficiently close to g. Then, similarly as for elliptic equations, one moves the hyperplane H~ to the left as far as possible with S()~) remaining valid. Denote by ~.*(u (., t)) the limit value of )~ in this process; that is, s
t)) --inf{~ > O: S()~) holds for any )~ 6 (~., s
I can be shown, using the maximum principle, that )~*(u(., t)) is a strictly decreasing function unless u(., t) is symmetric about the hyperplane /4o and nonincreasing in x l > 0. Thus )~* defines a Lyapunov functional for positive solutions. A variant of the LaSalle's invariance principle then implies the symmetry of all points in the co-limit set (see [99] for details). Different arguments are used in [16]. The authors prove there that any entire solution (that is solution defined on the whole real line) that is nonnegative, is symmetric about H0 for any t. This in particular applies to points in the co-limit set of any positive solution, as they always lie on the orbit of an entire solution (of a possibly different nonautonomous equation). This way one shows the asymptotic symmetrization of solutions and additional properties. For example, for dissipative equations where the positive cone is invariant the global compact attractor in the positive cone consists of symmetric functions. If Y2 is a ball centered at the origin, Theorem 5.1 can be applied with any hyperplane through the origin. We thus obtain: COROLLARY 5.2. A s s u m e the above hypotheses on f and let ~2 be the unit ball centered at the origin. Let u be as in Theorem 5.1. Then each z E w(uo) is radially symmetric: z = z ( r ) = z(Ixl). Moreover, either z - 0 on s or z(Ix[) > 0 f o r all x ~ ~ and in the latter case O,.z(r) < Of o r r ~ (0, 1).
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P. Pol6(ik
Having established the symmetrization of solutions as t --+ ~ , we next discuss their temporal asymptotics in case f is r-periodic in t and S2 is the ball. An underlying idea is that the large time behavior of positive solutions is "governed" by radially symmetric solutions. The latter satisfying a one-dimensional problem, one should be able to use the zero number arguments, as discussed in the previous section. The following theorem materializes the idea. THEOREM 5.3. Let the hypotheses o f Theorem 5.1 be satisfied Further let S-2 be a ball and f be r-periodic in t and o f class C l in u. Let u be as in Theorem 5.1. Then there exists a solution p(., t) of(5.3) and (5.4), that is r-periodic and such that
Ilu(.,t)- p(., t)ll
0
a s t ---> ec.
Theorem 5.3 and Corollary 5.2 give a complete space time asymptotics of positive bounded solutions on a ball. We remark that a similar result is not to be expected if (5.3) and (5.4) does not have the full radial symmetry. In fact, solutions that are merely reflectionally symmetric can be viewed as general solutions of a boundary value problem on the (still multidimensional) subdomain 1-20 = {x 9 S-2: x l > 0}. Most likely, there is no restriction as to how complicated such solutions can be in general (this has not been proved however). Theorem 5.3 is due to Chen and Polfi~ik [40] (for autonomous equations an analogous convergence result was proved in [93]). The conclusion of the theorem is equivalent to the statement that the discrete trajectory {u(.,nr)},, converges, as n ~ ~ , to a single element of X ~, necessarily a fixed point of the period map of (5.3) and (5.4). In other words, it states that coo, the co-limit set of this discrete trajectory, is a singleton. We sketch the main steps of the proof of the latter property. As before, we denote by F the period map of (5.3) and (5.4). By F~ we denote the restriction of F to Xr~, the subspace of X ~ consisting of radially symmetric functions. Note that F~ is the period map of the problem N-1 ut = Ur,- + - - u r
+ f (t, u),
r9
t>0,
(5.5)
r
u,.(1, t) = 0 = u(1, t),
t > 0.
(5.6)
By Corollary 5.2 we know that coo is a subset of X~, in fact it is a compact invariant set of Fr. The first step of the proof of Theorem 5.3 consists in showing that coo is a set of fixed points of F~. One starts by noting that all points of coo are chain recurrent points of Fr (not only of F, which is standard). Then a result of [39] can be used which says that all chain recurrent points of one-dimensional equations with separated boundary conditions are fixed points. The necessary zero number theorems that facilitate an adaptation of this result to singular equations (5.5) are given in [40] (see also [9]). Once it is known that coo consists of fixed points, one can use local analysis to prove that there is only a single fixed point in coo. This involves understanding of the spectrum of the linearization F' (~0), where q9 is a fixed point of F. Only relevant are fixed points that may occur as limit points of a nonnegative solution, which allows one to assume that q9 ~> 0 and
Parabolic equations
863
even 05(., t) ~> 0, where 4)(', t) is the r periodic solution of (5.3) and (5.4) with 05(., 0) = 99Of course, such a 0(', t) is radially symmetric. Here one finds another interesting property related to symmetry: all unstable modes of the linearization, and in most cases also the central modes, are radial. More precisely, the following relations hold for the spectra of the compact operators F' (q)) and Fr'(qg). (SP) Let q9 ~ 0, if ,k 9 C with Ikl > 1 is an eigenvalue of F'(q)) then its generalized eigenspace consists of radially symmetric functions. In other words, F'(qg) and F,! (qg) have the same generalized eigenfunctions corresponding to any eigenvalue i. with [)~l > 1. If moreover, q),.(1) =/=0 then the same assertion holds for any ,k with
IZl >/1. The property is proved in a more general context (reflectional symmetry) in [40]. We come back to it in a moment. Now, the spectrum of Fi(~0) is found by one-dimensional techniques. One obtains in particular that it consists of simple positive real eigenvalues (and 0). Using (SP) we thus obtain that if q)r(1) % 0 then cr(F' (qg)) A {Ikl -- 1} consists, if nonempty, of a single element )~ = 1 and it is then a simple eigenvalue. This situation is well fitted for application of an abstract convergence result of Hale and Raugel [89]. One concludes that if coo contains a ~0 with q),.(1) =/= 0 then it is actually equal to {q)}. As q),-(1) r 0 may fail for at most one radially symmetric fixed point (this is shown using zero number again), we obtain that coo is a singleton in any case. (Note that the abstract result of [89] is discussed in detail in the next section.) Symmetry of unstable modes of positive periodic solutions, as used in the proof, is a property of independent interest. For equilibria of autonomous equations this property has been around for a while; it has been employed in the study of symmetry breaking bifurcations at positive solutions (see [32,46,190,203]). We now formulate a more general theorem, due to Babin and Sell, which applies to positive solutions that are not necessarily periodic. They proved it for general nonautonomous fully nonlinear equations satisfying appropriate symmetry hypotheses, but we do not enter their general setting here. The theorem is somewhat similar in flavor to the exponential separation theorems stated in Sections 3 and 4. For an entire solution (that is solution defined for all t 9 R) of (5.3) and (5.4) that is positive and bounded we consider solutions of the following linear variational equation that are defined for t ~< 0
vt = Av + ft,(t, u)v, v = 0,
x 9 s
(5.7)
x 9 0s
(5.8)
Note that if both f and u are r-periodic in t then the unstable eigenspace of the period map of this equation consists of values v(., 0) of solutions v(., t) that converge to zero exponentially as t --+ - e c . Here the unstable eigenspace refers to the generalized eigenspace corresponding to the spectral set {k: ]k] > 1 }. THEOREM 5.4. Let hypotheses (D1)-(D3), (M1) be satisfied and let (M2) be satisfied with the interval [0, oc) replaced by ( - o c , oc). Let u(., t), t 9 II~, be an entire positive bounded solution of (5.3) and (5.4). Let v be a solution of (5.7) and (5.8), that is defined
864
P PoMgik
on ~ x ( - ~ , 0]. If IIv(', t)llL~ ~ 0 as t -+ - o o , then v(., t) is symmetric about the hyperplane Ho for any t <<.O. Under additional conditions one can also discuss more general solutions v (analogous to solutions in the center-unstable space for periodic equations) but we do not pursue it here. The crucial observation in the proof of Theorem 5.4 is that -Ux~ (., t) satisfies the linearized equation (5.7), it is positive in {x E ,{2: x l > 0} (this follows by the symmetry properties of entire positive solutions mentioned above) and uniformly bounded. One can thus use it in comparison arguments with other solutions of (5.7), see [16] (for linearizations along periodic solution the proof is also given in [40]). In case s is the unit ball and u, v are as is Theorem 5.4, we obtain that v(., t) is radially symmetric. Adapting the Floquet bundle theorem (see the previous section) to the singular linearized equations (5.5) and (5.6), one can establish existence of an unstable Floquet bundle for (5.7) and (5.8). This bundle consists of symmetric functions with a certain number of zeros. The proof of this property does not appear difficult but has not been carried out in detail. At present, the symmetry results for elliptic equations on unbounded domains do not seem to have been extended to parabolic equations, at least not in the generality similar to the bounded domain case. Some symmetrization and convergence results for equations on IR and for a special class of autonomous equations o n I~u can be found in [57,61 ] and [59,60], respectively. Related ideas appear in the work of Roquejoffre. In [175] (see also [148,174]), he discusses asymptotic properties of solutions on a cylinder. After first establishing their eventual monotonicity, he then shows that they approach a traveling wave.
6. Gradient-like systems In this section we discuss the following basic problem: knowing that the co-limit set of a relatively compact orbit contains an equilibrium z, find sufficient conditions for z to actually form the whole co-limit set. In other words, one wants to be guarantee that the orbit is convergent. The problem typically arises in gradient-like systems; that is, systems that have a Lyapunov functional. For example, the semilinear heat equation (1.4) and (1.5) has a Lyapunov functional defined by
V'qgv--~f~2(IV99(x)[22 -
f
{p(x)
)
f(x,~)d~
dx.
dO
This is to say that V is continuous on X ~ and decreases along any orbit that is not an equilibrium. If such an orbit is relatively compact, then V is constant on its co-limit set. Since the limit set is invariant, it may only contain equilibrium solutions. Even if a dynamical system admits no obvious Lyapunov functional, one may be able to show that the limit sets of a set of trajectories consist of equilibria. This occurs, for example, if the limit sets are contained in an invariant set where the dynamical system has a simple recurrence structure. More specifically, one needs that the chain recurrent
Parabolic equations
865
set of the restriction of the dynamical system to the invariant set consists of equilibria. This is usually equivalent to the restriction having a Lyapunov functional, see [39,45,105]. In this situation, one can show that the limit sets of the original trajectories consist of equilibria. This kind of arguing was used in [40] (see the previous section). There, positive solutions are shown to have the limit sets in the space of radially symmetric functions and the restriction of the underlying map to that space is gradient-like. We present two sufficient conditions for convergence. The first one involves the assumption of local normal hyperbolicity of a set of equilibria, the other one depends on gradient structure and analyticity of the underlying equation. Consider a C l map F on a Banach space Y. For an x E Y let co(x) denote the co-limit set of the trajectory {F 'l (x)},l. We consider the situation when co(x) contains a fixed point of F, with no loss of generality taken to be 0, such that the following conditions are satisfied: (H) a ( F ' ( O ) ) =or ~' U a c U a ~, where a ~', a C, a s are closed subsets of {,~ E C: ])~[ > 1}, {~. E C: I)~l = 1 }, {)~ E C: I~.l < 1 }, respectively. Let yi be the image of the spectral projection of K := F'(0) associated with the spectral s e t o - i , i = u, c, s (see [ 110]). All these spaces are K-invariant and Y = Y~' G yc | ys. We denote yc~, := y~, G yc. THEOREM 6.1. Let (H) be satisfied. Let x be a point in A s s u m e that either you is finite-dimensional or the orbit pact. Further assume that one o f the following properties (a) dim y c = 1, (b) dim y c = m < oo and there is a submanifold M 0 E M C Fix(F). Then co(x) = {0}.
Y such that 0 E co(x) C Fix(F). o f x, { F n (x) }, is relatively comis satisfied: C Y with dim M = m such that
This theorem has been proved, under slightly more restrictive assumptions, by Hale and Raugel [89] (a discussion of this result without proof was presented in [88]). A similar result is also found in [27]. An earlier theorem of Aulbach dealt with finite-dimensional maps (see [13]). The present formulation is taken from [28]. There is a natural continuous-time analog of the statement, which we do not formulate explicitly. Theorem 6.1 applies in particular to one-dimensional autonomous equations (note that the original convergence theorem of Zelenyak [212] is based on similar ideas), but for these equations one obtains better results using the zero number techniques. The theorem is of more use when the maximum principle does not apply or where it does not lead to convergence directly. Specific examples include wave equations [58,89,93], elliptic equations on infinite strips [27,93,138] and other evolution problems (see [53,57,59,60,91,213]). Let us sketch the argument for the conclusion of Theorem 6.1. Since 0 is in the limit set, the orbit of x eventually enters any neighborhood of 0. To show the convergence, one needs to understand how the orbit can possibly leave a small neighborhood of 0 (and show that that actually cannot happen). Relevant information can be obtained employing local
866
P. Pold(ik
center, unstable and center-unstable manifolds of 0. These are locally positively invariant manifolds of F tangent to the subspaces yc, yu and yet,, respectively; we denote them by W e (0), W" (0) and W TM (0). Under the hypotheses of the theorem, one can show that the orbit can leave small neighborhoods of 0 in "the direction of" the unstable manifold W" (0) only. More precisely, it is claimed here that either the {F" (x)} converges to 0 or its limit set contains an element of W" (0) \ {0}. To explain this property, assume that 0 is in the relative interior of a continuum of fixed points (the proof is easily reduced to this situation). The center manifold must contain this continuum, hence given the assumptions on the dimension, it actually coincides locally with the continuum. The unstable manifolds W" (z) of the fixed points in W e (0) then form the manifold W TM (0). Another property of W TM (0) is that it is locally exponentially attractive with asymptotic phase: for any orbit {x,, } near 0 there is an orbit {~n} in W cs (0), such that IIx,, - ~,, IIv ~< Cr" for some constants C > 0 and r 6 (0, 1). The estimate holds as long as x,, stays in a fixed neighborhood of 0 and the constants are universal for that neighborhood. Combining the above information, one sees that after the orbit of x gets close to 0 it follows an orbit in the unstable manifold of a fixed point z ~ W c (0). Since z can be taken arbitrarily close to 0, so that its unstable manifold is close to W u (0), the above claim follows. With the assumption that co(x) consists of fixed points one now obtains that F n ( x ) cannot leave and reenter an arbitrarily chosen neighborhood of 0 infinitely many times (indeed, W" (0) \ {0} contains no fixed points). This implies the convergence. The detailed arguments following the previous outline are given in [28]. In that paper the theorem is generalized so as to apply in some situations where the co-limit set cannot be a priori assumed to consist of fixed points. It is only needed that the fixed point 0 is stable for the restriction of the dynamical system to the center manifold and that co(x) does not contain any point of W u (0) \ {0} (see [61 ] for an application of this more general theorem in time-periodic parabolic equations on •). When verifying condition (i) of Theorem 6.1, one usually relies in some way on the Sturm-Liouville eigenvalue theorem, or its dynamic version (see Theorem 4.5). This of course fails in general gradient-like parabolic problems, such as (1.4) and (1.5), on higher dimensional domains. For a generic nonlinearity (in reasonable topologies) the equilibria are isolated thus one still obtains convergence of all bounded solutions. However, there do exist Equations (1.4) and (1.5) with nonconvergent bounded trajectories (see [162]). It is a remarkable property that there are no such equations with analytic nonlinearities. This follows from the following theorem, a second convergence criterion. THEOREM 6.2. Assume that f :12 x R --+ R is continuous and real analytic in u. Then any b o u n d e d solution o f (1.4) and (1.5) converges to an equilibrium o f (1.4) a n d (1.5). The result is due to Simon [184]. For recent extensions see [92,106]. The proof depends on the gradient (not merely gradient-like) structure of problem (1.4) and (1.5). In fact, if appropriate growth conditions are satisfied (which can be assumed without loss of generality if a single L ~ bounded orbit is examined, cf. Section 2), the right-hand side of (1.4) is the gradient of the functional - V with respect to the L 2 inner product. A crucial ingredient of the proof is an infinite-dimensional version of the Lo-
Parabolic equations
867
jasiewicz inequality for analytic functions which allows one to estimate V (u) in a neighborhood of its critical point in terms of the gradient of V. To demonstrate a basic idea of the proof, while avoiding all technical difficulties of the infinite-dimensional setting, we consider an analogous convergence problem in finite dimensions. Thus let x (t), t ~ 0, be a bounded nonconstant solution of a gradient ODE .~ - - V H ( x ) ,
(6.1)
where H ' R " --+ R is real analytic. We reparameterize x(t), t 6 [0, zx~), by s -- H ( x ( t ) ) , s E (s~, so], which is possible since t ~ H ( x ( t ) ) is decreasing. Note that in this parameterization H (x
-
Computing the derivative dx
dt
ds
ds
using (6.1), one finds dx(s)
1
ds
IVH(x(s))[
Now, when s --+ soc (which corresponds to t --+ ~ ) , x ( s ) approaches a compact connected set of critical points of H, and H takes the value s ~ on this set. Lojasiewicz inequality now yields the estimate
)H(x(s))-
>'
ClVH(x(s>)l
(s near s~,),
where C > 0 and ?, 6 (0, 1) are constant independent of s (cf. [ 184]). Combining the above and substituting H ( x ( s ) ) -- s we obtain that dx(s) ds
<~ C ( s - s ~ ) -•
Hence I d x / d s l is an integrable function on ( s ~ , so] which implies that u is convergent as s ~ s ~ (i.e., as t --+ zx~). See [92] for an adaptation of this ideas to systems of ODEs of second order in time. Detailed proofs in infinite dimensions can be found in [106,184].
7. D y n a m i c s on invariant m a n i f o l d s and realization of O D E s
Sections 3-6 describe the asymptotic behavior of solutions of (1.1) and (1.2) as t --+ ~ . We have treated general equations in one space dimension, whereas the theorems in higher
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dimension only apply to a restricted class of nonlinearities, domains or initial conditions. In this section we show that those restrictions are not merely technical and with sufficiently general equations (1.1) and (1.2) there is no hope for a universal description of the behavior of solutions. We employ the method of realization of vector fields. More specifically, we consider vector fields generated by equations of the form (1.1) and (1.2) on finite-dimensional invariant manifolds. Varying the data in the equation (the nonlinearity or the domain) we obtain a class of vector fields to which we refer as realizable vector field. Employing analytical tools, we show that "sufficiently general" vector fields are realizable so that interesting dynamical phenomena are encountered in (1.1) and (1.2). This method has proved useful in the study of dynamics of various classes of evolution equations. See [56,84,85,179] for applications in functional differential equations and [68, 163,200-202] for applications in nonlocal parabolic equations and systems of reactiondiffusion equations. References pertinent to (1.1) and (1.2) will be given below in the process of reviewing the results. We discuss the method of realization here with focus on the asymptotic behavior of bounded solutions of (1.1) and (1.2). Specifically, we want to answer the following questions. Can chaotic behavior be exhibited by solutions of (1.1) and (1.2)? Can a bounded orbit have its limit set of arbitrarily high dimension? Are the answers the same if various special class of Equation (1.1) (such as spatially constant equations, or equations with linear gradient dependence) are considered and are they the same independently of the dimension of the spatial domain $2? For gradient-like equations (1.4), are there nonconvergent bounded trajectories? In the whole section 12 always stands for a domain in ]~N, N ~> 2, that is bounded and has C 2'~ boundary. This minimal regularity requirement is assumed below with no further notice. When mentioning a smooth domain, we refer to C ~ smooth. We mostly consider autonomous equations in this section, time-periodic equations are discussed only briefly. For definiteness we assume Dirichlet boundary condition, other conditions can be treated in a similar way. Let C~ (S2 x R x R N) be the Banach space of all real functions (x, u, w) w-~ f ( x , u, w) that are continuous and bounded together with their first order partial derivatives with respect to u and w; we choose the standard supremum norm (the maximum of sup f , sup fu, fw~ . . . . . fu, u ) for this space. Let Y be a closed subspace of C~ ($2 • R x R N) equipped with the same norm. Specific examples of Y are discussed below. To the choice of Y, there corresponds a class of parabolic problems ut = A u + f (x, u, V u ) ,
u = 0,
x652, t>0,
(7.1)
x6012, t>0
(7.2)
with f 6 Y. Each of these equations defines a local semiflow on X ~ as in Section 2. Now consider an ODE -h(~),
~6B,
(7.3)
where B is an open set in R n for some n >~ 1. We say that (7.3) can be realized in (7.1) and (7.2) with f ~ Y if one can find a function f 6 Y such that (7.1) and (7.2) has the following
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property: there is a C 1 imbedding A f : B ~ X ~ and a C l increasing map r :R -+ IR such that if t ~ ~(t) is a solution of (7.3) with ~ = ~0 then t ~ u(., t) := A f ( ~ ( r ( t ) ) ) is a solution of (7.1) and (7.2) with u(., 0) = A f ( ~ o ) . In other words, the submanifold M I = {A/(~): ~ 6 B} of X c~ is locally invariant for (7.1) and (7.2) and the flow of (7.1) and (7.2) on M f is C 1 equivalent to the flow of (7.3): A f maps orbits onto orbits, preserving the orientation, but not necessarily parameterization, by time. We also say that, with f as in the previous definition, (7.1) and (7.2) realizes the vector field h on the invariant manifold M r . In this is the case, the behavior of solutions of (7.1) and (7.2) on M f is qualitatively the same as the behavior of solutions of (7.3). In the following theorem we state a density realization result for (7.1) and (7.2), f c Y, where Y is chosen to consists of functions of the form f ( x , u, tO) -- f ( x , u, tO) -- f l (x, u)(K . tO) + f2(x, u).
(7.4)
Here f l , f2 are C l functions and x is a fixed direction in IRN . In particular, f is linear in tO.
THEOREM 7.1. Let the integers n > 0 and N > 1 be chosen arbitrarily. Let 12 C R N be an arbitrary bounded domain and K any direction in R x. Let Y be the subspace of C 1(s R, R N) consisting of functions o f the form (7.4). Then the set of C l functions h" B --+ IR'~ such that (7.3) can be realized in (7.1) and (7.2) with f E Y contains (i) all linear functions, and (ii) a dense subset of Cl ( B, IR") endowed with the C l supremum norm. For statement (i), f can be taken linear in both u and w. This theorem is due to Prizzi and Rybakowski [ 169]. Their work builds on several earlier results, notably [ 159,168] which contain a proof for a special domain. By the theorem, any ODE, in any dimension, has an arbitrarily small perturbation that is realizable. This is true independently of the dimension of the spatial domain 1"2 (as soon as dim s > 1), even independently of s itself. The next theorem deals with spatially homogeneous equations. THEOREM 7.2. Let the integers n > 0 and N > 1 be chosen arbitrarily. There is a dense subset 7? C C 1(B, R") with the followingproperty. Forany h E 7? there is a smooth domain if2 C I~ u diffeomorphic to the ball and a smooth function f - f (u, w) (independent of x) such that (7.1) and (7.2) realizes (7.3) on some n-dimensional invariant manifold. This theorem is due to Dancer and Polfi~ik [51 ]. Note that the domain s is not arbitrary, not even fixed, in this theorem. We do not know whether the same theorem can be proved for an arbitrary fixed domain (probably it can). It is worthwhile to make a remark or two on the result on realizability of any linear ODEs in linear equations (7.1) and (7.2) (see (i) in Theorem 7.2). This is basically an inverse spectral theorem dealing with a finite number of eigenvalues. By Theorem 7.1(i), we can arbitrarily prescribe a finite number of complex eigenvalues for the operator A + a ( x ) + b ( x ) . V under Dirichlet boundary condition. A similar results fails for spatially
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homogeneous operators as they are symmetric with respect to an appropriate weighted L2 inner product. Realization of linear equations is a relatively easy way to show that a given PDE can have solutions whose w-limit sets have arbitrarily high dimension. Indeed, for (7.3) we take a system of m linear oscillators k-y,
~ +co2x--0
with rationally independent frequencies O ) i . The PDE that realizes this system has a orbit dense in an m-torus. This argument can be used for equations considered in Theorem 7.1, however, it does not apply to the spatially homogeneous equation. Fortunately, density realization can also be used to find trajectories with high-dimensional limit sets, as we show in a moment. Density realization theorems allow one to show that persistent phenomena (or robust properties) occur in the considered class of PDEs, for they occur for nonempty open sets of vector fields which necessarily contain a realizable vector field. We discuss two such phenomena: existence of a transverse homoclinic orbit and Anosov flows on hyperbolic invariant manifolds. Consider first an ODE (7.3) that has a nonstationary periodic orbit q with the following properties (i) q is hyperbolic, (ii) q has a transverse homoclinic orbit. Recall that q = q (t) is hyperbolic if its point q (0) is a hyperbolic fixed point of the Poincar6 map associated with q. This is equivalent to the property that the Floquet exponents of the linearized equation z=g'(q(t))z are all out of the imaginary axis except for the trivial exponent )~ = 0, which is simple. A transverse homoclinic orbit refers to the orbit ~ = ~ (t) of a point ~0 r O (q) = {q (t) } contained in the intersection of the stable and unstable manifold of q (thus dist(y(t), O ( q ) ) --+ 0 as t --+ 4-oo) where the intersection is transverse: W u (q) (n~o W s (q). Near such a homoclinic orbit, (7.3) has complicated dynamics: there is a compact cantorlike invariant set with shift dynamics imbedded in it. In particular, the orbits in this set exhibit sensitive dependence on the initial conditions and the set contains a dense orbit and periodic orbits of arbitrarily high periods (see [83,152,209]). Examples of ODEs with transverse homoclinics can be found in dimensions greater than two. Both the hyperbolic periodic orbit and its stable and unstable manifolds depend continuously in the C 1 sense on the vector field [151,172]. Thus if (7.3) has a transverse homoclinic orbit then so does any its sufficiently small C l perturbation. By the density realization theorem such a small perturbation can be chosen realizable. This leads to the following result.
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COROLLARY 7.3. Fix N >~ 2. l f Y is the class o f nonlinearities as in Theorem 7.1 then the following statement holds true f o r any bounded domain S-2 in R N. There is a function f ~ Y such that (7.1) and (7.2) realizes an ODE with a transverse homoclinic orbit. If Y is the class o f spatially homogeneous nonlinearities, as in Theorem 7.2, then the same statement holds f o r some smooth domain S-2 diffeomorphic to the ball in IRN. We next discuss examples of trajectories with high-dimensional co-limit sets. The idea is to use properties of normally hyperbolic invariant manifolds and Anosov flows. Consider an ODE (7.3) with the following properties: (i) (7.3) has a compact invariant C 1 manifold M C B of dimension m that is normally hyperbolic, and (ii) the flow of (7.3)on M is Anosov and transitive. We are not going to give precise definitions of these concepts. For thorough studies of normally hyperbolic manifolds see [ 17,24,64,103,210]. Important for our purposes is that normally hyperbolic manifolds are persistent in the sense that any small C 1 perturbation of (7.3) has an invariant manifold M given by a near identity imbedding of M in B. An Anosov flow on M is a flow that has a hyperbolic structure on the whole of M, its linearization expands some direction and contracts other directions, where these directions together with the flow direction span the whole tangent bundle of M (see [ 126,172] for the definition and background). For our arguments, it is only relevant that any Anosov flow ~p on M is structurally stable: any other flow ~ on M that is C 1 close to ~p (more precisely ~(., t) is required to be C I close to ~p(-, t) uniformly for t in a compact neighborhood of 0) is C o equivalent to ~ (see [172, Theorem 9.8.1 ]). A flow on M is transitive if it has a positive semiorbit that is dense in M. One can show that for any m ~> 3 there exits an n and an ODE (7.3) that has the properties (i) and (ii) (see [51 ]). Combining the persistence property of M with structural stability of Anosov flows, one then obtains that any sufficiently small C j perturbation of (7.3) has an invariant manifold M the flow of which is C o equivalent to the flow of (7.3) on M. In particular, the perturbation has a dense semiorbit in M. Density realization theorems imply that one can choose such a perturbation that is realizable in the given class of PDEs. We thus obtain the following result (see [51 ]). COROLLARY 7.4. Fix integers N >~ 2 and m >~ 3. If Y is the class o f nonlinearities as in Theorem 7.1 then the following statement holds true f o r any domain I-2 in IRN. There is a function f ~ Y such that (7.1) and (7.2) has an invariant manifold M o f dimension m such that the flow o f (7.1) and (7.2) on M is C o equivalent to an Anosov transitive flow. In particular, (7.1) and (7.2) has a trajectory whose co-limit set coincides with M. If Y is the class o f spatially homogeneous nonlinearities, as in Theorem 7.2, then the same statement holds f o r some smooth domain S-2 diffeomorphic to the ball in IRN. Let us now discuss the main ingredients of the proofs of Theorems 7.1 and 7.2. We first indicate a general scheme of the proof that can easily be adapted to other problems.
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Let Y be the space of functions as in Theorems 7.1 or 7.2. Consider problem (7.1) and (7.2) in the following form
u, = Au + a(x)u + g(x,u, Vu), u = 0,
xe,<2, t>0,
(7.5)
xeOs
(7.6)
t>0.
Here g e Y and a is a C l function which is constant if the class of homogeneous equations is considered. Suppose we are given an n (for the dimension of B in (7.3)) and want to prove the density realization theorem for ODEs (7.3). We make two assumptions on the linear part of the equation: (L 1) The operator A + a (x), under Dirichlet boundary condition, has kernel X I of dimension n. (L2) A set of LZ-orthonormal eigenfunctions ~Pl . . . . . ~o,, of A + a(x) that span Xl satisfy a certain algebraic independence condition. Assumption (L 1) allows us to construct invariant manifolds of 7.5 and 7.6 if g is close to 0 in Y. Condition (L2) comes out of computations of vector fields on the invariant manifolds. We will be more specific about it below. Of course, at some point we will have to worry about the existence of an operator satisfying (L 1) and (L2). The procedure is further summarized as follows. Step 1. The center manifold reduction. This yields a vector field V (g) representing the flow of (7.5) and (7.6) on an invariant manifold. Step 2. Investigation of the map g w-> V (g). The goal is to derive a sufficient condition for (local) density of the range of this map. This is where the second requirement, (L2), on the linear part comes about. Step 3. Construction of an operator A + a (x) that satisfies both (L 1) and the algebraic independence condition (L2) as computed in Step 2. Step 1 is a standard application of the center manifold theorem [42,94,177,204,205]. Taking g in a sufficiently small neighborhood U of 0 in Y, one finds an invariant manifold of (7.5) and (7.6) of the form
w+, - {+ + o-++(+). + c
}.
Here ag is a function defined on X l with values in a complement X2 of X I in X~; it is of class C l in ~ and g, and if g is of class C k in u and w then O-g(.) is also of class C k. Solutions of (7.5) and (7.6) in Wg have the form u(., t) = ~(t) +ag(~(t)), where ~(t) e XI. One can now write down an ODE for ~ (t) by substituting the last expression for u in (7.5) and applying to (7.5) the projection P : X --+ X1 with kernel X2 (this is a spectral projection of A + a(x)). This yields
-- V(g)(~)"- P~(~ + ~g(~)), where ~': X ~ ~
(7.7)
X is the Nemitskii operator of g (cf. Section 2). We refer to (7.7) as the
center manifold reduction of (7.5) and (7.6). It is an ODE on X l ~" ]K'~. In terms of our earlier definition, (7.5) and (7.6) realizes (7.7) on the invariant manifold Wg.
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Note that for g = 0 we have V (0) = 0 as X l is the center manifold for the linear equation consisting of equilibria. Passing to the second step, we consider the restriction of V (g) to an open bounded set in X~ (the reason is that the Weierstrass approximation theorem will be applied). This constitutes a map,
gl-> V(g):
Y --->
CI(B-, Xl),
whose range consists of realizable vector fields. It would be optimal if we could prove that the range of this map is the whole space C l (B, X l) or at least that the range contains a neighborhood of 0 in C I(B, X l). Any ODE on B would then be realizable (note that any vector field becomes small in the C l norm after a time rescaling). It is tempting (see [84,85]) to apply a local surjective mapping theorem to V to obtain the conclusion. This, however, is not so easy as one has to face the typical difficulties associated with self-composition of functions (see the definition of V (g)) and a loss of derivatives linked to it. See [178-180] for a more-detailed discussion of this problem and the N a s h - M o s e r technique that can sometimes help to deal with it. All these difficulties are avoided if the density realization, rather then full realization, is set as the goal. In this case we proceed as follows. For a small e, consider the vector field e - 1 V ( e g ) . It has the same dynamics as V (eg), with a different time scale, so the corresponding ODE is realizable. Further,
E-1V(eg)(~) -- P2(~ + a~e(~)) -- P~'(~) + o(~)
(~ 6 Xl)
because o-e~ ~ 0 as e ~ 0. For density realization it is therefore sufficient to neglect the small error term and show that the range of the linear map
g ~-> P~'(.) " Y --+ CI (-B, XI ) is dense in C I(B, X l). Regularity issues involved in necessary C l estimates of the error term are taken care of by considering a subspace of Y consisting of more regular functions. Using a little more sophisticated argument, relying on a scaling and the Weierstrass approximation theorem, one can find yet simpler sufficient conditions for density realization. Namely, it is suffices to show that for some fixed ~ E X l and any integer k, the map g w-> j .k ~ P~'(.) is surjective. Here j .k ~ stands for the k-jet; that is, the vector of coefficients of the kth order Taylor expansion of the given function at ~ . The map is defined on the subspace of Y consisting of C k functions and its target space is a finite-dimensional space ofjets (see [51,159] for the proof of sufficiency of this weaker condition). One can now formulate the surjectivity of the finite dimensional map in terms of eigenfunctions ~Pl . . . . . ~01, of A + a ( x ) . For example, in the simplest case, when Y is chosen to consist of all (not necessarily linear) functions of the form g(x, u, w) = g(x, w l ) , the surjectivity condition requires that the functions
q3i (0~., ~1 )/4'... (OXl ~On)/4'' be linearly independent. Here i ---- 1 . . . . . n and fl = (ill . . . . . /3,l) varies among all positive multiindices of norm not greater than k (if the k-jets are being prescribed). These algebraic
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P. Pold(ik
expressions appear in the computation of the k-jet of ~ ~ V ( g ) ( ~ ) at ~ = 0 when one uses the real coordinates on X l: = rl ~Ol + . . . + rn~Pn,
and the integral formula for the spectral projection:
Pu--~~oi(')fs2~oi(x)u(x)dx. i=l
The linear independence of the algebraic expressions is what we vaguely stated above as condition (L2). With different choices of the space Y one obtains different algebraic expressions; they are more involved with Y as in Theorem 7.1 and even more so for Y consisting of spatially homogeneous functions (see [ 168] and [51 ] for the corresponding formulas). After the algebraic independence conditions has been derived, it remains to find an operator A + a ( x ) that satisfies both requirements (L1) and (L2). This is the last step, usually the most difficult and technical part in the proof. To start up, one chooses a special domain where the Laplacian has eigenvalues of arbitrarily high multiplicity, so that (L1) holds for some constant a (x) = a0, and the eigenfunctions are known explicitly to some extent. Typically then, the algebraic independence condition is not satisfied. One therefore tries to achieve it using a perturbation. This is quite delicate as a high multiplicity of an eigenvalue is to be preserved at the same time. Moreover, in the spatially homogeneous case, one has the additional constraint that a (x) has to remain constant, so only the domain can be perturbed. A reasonably general procedure to deal with such perturbation problems has been developed by Dancer and Polfi6ik [51 ]. That procedure yields the operator as needed on some domain. In case a (x) is allowed to depend on x, as in Theorem 7.1, one can proceed further and find an operator satisfying (L1) and (L2) on an arbitrary domain. Such a construction is described in a recent work of Prizzi and Rybakowski (see [ 169]). Their results do not apply in the spatially homogeneous case; this is the reason for the difference in the formulations of Theorems 7.1 and 7.2 regarding the domain. We now proceed with a brief discussion of realization results for other special classes of Equations (1.1). First at hand are equations, both autonomous and time-periodic, that do not depend on Vu. The semilinear heat equation ut = A u + f ( x , u ) ,
u = 0,
x6~,
t>0,
x E 0 ~ , t > 0,
(7.8) (7.9)
defines a gradient-like semiflow on any invariant manifold. This of course constraints the class of realizable vector fields. Following the scenario of the density realization given above, one can prove that any gradient vector field on B has an arbitrarily small perturbation that is realizable in (7.8) and (7.9). Although this result does have interesting consequences on the global dynamics (see [ 160]), from the point of view of the asymptotic
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behavior of single trajectories it is probably of no use; in robust gradient systems all trajectories converge to an equilibrium. So, for example, nonconvergent bounded trajectories cannot be found by density realization, not even in its strengthened version, realization of families of vector fields, as discussed in [ 160]. An ad hoc realization procedure to show the existence of nonconvergent trajectories was found by Polfi~ik and Rybakowski [ 162]. They proved the following theorem. THEOREM 7.5. Let F2 be a disk in ~2. For any positive integer k, there exists a C k function f : #2 x ~ --+ R such that problem (7.8) and (7.9) has a bounded solution that approaches, as t ---> ~ , a subset o f X ~ homeomorphic to the circle. As discussed in the previous section, examples of nonconvergent bounded trajectories do not exist if f is analytic. They probably do exist with f e C a but the results of [162] do not imply that. Also, no such example has been found for a spatially homogeneous equation (7.8). If the nonlinearity in (7.8) is allowed to depend on time periodically, f = f ( t , x, u), then the corresponding dynamical system is no longer gradient-like. Introducing the notion of realizability of time-periodic ODEs (see [158]), one can prove both linear and density realization theorems for such equations (the detailed proof is to appear). Similarly as for autonomous equations, these results imply existence of chaos and high dimensional w-limit sets for equations with f -- f ( t , x , u); the dimension of the domain #2 can again be arbitrary (greater than 1). Previously, the existence of 2-dimensional co-limit sets was shown by Dancer [47]; see also [71 ] for related results for equations on S l . No such realization result has been proved for spatially homogeneous time-periodic nonlinearities f -- f ( t , u). We have mentioned above that moderating our goal from complete realization to density realization made the whole process feasible. It is nice about the procedure that it has many general features that can be used in other classes of equations. Complete (not just density) realization theorems are still of interest, however. We mention the following one. THEOREM 7.6. Let #2 be an arbitrary domain in ]~N, N >~ 2. For any function h in C ~ ( ~ X + l , ~'~x § ) the ODE = h(~) on ]~N-+-I can be realized in (7.1) and (7.2) with f ~ C~ (#2 x R x ]~N). Note that this realization result is not independent of the spatial dimension, only ODEs on •X+! can be realized. With this limitation, the result shows that arbitrary dynamics can be found on finite dimensional invariant manifolds of (7.1) and (7.2) with a general nonlinearity. Theorem 7.6 was proved by Polfi6ik and Rybakowski [161] for special domains #2; it was extended to any domain by Prizzi and Rybakowski [169], see also [167]. Other complete-realization theorems can be found in [157,178]. As we know from the theory of monotone dynamical systems, most bounded solutions of (7.1) and (7.2) converge to an equilibrium (or to a periodic solution if time-periodic equations are considered). Thus the invariant manifolds with complicated dynamics, as obtained
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by the realization techniques, are necessarily unstable. An interesting problem arises as to what is the minimal possible "unstable dimension" of such manifolds. More specifically, in the above construction, where the manifolds are small perturbations of ker(A + a (x)), the unstable dimension can be defined as the number of positive eigenvalues of A + a(x). This number coincides with the dimension of the leaves of the unstable foliation near 0. When N = 2, there are restrictions, at least when using self-adjoint linear part, on how small the unstable dimension can be. Indeed, there must be many positive eigenvalues of A + a(x) should the kernel have high dimension, see [23,41,104] for precise estimates. In higher dimensions, however, it is likely that the whole construction can be done with A + a (x) having high dimensional kernel and only one positive eigenvalue. Phrased differently, there seem to be no restriction on the possible multiplicity of the second eigenvalue of A + a(x) (see [23] for a closely related discussion of the multiplicity problem for operators on manifolds without boundary). However, one still has to meet the algebraic independence condition on the corresponding eigenfunctions. This has not been done so far.
Acknowledgment The author is indebted to Peter Bates, Xu-Yan Chen, Bernold Fiedler and Kening Lu for their remarks and suggestions. He is particularly thankful to Pavol Brunovsk3~ and Jack Hale for their continuous encouragement and for their comments on the manuscript. This paper was written while the author was enjoying the excellent working environment at the School of Mathematics, Georgia Institute of Technology.
References [1] N.D. Alikakos and E Hess, Liapunov operators and stabilization in strongly order-preserving dynamical systems, Differential Integral Equations 4 (1991), 15-24. [2] N.D. Alikakos, E Hess and H. Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equations, J. Differential Equations 82 (1989), 322-341. [3] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620-709. [4] H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math. 360 (1985), 47-83. [5] H. Amann, Quasilinear parabolic systems under nonlinear boundary conditions, Arch. Rational Mech. Anal. 92 (1986), 153-192. [6] H. Amann, Linear and Quasilinear Parabolic Problems, Birkh~iuser, Berlin (1995). [7] S. Angenent, The Morse-Smale proper~ for a semilinear parabolic equation, J. Differential Equations 62 (1986), 427-442. [8] S. Angenent, The zeroset of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79-96. [9] S. Angenent, Solutions of the 1-d porous medium equations are determined by their free boundary, J. London Math. Soc. 43 (1990), 339-353. [ 10] S. Angenent, Parabolic equations for curves on surfaces. II. intersection, blow-up and generalized solutions, Ann. of Math. 133 ( 1991), 171-215. [ 11 ] S. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Amer. Math. Soc. 307 (2) (1988), 545-568. [ 12] R. Aris, Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Clarendon Press, Oxford (1975).
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[ 184] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. 118 (1983), 525-571. [185] H.L. Smith, lnvariant curves for mappings, SIAM J. Math. Anal. 17 (5) (1986), 1053-1067. [ 186] H.L. Smith, Monotone Dynamical Systems, Amer. Math. Soc., Providence, RI (1995). [ 187] H.L. Smith and H.R. Thieme, Quasi convergence and stabili~' for strongly order-preserving semiflows, SIAM J. Math. Anal. 21 (3) (1990), 673-692. [ 188] H.L. Smith and H.R. Thieme, Convergence for strongly order-preserving semiflows, SIAM J. Math. Anal. 22 (4) (1991), 1081-1101. [ 189] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, Berlin (1967). [ 190] J. Smoller and A. Wasserman, Symmet~-breaking for positive solutions of semilinear elliptic equations, Arch. Rat. Mech. Anal. 95 (1986), 217-225. [191] E Tak~i6, Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl. 148 (1) (1990), 223-244. [192] P. Tak~i6, Asymptotic behavior of strongly monotone time-periodic dynamical processes with symmetry, J. Differential Equations 100 (2) (1992), 355-378. [193] E Tak~i6, Domains of attraction of generic w-limit sets for strongly monotone discrete-time semigroups, J. Reine Angew. Math. 423 (1992), 101-173. [194] P. Tak~i~, Linearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems, Proc. Amer. Math. Soc. 115 (3) (1992), 691-698. [195] E Tak~i6, A construction of stable subharmonic orbits in monotone time-periodic dynamical systems, Monatsh. Math. 115 (3) (1993), 215-244. [196] E Tak(16, A short elementary proof of the Kre~n-Rutman theorem, Houston J. Math. 20 (1) (1994), 93-98. [197] I. Tere~6Lk, Dynamics of C 1 smooth strongly monotone discrete-time dynamical systems, Preprint. [198] I. Tereg6~ik, Dynamical systems with discrete Lyapunov functionals, Ph.D. thesis, Comenius University (1994). [199] W.C. Troy, Symmetr3' properties in systems of semilinear elliptic equations, J. Differential Equations 42 (3) (1981), 400--413. [200] S.A. Vakulenko, Existence of Ruelle-Takens transition to chaos for some evolution equations, C. R. Acad. Sci. Paris 316 (1993), 1015-1018. [201] S.A. Vakulenko, Reaction-diffusion systems with prescribed large time behavior, Ann. Inst. Henri Poincar6 66 (1997), 373-410. [202] S.A. Vakulenko, Dissipative systems with controllable inertial forms, Preprint. [203] A. Vanderbauwhede, Symmetry-breaking at positive solutions, Z. Angew. Math. Phys. 38 (2) (1987), 315326. [204] A. Vanderbauwhede, Centre manifolds, normal forms and elementary, bifurcations, Dynamics Reported, Vol. 2, Wiley, Chichester (1989), 89-169. [205] A. Vanderbauwhede and G. Iooss, Center manifold theory, in infinite dimensions, Dynamics Reported: Expositions in Dynamical Systems, Springer, Berlin (1992), 125-163. [206] M.P. Vishnevskii, Stabilization of solutions of weakly coupled cooperative parabolic systems, Mat. Sb. 183 (1992), 45-62. [207] M.P. Vishnevskii, Monotone solutions ofquasilinear parabolic equations, Sibirsk. Mat. Zh. 34 (4) (1993), 50-60, ii, viii. [208] E Vuillermot, Almost-periodic attractors for a class of nonautonomous reaction-diffusion equations on R n, III: Center curves and Lyapunov stabili~, Nonlinear Anal. 22 (5) (1994), 533-559. [209] S. Wiggins, Global Bifurcations and Chaos, Springer, New York (1988). [210] S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer, New York (1994). [211] J. Wu, Theory and Applications of Partial Functional-Differential Equations, Springer, New York (1996). [212] T.I. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Differential Equations (transl. from Differencialnye Uravnenia) 4 (1968), 17-22. [213] T.I. Zelenyak, M.M. Lavrentiev and M.P. Vishnevskii, Qualitative Theory of Parabolic Equations, Part 1, VSP (1997).
CHAPTER
17
Global Attractors in Partial Differential Equations
G.
Raugel
CNRS et Universitd de Paris-Sud, Analyse Numdrique et EDP, UMR 8628, B6timent 425, F-91405 Orsay cedex, France E-mail: genevieve, raugel@math, u-psud.fr
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
887
2. F u n d a m e n t a l concepts
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889
2.1. S o m e definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. co and or-limit sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
891 893
2.3. Global attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
898
2.4. E x a m p l e s of asymptotically s m o o t h semigroups
907
2.5. M i n i m a l global B-attractors
............................
.......................................
912
2.6. Periodic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
913
3. General properties of global attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
915
3.1. D e p e n d e n c e on parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
915
3.2. D i m e n s i o n of c o m p a c t global attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
923
3.3. Regularity of the flow on the attractor and d e t e r m i n i n g m o d e s
927
3.4. Inertial manifolds
....................
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4. Gradient systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. General properties of gradient systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Retarded functional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Scalar parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. O n e - d i m e n s i o n a l scalar parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. A d a m p e d hyperbolic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
934 936 936 945 948 952 958
5. Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Acknowledgements
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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974
References
H A N D B O O K O F D Y N A M I C A L S Y S T E M S , VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 885
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1. Introduction This survey is devoted to an introduction to the theory of global attractors for semigroups defined on infinite dimensional spaces, which has mainly been developed in the last three decades. A first purpose here is to describe the main ingredients leading to the existence of a global attractor. Once a global attractor is obtained, the question arises if it has special regularity properties, a particular shape etc. or if it has a finite-dimensional character. The second objective is thus naturally to give some of the most important properties of global attractors. Finally, we want to show the relevance of the abstract theory in applications to evolutionary equations. Clearly, we can here neither describe all the related questions and results, nor give the detailed proofs of the main statements, although they are often very instructive. To keep the text elementary and self-contained, we have recalled all the needed basic concepts in the theory of dynamical systems and have included some proofs. In order to illustrate the general abstract results, we have chosen to discuss few equations, but in details, rather than to give a catalogue of applications to partial differential equations and functional differential equations. The dynamical systems that arise in physics, chemistry or biology, are often generated by a partial differential equation or a functional differential equation and thus the underlying state space is infinite-dimensional. Usually these systems are either conservative or exhibit some dissipation. In the last case, one can hope to reduce the study of the flow to a bounded (or even compact) attracting set or global attractor, that contains much of the relevant information about the flow and often has some finite-dimensional character. It is difficult to trace the origin of the concepts of dissipation and attractor. The word attractor, applied to a single invariant point, is ancient and probably appeared at the beginning of the century. One can find it, for example, in the book of Coddington and Levinson in 1955 [42] or in a paper of Mendelson in 1960 [156]. For a flow on a locally compact metric space, attractors consisting of more than one point have been studied by Auslander et al. in 1964 [6] (see the paper of [162] for various definitions in the finite-dimensional case). Several different notions of attractors are already found in the lecture notes of Bhatia and Hajek [20] in 1969, for a semi-flow on an infinite-dimensional space. In 1968, Gerstein and Krasnoselskii [78] studied the existence and properties of a maximal compact invariant set for the discrete system generated by a compact map S on a Banach space. In 1971, Billotti and LaSalle described the maximal compact invariant set and proved stability results for maps whose iterates were eventually compact. The specific notion of compact global attractor, as used in this review, appeared in the papers of Oliva in the early 1970s (see [103]). The work of Ladyzenskaya [133,134] in 1972 implied the existence of the compact global attractor for the semi-flow generated by the two-dimensional Navier-Stokes equations. In the same year 1972, Hale et al. [100] gave general existence results of maximal compact invariant sets and introduced the concept of asymptotically smooth systems. Let us now describe more precisely the concepts of dissipation and global attractor. In his study of the forced van der Pol equation, Levinson [144] introduced the concept point dissipative for maps S on the space R '7. A map S is point dissipative if there exists a bounded set B0 C R" such that, for each x ~ R", there exists an integer no(x, Bo) so that S"(x) ~ Bo, for n >/no. Due to the local compactness of R",
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G. Raugel
any point dissipative map S is also bounded dissipative (or equivalently uniformly ultimately bounded); that is, there exists a bounded set Bo C R n such that, for each bounded set B C 1R'7, there exists an integer no(B, Bo) so that S n (B) C B0, for n ~> no. If S is bounded dissipative, the local compactness of R 'l also implies that the co-limit set co(B) = f"]m>~OC I ( U j - I SJ(B)) of any bounded set B is compact, invariant (i.e., S(B) = B) and attracts B, that is, 6•,,(S"(B),CO(B)) --+ 0 as n --+ +oc, where 6R,,(Sn(B),CO(B)) = SUPxes,,(8)inf~,e~o(8) Ilx - yIIR,,. Therefore, if S is point dissipative, A = co(B0) is the global attractor, that is, A is bounded, invariant and attracts every bounded set B of N n. Here, in addition, A is compact. Thus, in finite dimensions, point dissipativness implies the existence of a compact global attractor. Note that this definition of the global attractor implies that A is maximal with respect to inclusion and hence is unique. Unfortunately, when the underlying space X is not locally compact, there are examples where point dissipativness does not imply that the orbits of bounded sets are bounded and where bounded dissipativness does not imply the existence of a global attractor. So the following question arises: are there interesting classes of dynamical systems on non locally compact spaces that have properties similar to the ones mentioned above for dynamical systems on R"? To have a theory comparable to the one for maps on R", one must impose a type of smoothing property on the operator S : X ~ X. This is done by assuming, for example, that S : X ~ X or an iterate of S is compact (see [21]). More generally, it is sufficient to suppose that S is asymptotically smooth in the terminology of Hale et al. [ 100] or equivalently, asymptotically compact in the terminology of Ladyzhenskaya [ 137]. In Section 2, we recall all the needed precise definitions, introduce the above concepts of dissipativness and asymptotic smooth or compact systems. We discuss some implications between these notions. The fundamental theorem of existence of a compact global attractor is stated and proved. Some basic properties like invariance, stability and connectedness of compact global attractors are also discussed. Finally, a large part of the section contains examples of asymptotically smooth systems. Section 3 is devoted to a presentation of the most important properties of compact global attractors. Compact global attractors are robust objects with respect to perturbations. We give several continuity properties of the global attractors with respect to perturbation parameters and recall the stability of the flow on the global attractor under perturbations for Morse-Smale systems. The mentioned properties play an important role in the study of systems depending on several physical parameters and also in numerical approximations of these systems. Finally, we discuss the possibility of the flow on the compact global attractor A being finite-dimensional by first showing that, in most of the cases, A has finite Hausdorff or fractal dimension. The next question of interest is the reduction of the study of the flow on A to the discussion of the flow of some system on a finite-dimensional space. One effort in this direction is to assert the existence of an inertial manifold, that is a finite-dimensional Lipschitzian positively invariant manifold, that contains the global attractor. Unfortunately, the existence of inertial manifolds is rare in the general class of systems arising in applications. Another approach is to show the existence of a finite number of "modes", on which the corresponding dynamics approximates the dynamics of the original system on ,4 (for example, Galerkin approximations). For evolutionary equations, this approach gives regularity with respect to time of the flow on A and regularity in "the spatial variables" when PDE's are involved.
Global attractors in partial differential equations
889
So far, one has not yet given a description of the flow on the global attractor. In the general case, a qualitative description of the global attractor seems difficult. Section 4 is devoted to the class of gradient systems, that is systems which admit a Lyapunov functional. In this case, due to the invariance principle of LaSalle, the global attractor A, if it exists, is the unstable set of the set of equilibrium points. If the equilibrium points are all hyperbolic, then A is the union of the unstable manifolds of each equilibrium point. Applications of the general abstract theory, in the frame of gradient systems, are then given to FDE's and to two representative classes of scalar partial differential equations, the reaction-diffusion equations and the (weakly) damped wave equations. Special emphasis is made on the scalar reaction-diffusion equation defined on a bounded interval of and provided with separated boundary conditions. In this case, a result of Henry [ 123,3] says that the stable and unstable manifolds are always transversal, which means that the global dynamical behaviour can only change by bifurcations of the equilibria. This important property was the starting point for the precise qualitative description of the flow on the global attractor. In this one-dimensional case, special properties like the strong maximum principle, the Sturm-Liouville theory and the Jordan curve theorem play a primordial role. Finally, in Section 5, we illustrate the abstract theory of global attractors given in Section 2, by studying weakly damped dispersive equations, the prototype of which is the weakly damped Schr6dinger equation. Many topics have been left on the side, including the non autonomous evolutionary equations leading to the notions of processes and skew-product semi-flows (see [49,197,161, 118,204,34], etc.), the generalization of the concept of attractor to multivalued mappings (see [ 15] for instance), the notion of random attractors for dissipative stochastic dynamical systems (see [48,51] for basic properties). Only few applications to the class of retarded functional differential equations have been given below (see [116,174]). Finally, for further readings on global attractors and more examples, the reader should consult the books [ 13,94,202,140,35,198], for example.
2. Fundamental concepts In this section, (X, d) (or simply X) denotes a metric space, with distance d. We use the semi-distance 6x(., .) defined on the subsets of X by
6x(x, A) =- inf d(x, a),
VxEX, VACX,
aEA
8x(A,
B) --
sup inf aEA bEB
d(a,
b) -
sup 6x(a, B),
VA, B C X .
aEA
For any subset A of X and any positive number e, we introduce the open neighbourhood N x ( A , e) = {z E X ] 3x(z, A) < e} (respectively the closed neighbourhood N x ( A , E) = {z ~ X ] 3x(z, A) <~ e}). Finally, we define the Hausdorffdistance Hdistx(A, B), for any subsets A, B of X by Hdistx (A, B) -- max(Sx (A, B), 3x(B, A)).
G. Raugel
890
In what follows, we mainly concentrate on continuous dynamical systems, or continuous semigroups, S(t), t >~0 on X, whose definition we now recall. DEFINITION 2.1. A continuous dynamical system or continuous semigroup on X is a oneparameter family of mappings S(t), t >~0, from X into X such that (1) S ( 0 ) = I; (2) S(t + s) = S(t)S(s) for any t, s/> 0; (3) for any t >1 O, S(t) E C~ X); (4) for any u E X, t ~ S(t)u ~ C~ +c~), X). If the mappings S(t) from X into X are defined for t E ~, if the properties (2), (3) hold for any t, s E R and if, in (4), (0, +cx~) can be replaced by ~, then S(t), t E 1t< is a continuous group. A one-parameter family of mappings S(t), t >~ O, satisfying only the properties (1), (2) and (3) will be simply called "a semigroup".
We recall that, if S E C~ X), the family S", n E 1~, is called a discrete dynamical system or discrete semigroup. If S is a C~ from X to X, then the family S m, m E Z, forms a discrete group. Most of the properties described below are also valid for discrete dynamical systems. In the sequel, if we do not want to distinguish between discrete dynamical systems and (non discrete) semigroups, we simply refer to a semigroup S or S(t), t E G +, where G + is either [0, +cx~) or the set of nonnegative integers N. Hereafter, G denotes either R or Z. The first example of continuous semigroups is given by ordinary differential equations k = f (x), x E •", where f : x E R" w-~ f ( x ) E It~'~ is a globally Lipschitzian mapping. Another basic example is the class of retarded functional differential equations (see [116]). Evolutionary partial differential equations also give rise to continuous semigroups as shown in the following model example. EXAMPLE 2.2. Let X, Y be two Banach spaces, such that Y C X, with continuous injection. Let r 0 ( t ) be a linear Co- semigroup in X with infinitesimal generator A and f : Y --+ X be a Lipschitzian mapping on the bounded sets of Y. We assume that, either, (1) Y = X ,
or (2) S0(t) is an analytic semigroup on X and Y = X ~ = D((~. I d - A)~ where c~ E [0, 1) and ~. is an appropriate real number. We consider the semilinear differential equation in Y, du(t) dt
= Au(t) + f ( u ( t ) ) ,
t > O, u(O) -- uo E Y.
(2.1)
It is well known that this equation has a unique mild solution u E C~ T*(u0)), Y), where T* (u0) E (0, +cx~]. If T* (u0) = +cx~, for any u0 E Y, then the family of mappings S(t) defined by S(t)uo - u(t) is a continuous semigroup on Y. In particular, the mapping (uo, t) ~-+ S(t)uo is continuous from [0, +cx~) • Y into Y. We recall that u(t) is a mild solution of (2.1) if, for t ~> 0,
S(t)uo - r o ( t ) u o +
f0 t Zo(t
- s) f (S(s)uo) ds =- r o ( t ) u o + U(t)uo.
(2.2)
Global attractors in partial differential equations
891
Under the same hypotheses as above, assume now that there exists a subset Z0 of Y such that T* (u0) = + o c , for any u0 9 Z0, and that there is a positive constant Co such that
II s(,).o II Y
Co,
V.o
Zo, v,
o.
Ut)o {S(t)uO}r,
We thus introduce the set Z -- U,0ez0 equipped with the distance d induced by the norm of Y. Then, (Z, d) is a complete metric space, S(t)lz defines a continuous semigroup on Z and IIS(t)uollr <~ Co for any u0 9 Z, t >~ 0. REMARK. In the definition of continuous dynamical systems, several authors require also that, for any u E X, the mapping t E [0, + o o ) ~ S(t)u E X is continuous at t -- 0. Actually, this hypothesis is unnecessary most of the time. In Section 4 below (see Equation (4.69)), we study an example where S(t) is not continuous at t = 0. A result of [37] implies that, if S(t) is a continuous dynamical system in the sense of Definition 2.1, then the mapping (t, u) E (0, +oc) x X w-~ S(t)u E X is continuous. If, moreover, the space X is locally compact and if, for any u E X, t E [0, + o c ) ~ S(t)u E X is continuous at t = 0, then, by a theorem of [53], the mapping (t, u) E [0, +oc) x X ~ S(t)u E X is continuous. If the space X is not locally compact, the joint continuity of S(t)u at t = 0 may not be true (see the examples of [36,15]).
2.1. Some definitions In this subsection, we assume that S(t), t E G +, is a semigroup on X. Here we define carefully the notions of invariance and attraction, which play a crucial role in the theory of global attractors. DEFINITION 2.3. A set A is positively invariant if S(t)A C A, for any t e G +. The set A is invariant if S(t)A = A, for any t e G +. The following concept dealing with invariance and connectedness has been introduced in [141] and will be used later. DEFINITION. Let S be a semigroup of continuous maps from X into X. A closed invariant subset A of X is said to be invariantly connected if it cannot be represented as the union of two nonempty, disjoint, closed, positively invariant sets.
The positive orbit of x E X is the set g +(x) = { S ( t ) x l t E G+}. If E C X, the positive orbit of E is the set
•
-- U SU)E = U • (z). tcG+
z.eE
More generally, for r E G +, we define the orbit after the time r of E by
y+(E)--v+(S(r)E).
G. Raugel
892
Let now I be an interval of IR and S(t), t ~> 0, a semigroup. We recall that a mapping u from I into X is a trajectory (or orbit) of S(t) on I if u(t + s) = S(t)u(s), for any s 6 I and t ~> 0 such that t + s 6 I. In particular, if I = ( - ~ , 0] and u(0) = z 6 X, u is called a negative orbit through z and is often denoted by V-(z). If I = R and u(0) = z, then u is called a complete orbit through z and is often denoted by y(z). We let F - ( z ) be the set of all negative orbits through z. If F - ( z ) is not empty, it may contain more than one negative orbit, because we have not assumed the property of backward uniqueness. We also let F ( z ) = F - ( z ) U y+(z) be the set of all complete orbits through z. In the same way, we define the sets y - ( E ) , F - ( E ) and F ( E ) , for any subset E of X. For later use, for any z 6 X, we introduce the following set:
H(t, z) -- {y 6 X I there exists a negative orbit uz through z such that uz(O) -- z and U z ( - t ) -- y }. We remark that F - ( z ) = [.-Jt/>0 H (t, z). Likewise, if E C X, we define the set H (t, E) =
~ z c E H(t, z) and remark that F - ( E ) -- ~t>~o H(t, E). In a similar way, replacing ( - c o , 0] (respectively R) by (-cx~, 0] A Z (respectively Z), we define the negative and complete orbits of maps S. In the framework of maps, it is very easy to give examples of non backward uniqueness. Consider the non injective logistic map S:[0, 1] ~ [0, 1], Sx = )~x(1 - x), with 2 < ~. ~< 4. The point x0 = (,k - 1)/~. is a fixed point of S and the point y = ~ - l satisfies Sy = xo. The iterates S - n y ~ (0, x0) are well defined and y ( y ) = { S - n y I n = 0, 1,2 . . . . } U {x0} is a complete orbit trough x0. The proof of the following lemma is elementary. LEMMA 2.4. The set A C X is invariant for the semigroup S(t), t ~ G +, if and only if for any a ~ A, there exists a complete orbit Ua through a, with ua(G +) C A. If the semigroup S(t), t >~O, is continuous, the complete orbits belong all to C~ A). In general, there may exist an invariant set A, which does not contain all complete orbits of S through each point in A. In the above example of the logistic map, the invariant set A -- {x0} does not contain the complete orbit y ( y ) . PROPOSITION 2.5. Let S(t) be a continuous semigroup on X and A be a compact invariant set. If the operators S(t) are injective on A, for t ~ O, then S(t)la is a continuous group of continuous operators on A. PROOF. By Lemma 2.4, for any a E A, there exists a complete orbit Ua E C0(I~, A) such that S ( t ) U a ( - t ) - a, for any t ~> 0. Since S(t)IA is one-to-one, we can set, for a 6 A"
S ( - t ) a - S ( t ) - l a - Ua(-t).
Global attractors
893
in p a r t i a l d i f f e r e n t i a l e q u a t i o n s
Clearly, S ( t ) S ( s ) - S(t + s), for any t, s e IK. Moreover, for any t >~ 0, S ( t ) ' A --+ A is a continuous bijection on the compact set A, and therefore is an isomorphism from A to A. [2 Of primary importance in the theory of dynamical systems is the set ff -- {bounded complete orbits of S}. If this set J is bounded, then, by L e m m a 2.4, it is the maximal bounded invariant set that is; it is invariant, bounded and contains each bounded invariant set. If S has a global attractor A, then ./4 coincides with J . However, in the general case, ff needs not to be a global attractor, even if ,.7 is compact and attracts compact sets (see Example 2.24 below; examples involving continuous dynamical systems are also found in [98]).
2.2. co and a-limit sets As indicated in the introduction, we are going to construct global attractors as co-limit sets of bounded sets. For this reason, we now recall the definition and main properties of co and a-limit sets. DEFINITION 2.6. Let E be a nonempty subset of X. (i) We define the co-limit set co(E) of E as X
N
N ( U
seG +
scG +
.
t>/s. t e G +
(ii) We define the a-limit set a ( E ) of E as X
N( U
seG +
H (t, E ) )
.
(2.4)
t>/s, t e G +
REMARK. Let z e X be such that there exists a negative orbit u= through z. We define the a,=-limit set at,= (z) of the orbit u= as X
(2.5) scG +
An equivalent description of the co and a-limits sets is given in terms of limits of sequences as follows: LEMMA 2.7 (Characterization Lemma). Let E be a nonempty subset o f X. Then, I
co(E) -- I y ~ X ]there exist sequences t,, ~ G + and z,l ~ E such that Ik
t,~
> + o o and S(t,~)z,,
n --~ + o o
> y ~,
n ~ +oo
I
G. Raugel
894 [
~ ( E ) = I y E X I there exist sequences tn E G +, t,,
> +oc,
x,,
n --+ + o c
Xn E
> y where
X and
Zn E
E such that
x,, = Uz,, ( - t n )
n --+ + oc
and u z,, is a negative orbit through z,, 1. Likewise, if z is a point in X such that there exists a negative orbit u~ through z, then ~u: (z) -
[
I y E X I there exists a sequence t,, E G + such that t,7 and u z ( - t n )
> y].
> +oo n--++oo
(2.6)
tz ---+ - q - ~
We remark that if E is a nonempty subset of X, we have the following inclusions, for t E G +,
co(E) -- co(S(t)E), S(t)co(E) C co(E),
~(E) C ~(S(t)E), S(t)ot(E) C or(E).
(2.7)
REMARK. If E is a nonempty subset of X, then, generally, co(E) :/: U:EE co(z). Indeed, let us consider the flow S(t) generated by the following ordinary differential equation
y= y(1-
y)(2 + y).
For any Yo E ]R, l i m , ~ + ~ S(t)yo exists and limt~+e~ S(t)yo = 1 if Yo > 0, S(t)O = 0 and l i m t ~ + ~ S(t)yo = - 2 if yo < 0. Thus, co(yo) = 1 if Y0 > 0, co(0) = 0 and co(Y0) = - 2 if yo < 0. However, for any t ~> 0, S ( t ) [ - 2 , 1] = [ - 2 , 1] and therefore co(E) = [ - 2 , 1]. EXAMPLE 2.8. The co-limit set can be empty as the following example, which appeared in the thesis of Cooperman [47] (see also [38]) shows. Let Ho be the Banach space of all real sequences x = {xi, i >>, 1 I xi --+ 0 as i ~ +oo}, equipped with the norm IlxllHo = sup//> ! Ixil. We introduce the map T : x = (x l, x2 . . . . ) E Ho w+ (1, x l, x2 . . . . ) E HO and define the map U : Ho --+ Ho by U(x) = x/IIx III-/o if IIx IIm > 1 and U(x) = x if IIx IIm ~< 1. Finally, we let S = T o U. We remark that, since S" = T" o U, for any x E Ho, the first n terms in the sequence S n (x) are equal to 1. Clearly, for any x0 E H0, the co-limit set of x0 is empty. Indeed, by the characterisation lemma, if co(x0) :/: 0, there exists y ~ H0 and a sequence nj E l~, nj ---+ -+-oo, such that S"J x0 --+ y. Since y E Ho, there exists i0 E N such that, for i >~ io, ]yi] <~ 1/2. But, for nj >~ io, IIS"J (xo) - yll/40 ~> 1/2, which contradicts the convergence of sn~ xo towards y. Likewise, co(E) = 0 for any subset E of H0. Another obvious example is given by the flow generated on It~ by the ordinary differential equation ~ - 1. Thus, we can wonder when the co-limit sets are nonempty and which are their properties. DEFINITION. Let A, E be two (nonempty) subsets of X. The set A is said to attract E if
6x(S(t)E, A)
) O, t -+ -t-oo
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that is, for any e > 0, there exists a time r -- r (e, A, E) >~ 0 such that S(t)E C Nx(A,E),
t~>r.
The following properties are elementary, yet fundamental. LEMMA 2.9. Let E be a nonempty subset o f X and S a semigroup on X. Assume that co(E) is nonempty, compact and attracts E, then the following properties hold: (1) co(E) is invariant. (2) If moreover E is connected, co(E) is invariantly connected. If in addition, either co(E) C E or S(t) is a continuous semigroup, then co(E) is connected. PROOF. Statement (1) as well as the connectedness of co(E) in the case of a continuous semigroup are well known and their proofs can be found, for instance, in [94, Chapters 2 and 3]. The connectedness of co(E) in the case where co(E) C E is shown in [86, Lemmas 4.1 and 4.2], for example. Here, adapting arguments given in [86], we prove Statement (2). (i) We begin by showing by contradiction, that, if co(E) is a nonempty, compact set, which attracts E, then co(E) is invariantly connected, if E is connected. To simplify the notation, we assume, without loss of generality, that S is a discrete dynamical system. If co(E) is not invariantly connected, then co(E) -- F1 U F2, where Fl, F2 are disjoint, nonempty, compact, positively invariant sets. We fix e > 0 so that N x (Fl, e) O N x (F2, e) -0. Since Fi and F2 are invariantly connected, the continuity of the mapping S implies that there exists 6, 0 < 6 < ~, such that S ( N x ( F i , 6)) C N x ( F i , e), for i -- 1,2. As co(E) attracts E, there exists no c l~l, such that,
S"E C Nx(co(E),~),
Vn >~no.
(2.8)
In particular,
S"~ C (Nx(FI,6) NS"~176
(2.9)
We note that, if x 6 E and S"~ ~ N x ( F i , 6), then S"~ E N x ( F i , e) N N x ( c o ( E ) , 6), which implies that S"O+lx ~ N x ( F i , 6 ) . Thus, by recursion, S"x ~ N x ( F i , 6 ) , for n ~> no. It follows that, if there exists j , j -- 1 or 2, such that N x (Fj, 6) N S ''~ E -- 13, then N x ( F j , 6) n S" E -- 13, for n ~> no, which means that Fj -- 13. Hence 9t'/~ ~ N x ( F i , 6) n S 'z0 E is nonempty, for i -- 1,2. Thus, we have just proved that the connected set S ''~ E is the union of the two nonempty, closed, disjoint subsets 5t-~ and ~ 0 , which is a contradiction. Therefore, co(E) is invariantly connected. (ii) To prove that co(E) is connected when E is connected and co(E) C E, we again argue by contradiction. If co(E) is not connected, then co(E) -- Fl U F2, where Fl, F2 are disjoint, nonempty, compact sets. We fix 0 < 6 ~< e so that N x ( F j , e) n N x ( F 2 , e) -- 0. As in (i), there exists no 6 1~1,such that the inclusions (2.8) and (2.9) hold. The property co(E) C E together with the invariance of co(E) yields that co(E) C S ''~ E. Thus, we deduce from (2.9) that Fi C N x (Fi, 6) N S "~ E, for i -- 1,2. Hence, the connected set S ''~ E is the union of the
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two nonempty, closed, disjoint subsets .T"~ and .T"~ which is a contradiction. Therefore, co(E) is connected. (iii) Finally, we prove by contradiction that co(E) is connected, when S(t) is a continuous semigroup. Arguing as in (ii), we show that there exists no 6 N, such that the inclusions (2.8), for any t >~ no, and (2.9) hold. To obtain a contradiction, we need to prove, as in (ii), that ~ 0 __ N x ( F i , 6 ) N S'~OE is nonempty for i -- 1,2. Since S(t) is a continuous semigroup and that Fl, F2 are compact sets, there exists a positive time r such that, for 0 ~< t ~< r, S ( t ) ( N x ( F i , 6)) C Nx(Fi, e), for i -- 1,2. Let x 6 E be such that S(no)x ~ N x ( F i , 6). Then, for 0 ~< t ~< r, S(no + t)x E Nx(co(E), 6) N Nx(Fi, e), which implies that, for0 ~< t ~< r, S ( n o + t ) x ~ N x ( F i , 6). Thus, by recursion, S(t)x ~ Nx(Fi, 6), for t ~> no. We now conclude like in (i) that ~/0 _ N x ( F i , 6 ) A ShOE is nonempty, for i - - 1,2. REMARKS 2.10. (i) We deduce from the above lemma that, if E is connected and if co(E) contains only fixed points of S, then co(E) is connected, which had been proved in [108, Lemma 2.7]. This result is useful when one wants to show, in the case of gradient systems, that the co-limit set co(x0) of an element x0 6 X is a single equilibrium point (see [108] and [26] as well as Section 4.1 below). (ii) Adapting the proof of Lemma 2.9 one shows that, if the or-limit set c~(E) of the nonempty subset E of X is nonempty, compact and 6 x ( H ( t , E), ~(E)) --+ 0 as t ~ + ~ in G +, then or(E) is invariant. If moreover, H(t, E) is connected for any t E G +, or(E) is invariantly connected. If, in addition, either or(E) C E or S(t) is a continuous semigroup, then or(E) is connected. Of course, similar properties hold for the ot,:-limit set of negative orbits Uz through z~X. The following property had already been proved for instance by Hale in 1969 [92]. PROPOSITION 2.1 1. Let S be a semigroup on X. If E is a nonempty subset of X and there exists r ~ G + such that V+ (E) is relatively compact, then co(E) is nonempty, compact and attracts E. In the case X - •", the hypotheses of Proposition 2.11 hold if we only assume that y + ( E ) is bounded. If there exists to 6 G + such that S(t) is compact for t > to in G +, then the hypotheses of Proposition 2.11 still hold, when y+ (E) is bounded, even if X is infinite-dimensional. Semigroups that are compact for t > 0 occur in the study of parabolic equations or retarded differential equations etc. However, there are examples, like the damped wave equation, where the associated semigroup is not compact and yet the properties given in Proposition 2.11 hold. For this reason, we consider the more general class of asymptotically smooth semigroups, which has been introduced in 1972 by Hale et al. [100]. DEFINITION 2.12. The semigroup S is asymptotically smooth if, for any nonempty, closed, bounded set B C X, there exists a nonempty compact set J -- J (B) such that J attracts {x e B lS(t)x 9 B, u ~ G+}.
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One remarks that S is asymptotically smooth if and only if, for any nonempty, closed, bounded set B C X for which S ( t ) B C B, for any t E G +, there exists a compact set J C B such that J attracts B (see [94]). Obviously, if the semigroup S(t) is compact for t > to ~> 0, then S(t) is asymptotically smooth. Asymptotically smooth semigroups have the following important property (see [94, Chapters 2 and 3]). PROPOSITION 2.13. If S is an asymptotically smooth semigroup on X and E is a nonempty subset of X such that v + ( E ) is bounded for some r E G +, then co(E) is nonempty, compact, invariant and attracts E. In 1987, Ladyzhenskaya [ 137] introduced the notion of asymptotically compact semigroups. DEFINITION 2.14. The semigroup S is asymptotically compact if, for any bounded subset B of X such that V+ (B) is bounded for some r E G +, every set of the form {S(t,,)z,, }, with z, E B and t, E G +, t,, ---~,~--,+~c +oc, t, ~> r, is relatively compact. We remark that Proposition 2.13 at once implies that every asymptotically smooth semigroup S is asymptotically compact. On the other hand, Ladyzenskaya [ 137] had proved that, if S(t) is an asymptotically compact semigroup on X and E is a nonempty subset of X such that v + ( E ) is bounded for some r E G +, then co(E) is nonempty, compact and attracts E. From this result, one immediately deduces that any asymptotically compact semigroup is asymptotically smooth, obtaining thus the following result. PROPOSITION 2.15. Let S be a semigroup on X. Then, S is asymptotically smooth if and only if it is asymptotically compact. Since the concepts of asymptotically compact and asymptotically smooth are equivalent, I will not distinguish them in the sequel. I prefer to use the term asymptotically smooth, because it appeared first. Moreover, the term asymptotically compact is now misleading, because some authors, like Ball [15], Sell and You [198], call a semigroup asymptotically compact if for any bounded subset B of X, any set of the form {S(t,,)z,,}, with z,, E B and t,, E R, t,, --+,,-~+oc +cx~, t,, ~> r, is relatively compact. The property of eventual boundedness of orbits of bounded sets is included in this definition, whereas, this is not the case for asymptotically smooth semigroups. In 1982, in his study of the homotopy index for semiflows in non locally compact spaces, Rybakowski introduced the related concept of admissibility (see [193] and also the appendix of [ 103]). Let S(t) be a (local) continuous semigroup on X and N be a closed subset of X. The subset N is called S-admissible if for every sequence {z,,} C N and every sequence t,, E R, t,, --+,~+~c +oc, such that
{S(t)z,, It E [0, t,,]} C N,
for any n E N,
the set {S(t,,)z,, In E N} is relatively compact. As pointed out by Rybakowski, the notions of admissibility and asymptotically smooth semigroups are not equivalent [ 103].
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REMARK 2.16. (i) One can also show that, if S(t) is an asymptotically smooth semigroup on X and E is a nonempty subset of X such that F - ( E ) is nonempty and bounded, then o~(E) is nonempty, compact, invariant, and 8 x ( H ( t , E), or(E)) --+t~+~ 0 (for a proof, see [94, Chapters 2 and 3] or [77]). (ii) Likewise, one shows that, if S(t) is asymptotically smooth and there exists a bounded negative orbit u z E C ~ 0], X) through some z 6 X, then the C~u.limit set ot,: (z) is nonempty, compact, invariant and 8 x ( u z ( - t ) , c~,=(z)) --+t~+oo O. Statement (ii) of Remark 2.10 implies that otu=(z) is invariantly connected. If, moreover S(t) is a continuous semigroup, then ot,,=(z) is connected. Proposition 2.13 indicates that, if a global attractor A exists, then A contains the co-limit set of any bounded set.
2.3. Global attractors We are now ready to recall the definition of a global attractor and state its basic properties. In this paragraph, we also give the fundamental theorem of existence of compact global attractors. DEFINITION 2.17. A nonempty subset A of X is called a global attractor of the semigroup S if: (1) A is a closed, bounded subset of X, (2) A is invariant under the semigroup S, (3) A attracts every bounded subset B of X under the semigroup S. In the same way, one defines local attractors. A nonempty subset J of X is a local attractor if J is closed, bounded, invariant and attracts a neighbourhood of itself. In the past, a special class of global attractors has mainly been studied: it is the class of compact global attractors. If S(t) is a continuous semigroup and the global attractor A is compact, it is straightforward to show that, given a trajectory y+(x0), there exist sequences of positive numbers e,l, t,z and a sequence of points Yn E A, such that e,7 --+,,~+ec 0, tn+l > t,,, t,+l - t,, --+,,~+oc + o c , and 3x(S(t)xo, S(t - t,,)yn) <<.en, for any tn <<.t <<.t,,+l. Moreover, 6x(y,,+l, S(tn+l - tn)y,,) decays to 0. We also remark that if A is the global attractor of a semigroup S(t), t E R, then, for any to > 0, A is the global attractor of the discrete semigroup generated by S(to). Conversely, if S(to) has a compact global attractor A0 and that S(t) is a continuous semigroup, then A0 is also the global attractor of S(t), t ~ ]I{. Before giving the main theorem of existence of global attractors, we describe some fundamental properties of the global (and local) attractors.
Properties of global attractors. The following properties of a global attractor are a direct consequence of its definition and of Lemma 2.4.
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LEMMA 2.18. If the semigroup S admits a global attractor A, the following properties
hold: (a) If B is an invariant bounded subset of X, then B C A (maximality property). (b) If B is a closed subset of X, which attracts every bounded subset B of X, then
A C B (minimality property). (c) A is unique. (d) A = {bounded complete orbits of S(t) }. In the case of a Banach space X, as an immediate consequence of L e m m a 2.9(2), we have the following connectedness result, due to Massatt [ 151 ]" PROPOSITION 2.19. Let S be a semigroup on a Banach space X and A be a compact,
invariant set attracting any compact set of X, then A is connected. In particular, if S has a compact global attractor A, then A is connected. Even when X is not a Banach space, we obtain some connectedness properties of the global attractor. The next proposition generalizes T h e o r e m 4.2 of [ 15] as well as Theorem 3.1 of [86]. PROPOSITION 2.20. If S is a semigroup on a connected metric space X and if A is a compact, invariant set attracting any compact set of X, A is invariantly connected. If in addition, S(t) is a continuous semigroup on X, then A is connected. PROOF. We begin by showing by contradiction that if A is a compact, invariant set attracting any compact set of X, A is invariantly connected. To simplify the notation, we assume, without loss of generality, that S is a discrete semigroup. If A is not invariantly connected, then A -- A l U A2, where A l, A2 are nonempty, disjoint, compact, positively invariant subsets of X. We fix e > 0 so that N x ( A l , e) n N x ( A 2 , e) = ~J. Since Al and A2 are invariantly connected, the continuity of the mapping S implies that there exists 3, 0 < 6 < e / 2 , such that S ( N x ( A i , 3)) C N x ( A i , ~), for i = 1, 2. We set, for i = 1, 2,
Xi - {x e X [there exists no - no(x) such that S(n)x ~ N x ( A i , 6), n >~no}. Clearly, X I n X 2 - - ~. Moreover, the following properties hold: (i) X- X! U X2. Indeed, let x E X. As x is attracted by A, there exists no(x) = no such that {S(n)x In >~ no} C N x ( A , 6). Assume that S(no)x c N x ( A l , 6). Then, S(no + 1)x 6 N x ( A , 3) n N x ( A l , e), which implies that S(no + 1)x 6 N x ( A I , 3). By recursion, it follows that S(n)x c N x ( A I , 6), for n ~> no and thus x E Xl. (ii) Xi :fi 0, for i - 1,2. Indeed, due to the positive invariance of Ai, we have the inclusion Ai C Xi. (iii) X I and X2 are closed sets. If x l E X l, there exist Ym c X l, m 6 I~, such that Ym converges to x t as m goes to oe. Since A attracts every compact set of X, there exists no > 0 such that {S(n)K I n ~ no} C N x ( A , 6), where K -- {xt} U {Ym ] m 6 1~}. The arguments of (i) imply that S(n)ym E N x ( A I , 6), for n >~ no and m 6 N. Due to the continuity of the map S(no), there exists m0 > 0 such thatd(S(no)ym, S(no)xi) <~3, f o r m / > m0. It follows
G. Raugel
900
that xl ~ Nx(A1, 28) C N x ( A l , e). Hence, {S(n)xl I n ~ no} C N x ( A I , 8) and xl 6 X1. We have thus proved that the connected set X is the union of the two closed, disjoint compact subsets X1 and X2, which is a contradiction. Therefore A is invariantly connected. To prove that A is connected when, in addition, S(t) is a continuous semigroup, we argue by contradiction and follow the lines of the above proof. If A is not connected, then A = A l U A2, where A l, A2 are disjoint, nonempty, compact sets. Like before, one introduces two positive constants e and 8 and, for i = 1, 2, the set
Xi -- {x ~ X I there exists r0 - r0(x) such that S(t)x ~ N x ( A i , 8), t >>,r0]. Clearly X I A X2 = ~ and the following properties hold: (a) X = X l U X2. Let x 6 X. Since x is attracted by A, there exists r > 0 such that {S(t)x It >1 r} C N x ( A , 6). As S(t) is a continuous semigroup, {S(t)x l t >~ r} is connected and therefore is either completely contained in N x ( A l , e) or Nx(A2, e). (b) Ai C Xi, for i = 1, 2. Indeed, let a E Ai. Due to the invariance of A, there exists b E A and T > 0 such that S ( T ) b = a and S(t + T)b = S(t)a ~ A, for any t ~> 0. Since S(t) is a continuous semigroup, {S(t)a = S(T + t ) b l t >~ 0} is connected and therefore completely contained in Ai. (c) One shows like in (iii) that X l and X2 are closed subsets of X. Thus, X is the union of the two closed, disjoint compact subsets X1 and X2, which is a contradiction. Hence A is connected. Fq REMARKS. (1) As an immediate consequence of Proposition 2.20, we notice that if the set A satisfies the hypotheses of Proposition 2.20, if, for any x 6 X, the map t 6 [0, + c ~ ) S(t)x ~ X is continuous and if X is not connected, then every connected component of X contains exactly one connected component of A. (2) If S(t) is not a continuous semigroup on X, then, in general, under the hypotheses of Proposition 2.20, the set A is only invariantly connected. Gobbino and Sardella [86] give an example of a discrete semigroup defined on a connected metric space X which has a non connected compact global attractor. Modifying the proof of [86, Proposition 4.3] in the above way, one shows that, if S(t) is any semigroup on X and if A is a compact, invariant set attracting every compact set of X, then either A is connected or A has infinitely many connected components. This fact directly implies that, when X is connected and locally connected, the compact global attractor r is connected. Before giving the fundamental theorem of existence of compact global attractors, we introduce the important notions of stability and dissipativness. Let S be a semigroup on the metric space X. A set J Q X attracts points locally if there exists a neighbourhood U of J such that J attracts each point of U under S. We recall that E Q X is (Lyapunov) stable if, for any neighbourhood V of E, there exists a neighbourhood W Q V of E such that
S ( t ) W Q V,
Yt 6 G +.
(2.10)
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We say that E is stable for t ~> r in G + if the above inclusion (2.10) holds only for any t E G +, t ~> r. The set E is asymptotically stable if E is stable and attracts points locally. The set E is uniformly asymptotically stable if it is asymptotically stable and attracts a neighbourhood of itself. The next theorem describes a stability result for compact global attractors. THEOREM 2.21. Let S be a semigroup on X. If A is a compact, positively invariant set, which attracts a neighbourhood of itself the following properties hold: (i) If the mapping (t, z) E G + x X w-~ S(t)z E X is continuous, then A is stable and thus uniformly asymptotically stable. In particular, A is uniformly asymptotically stable if S is a discrete semigroup. (ii) If S(t) is a continuous semigroup, then, f o r any r > O, A is stable f o r t >~ r. PROOF. (i) Here, for any r / > 0 and any E C X, we set V,I(E) = N x ( E , 7). Let e > 0 be a fixed positive number. By assumption, there exists a neighbourhood Wl of A and ti E G + such that, for any t E G +, t ~> ti,
S(t)Wi C Ve(A). Moreover, the joint continuity of the mapping (t, z) E G + x X ~ S(t)z E X implies that, for any aoi E A, there exists JT(aoi) > 0 such that, for any z E X with 6x(z, aoi) < rl(aoi) and any t E G + A [0, ti ],
•x(S(t)z, S(t)aoi) < e, and hence, due to the positive invariance of A
S(t)z E Vc(S(t)A) C V~(A).
(2.11)
Since A is compact, it can be covered by a finite number k of neighbourhoods Vrl(aoi)(aoi). Hence, there exists J7 > 0 such that a C V,I(A) C Uii-~ V,1iaoi)(aoi). We deduce from 2.11 that
S(t)V,I(A) C Ve(A),
Vt E G + M[0, tl].
If we set W = Wl M V,7(A), then
S ( t ) W C Ve(A),
Vt E G +.
We finally remark that, for any discrete dynamical system, the mapping (t, z) E l~l x X S(t)z E X is continuous. (ii) We recall that, due to a result of [37], if S(t) is a continuous semigroup, the mapping (t, z) E It, +cx~) x X ~ S(t)z E X is continuous, for any r > 0. The arguments used in (i) then show that A is stable for t 7> r. D Other results involving stability properties are given in [94]. We finally introduce the concept of dissipativness.
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DEFINITION 2.22. The semigroup S is point (compact) (locally compact) (bounded) dissipative on X if there exists a bounded set B0 C X, which attracts each point (compact set) (a neighbourhood of each compact set) (bounded set) of X. If the semigroup is bounded dissipative, there exists a bounded set B1 C X with the property that, for any bounded set B C X, there exists r = r ( B ) 6 G + such that y + ( B ) C B1. Such a set B I is called an absorbing set for S. If S is a semigroup on X, which admits a global attractor, then S is bounded dissipative. Of course, if S is bounded dissipative, it has not necessarily a global attractor, as it is easily seen in Example 2.8, where the unit ball in H0 is an absorbing set and nevertheless there does not exist a global attractor. An interesting implication of dissipativness for asymptotically smooth semigroups is as follows [99,38]: THEOREM 2.23. Let ~ be a family o f subsets o f X. If S is an asymptotically smooth semigroup on X and there is a bounded set B in X that attracts each element of.T, then there exists a compact invariant set which attracts every element o f ~ . EXAMPLE 2.24. The notions of point dissipative, compact dissipative and bounded dissipative are not equivalent in general, as is shown in this example, which is a modification of an unpublished example of the thesis of Cooperman [47] and is described in [38]. Let H = 12 be the Hilbert space of square summable series {x = (x l, X2 . . . . ) I xi ~ IR, i = 1, 2 . . . . . ilxll2 = ~--2~ioo1 ix i 12 < CO} with the orthonormal basis ej -- (0 . . . . . O, 1, 0 . . . . ), j -- 1, 2 . . . . . where the number 1 appears at the j th position. Also, we denote by 0 the zero element of 12 and introduce the following points Xl.j
-
-
1 -~ej,
j ~> 2,
and
1 211_2 xn,j -- - ~ e j + el,
j ~ n ~ 2.
We define an auxiliary map T : H --+ H by first setting 1 T(x,,,j) -- ~-s
+ 2nel,
n ~ 1, j ~> max(n, 2).
(2.12)
Then, we extend T to a map from H to H as follows. I f x belongs to the ball BH(Xn,j, On.j) 1 of center x,,,j and radius c , . j , n >~ 1, j >1 max(n, 2), where cn,j = 41 2J' we set, using (2.12):
T(x) =
Ilx - Xn,jllx,,j + c , . j ( x - xn,j) + (c,,j - Ilx - x,,,jl[)T(xn,j) Cll, j
otherwise, we simply set T (x) = x. We finally introduce the mapping S : H --+ H given by
x~H.
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Cholewa and Hale [38] have shown that S ' H ~ H is continuous and asymptotically smooth. Moreover, each point of H has a neighbourhood which is attracted by the zero element 0 in l 2, which implies that S is compact dissipative. Yet, there is no bounded set in H attracting each bounded set of H. Indeed, one at once proves by recursion that 1 S j (e j) -- ~-fej + 2 j e l ,
j >~ 2,
which implies that, for each k >~ 1, the set y1,+ ({x 6 H [ Ilx ]] -- 1 }) is unbounded in H. However, the different notions of dissipativness can be equivalent in some cases. For example, it has been proved by Massatt ([ 151 ], see also [ 116]) that point dissipativness and compact dissipativness are equivalent for certain classes of neutral functional differential equations which arise in connection with the telegraph equation. More generally, if the semigroup S is asymptotically smooth, one can relate the different types of dissipativness. Assume that S is asymptotically smooth and point dissipative. If, for any compact subset K of X, there exists ~ >~ 0 such that y + (K) is bounded, then S is locally compact dissipative. If, for any bounded subset B of X, there exists T ~> 0 such that y + (B) is bounded, then S is bounded dissipative (see [ 151 ] and also [94]). The existence of compact global attractors is actually a consequence of the latter property and of Theorem 2.23, although we shall give a different proof below. Existence o f global attractors. Let us recall that a set A is said to be a maximal compact invariant set if it is a compact invariant set under S and is maximal with respect to these properties. One of the basic results concerning the existence of a maximal compact invariant set (see [100]) is the following:
THEOREM 2.25. If S is a semigroup on X such that there is a nonempty compact set K that attracts each compact set o f X, and A -- ("]tcc+ S ( t ) K , then (i) A is independent o f K, (ii) A is the maximal compact invariant set in X, (iii) A attracts compact sets. A n d the connectedness properties given in Propositions 2.19 and 2.20 hold. REMARK. The above set A is invariant and attracts compact sets. It is expected to be the compact global attractor. Unfortunately, without additional hypotheses, this is not the case, as we have already seen in Example 2.24. A simpler example where A satisfies the properties of Theorem 2.25 and is not a compact global attractor is described in [98]. Let H be a separable Hilbert space with orthonormal basis e j, j ~> 1, and let ~j, j ~> 1, be a sequence of real numbers, 0 < ~j < 1, for j ~> 1. We introduce the linear mapping defined on the basis vectors by S e j -- ~.jej. Since IlSxllH <<, Ilx ][H, for all x 6 H, y + (B) is bounded if B is bounded. Clearly, S"x --+ 0 as n --+ +cx~, for any x 6 H; and {0} attracts a neighbourhood of every point and thus every compact set. If ~.j ~ ~ < 1, then {0} is the global attractor of S. But, if, for instance, ~j ~ 1,
904
G. Raugel
when j --+ + ~ , then {0} cannot be a global attractor. One remarks that, in the first case, the radius r ( a ( S ) ) of the spectrum of S is strictly less than 1, while in the second case r ( a ( S ) ) = r(aess(S)) is equal to 1, where r(aess(S)) is the radius of the essential spectrum of S. This example is actually a simple illustration of the general theorem (see [98] and also [94, Section 2.3]), which states that if S is a bounded linear point dissipative map on a Banach space X, then {0} attracts compact sets. It is the compact global attractor if and only if r ( a ( S ) ) < 1 or also if and only if r(aess(S)) < 1. We now state and prove the fundamental theorem of existence of a compact global attractor. A first version of it is due to [ 100]. Several different proofs of the statements below can be found in [93, Theorem 2.2], [94, Theorem 2.4.6], [ 137, Theorem 3.1 ] (see also [202] and, for a generalization to multivalued maps, [15]). Here we give a simple proof, using mainly Proposition 2.13. THEOREM 2.26 (Existence of a compact global attractor). The semigroup S(t), t E G +, on X admits a compact global attractor A in X if and only if (i) S(t) is asymptotically smooth, (ii) S(t) is point dissipative, (iii) For any bounded set B C X, there exists r E G + such that V+ (B) is bounded. T Moreover, (2.13)
A -- U { w (B) ] B bounded subset of X }.
And the connectedness properties given in Proposition 2.19 and Proposition 2.20 hold. In the proof of Theorem 2.26, we use the following auxiliary result. LEMMA 2.27. Assume that the semigroup S(t), t ~ G +, on X is point dissipative and that, f o r a n y boundedset B C X, there exists r ~ G + such that v + ( B ) is bounded. Then, there is T a bounded set Bl C X such that, for any compact subset K of X, there exist e -- e ( K ) > 0 and t l -- t l (K) ~ G + such that S ( t ) ( N x ( K , e ) ) C B1,
Vt ~ t l ( K ) , t 9 G +.
(2.14)
PROOF. Without loss of generality, we may assume that G + - [ 0 , + ~ ) . Since S(t) is point dissipative, there exists a bounded set B0, which may be assumed to be open, such that, for any x0 E X, there is a time t* (x0)/> 0 with S(t)xo C Bo,
Vt ~ t* (xo).
As S(t* (x0)) is continuous from X into X, we can find e(xo) > 0 such that S(t*(xo))(Nx(xo, e(xo))) C BO, and thus, for s ~> ro , where ro is chosen so that g l-+(Bo) is bounded, 0
s(s
(zo
c y+(8o r0
= 8l
Global attractors in partial differential equations
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If K is a compact set in X, we can cover K by a finite number of neighbourhoods N x ( x o i , 8(xoi)), 1 <<,i <~k, where xoi e K. Moreover, there exists e(K) > 0, such that K C N x ( K , e ( K ) ) C U /i--k = l Nx(xoi, e(xoi)). Finally, if we set tl (K) -- maxl <~i~
S ( t ) ( N x ( K , a ( K ) ) ) C Bl,
'v't ~> tl.
(2.15) Q
PROOF OF THEOREM 2.26. Clearly, if A is a compact global attractor, then the properties (i), (ii), (iii) hold. Conversely, we assume now that the properties (i), (ii), (iii) are satisfied; without loss of generality, we may also suppose that G + -- [0, +cx~). Let Bl be the bounded set which has been constructed in Lemma 2.27. We set A = co(Bl ). Due to Proposition 2.13 and Lemma 2.9, A is nonempty, compact, invariant and attracts B l. Actually, A attracts any bounded set B of X. Indeed, again according to Proposition 2.13, the co-limit set K = co(B) is nonempty, compact and attracts B. Let e be a real number such that 0 < e < e(K), where a ( K ) has been introduced in Lemma 2.27. Since K attracts B, there exists to > 0 such that
S(t)B C N x ( K , a ) ,
t ~ to,
and therefore, for t ~> 0,
S ( t ) S ( t i ( K ) + t o ) B C S ( t ) S ( t I ( K ) ) N x ( K , a ) C S(t)Bl, where tl (K) has been defined in Lemma 2.27. Since A attracts Bl, it follows that A also attracts B. Due to Lemma 2.18, A contains any bounded invariant set and, in particular, the w-limit set co(B) of any bounded set B. Since A - co(A), the equality (2.13) holds. [] EXAMPLE 2.28. The existence of a global attractor does not necessarily imply that, for any bounded set B C X, the orbit v + ( B ) is bounded. For instance, if S(t) is a continuous semigroup, which is not continuous at t -- 0, then, due to the lack of continuity at t -- 0, the size of V+ (B) can grow to oo, when r ~ 0; such a semigroup generated by an evolutionary equation will be introduced in Section 4.5. Here we construct a continuous map S" 12 --+ 12, which has a compact global attractor and yet the image through S of the unit ball is unbounded. Like before, we consider the Hilbert space H -- 12 of square summable series, with the orthonormal basis e j, j -- 1,2 . . . . . Let ~p's e [0, +oo] ~-+ cp(s) E [0, 1] be a continuous function such that ~p(s) -- 1, if 0 ~< s ~< 89and ~0(s) - 0, if s ~> 88 We define the map S in the following way: 1
S(x) = jcp( -~ej - x S(x)
-
o,
el,
(, 1)
Vx E BH -~ej, -~ , j >/ 2,
otherwise,
where 0 is the zero element of 12. Clearly, Sk(x) - - 0 , for any x c H and k ~> 2; thus {0} is the compact global attractor. On the other hand, S(BH (0, 1)) is unbounded in H.
G. Raugel
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We recall that an equilibrium point of the semigroup S(t) is a point x E X such that S(t)x - x, for any t E G +. The following result had already been proved by Billotti and LaSalle in 1971 (see [21]). Here, we deduce it from Theorem 2.26. THEOREM 2.29. Assume that S(t) is either a continuous semigroup or a discrete semigroup and that S is point dissipative. If there exists tl E G + such that the semigroup S(t) is compact for t E G +, t > tl, then there exists a compact global attractor A in X. Moreover, if X is a Banach space and tl = O, there exists (at least) an equilibrium point of S(t). PROOF. We give the proof in the case where G + - [0, +cx~). Theorem 2.29 is a direct consequence of Theorem 2.26, if we show that, for any bounded set B C X, there exists r ~> 0 such that y + ( B ) is bounded. Since S(t)B is relatively compact, for t > tl, it is sufficient to show this property for any compact set. As S(t) is point dissipative, there exists a bounded set B*, which may be assumed to be open, which attracts every point of X. Let t2 -- tl + 2 and Bo -- N x ( B * , e0), where e0 > 0. Arguing as in the proof of L e m m a 2.27, one shows that, for any compact subset H of X, there exists t3(H) > t2 such that S ( t ) ( H ) C y+(S(t2)(Bo)), for any t ~> t3(H). Thus, it remains to show that y+(S(t2)(Bo)) is bounded. Let Ko -- S(t2)Bo; as in the proof of L e m m a 2.27, for any x0 6 K0, there are e(xo) > 0 and t(xo) > 0 such that S(t(xo))Nx(xo, e(xo)) C Bo. Since K0 is compact, we can cover it by a finite number k of neighbourhoods N x (xoi, e(xoi)), 1 ~
t3(K0)-
max (t2 + t(xoi))
and
K3-
U
S(s)S(tl -Jr-1)B0.
s--I
If S(t) is a continuous semigroup on X, then K3 is a compact set. Moreover, one at once checks that S(s)S(t2)Bo C K3, for any s ~> 0. The last statement of the theorem can be found in [78] and in [21 ] (see also [ 102]). The proof of the above theorem is similar when S is a discrete semigroup. U] REMARK 2.30. Often, one can define a global attractor in a weak sense, when the semigroup is bounded dissipative. For example, let X be a separable, reflexive Banach space; we denote by Xw the space X endowed with the corresponding weak topology. Assume that S(t) is a bounded dissipative continuous semigroup, such that, for each fixed t, S ( t ) : X w --+ Xu, is continuous. Then there is a positive number r such that the ball Bo = Bx (0, r) of center 0 and radius r is an absorbing set for S(t). In particular, there exists r >~ 0 such that S(t)Bo C Bo, for t >~ r. Since X is reflexive and separable, the weak topology on B0 weak is metrizable. We denote by dw the corresponding metric. Since B0 weak is compact for the weak topology, the semigroup S(t)lr restricted to Y = y + B0 weak is obviously compact on Y for the weak topology and point dissipative. Therefore, by Theorem 2.29, there exists a global attractor A, bounded in X, compact in Xw, invariant, such that, for any bounded set B C X, lim 6w(S(t)B, .A) -- O.
t---~-t-o~
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2.4. Examples o f asymptotically smooth semigroups We now give some examples of asymptotically smooth semigroups. The motivation of the hypotheses in the first example probably comes from the D u h a m e l formula for evolutionary equations. If B is a bounded subset of a Banach space X, we set IIBllx - suphc8 Ilbllx. THEOREM 2.3 1. Let X be a Banach space and S(t), t 9 G +, be a semigroup defined on a closed positively invariant subset M o f X. Assume that one can write, f o r any t 9 G + and any u 9 M,
(2.16)
S(t)u - U ( t ) u + V ( t ) u ,
where U (t) and V (t) are mappings o f M into X, with the property that, f o r any bounded set B C M, there exists r0(B) 9 G + such that: (i) U ( t ) B is relatively c o m p a c t f o r any t in G +, t > r0(B), (ii) and, f o r any t in G +, t > r0(B), [V(t)u Ix ~ k(t, IIBIIx),
(2.17)
Vu r B,
where k" (s, r) 9 [0, + o e ) x [0, + o e ) ~ [0, + o e ) is a function such that k(s, r) ---->0 as s--+ +oc. Then S(t) is asymptotically smooth. Conversely, if a semigroup S(t), t 9 G +, admits a compact global attractor A on a Banach space X, then S(t) must have the representation (2.16) with U(t), V ( t ) satisfying (i) and (ii). PROOF. To simplify the notation, we assume that G + - [0, +oQ). Let B be a b o u n d e d subset of M and r - r (B) ~> 0 such that y + (B) is bounded. We consider a set of the form {S(t,,)z,,}, with z,, 9 B and t,, 9 [0, + o e ) , t,, ----~,,~+ec + o e , t,,/> r. It suffices to show that, for any e > 0, we can cover the set {S(t,)zn} by a finite n u m b e r of balls of radius r ~< e. Due to the condition (2.17), there exists tl -- tl (~, B) > 0 such that
IIv(,)u Ix k(,,
e/2,
Vu 9 y~+(B), Vt ~> t,.
(2.18)
We set t2 -- r o ( g + ( B ) ) and choose t3 > sup(ti, t2). Let nl 9 l~l be such that, for n ~> n l , t, ~> r + t3" the set {S(t,,)z,, I n <~ nl} can be covered by a finite n u m b e r of balls o f r a d i u s e. It remains to show that the set Bl -- {S(t,,)y I Y 9 B, t,, ~> r + t3} can be covered by a finite n u m b e r of balls of radius r ~< e. Each element of B1 can be written as
S(t3)S(t,, - t3)y - U(t3)S(t,, - t3)y + V(t3)S(t,, - t3)y. But U(t3)S(t,, - t3)y C U ( t 3 ) y + (B), which is compact and thus can be covered by a finite n u m b e r of balls of radius ~/2. Furthermore, due to (2.18), IIw(t3)S(t,, - t3)yllg <~ ~/2, for any n ~> n l. Thus B l can be covered by a finite n u m b e r of balls of radius ~. Conversely, assume that the semigroup S(t) admits a compact global attractor A on the Banach space X. Then, due to the compactness of A, for any z E X, there exists at least
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G. Raugel
one element a E .A such that d x ( z , a) = 6x(z, A); we thus choose such an element a and denote it by a = Pz. The mappings U ( t ) u = P S ( t ) u and V ( t ) u = ( l d - P ) S ( t ) u clearly satisfy the conditions (i) and (ii) of Theorem 2.31. N REMARK 2.32. In the case of a Hilbert space (see [202, Remark 1.1.5]) and more generally, in the case of a uniformly convex Banach space, we can choose U (t)u = PoS(t)u and V (t)u = ( l d - Po)S(t)u, where P0 is the projection onto the closed convex hull ~--6(A) of the global attractor A. In this case, U (t) and V (t) are continuous functions of u. REMARK 2.3 3. The following typical example of semigroup satisfying the hypotheses of Theorem 2.31 is often encountered in the study of semilinear equations. It has been first studied by Webb [207]. Let S(t), t >1 O, be a continuous semigroup on a Banach space X such that, for t ~> 0, S(t)u-
Z(t)u +
f0 t S ( t -
s)K(S(s)u)ds,
for any u E M, where M is a closed positively invariant subset of X, r ( t ) is a C ~ semigroup of linear mappings from X into X and K is a compact map from X into X. We assume that r ( t ) = ,~v' 1 (t) -~- r z ( t ) , t ~> 0, where r l (t) is a compact linear map for t > to ~> 0 and the linear map r z ( t ) satisfies (2.19) with k2:t E [0,-k-cx~) w-~ [0, + ~ ) is a function such that k2(t) --+ 0 as t --+ +oo. If the positive orbit of any bounded subset B of M is bounded, one easily shows ([207,208, 94]) that then S(t) is written as a sum S(t)u = U ( t ) u + V (t)u, with U, V satisfying the hypotheses of Theorem 2.31. In the applications, it is often difficult to determine the decomposition U (t) + V (t) given in Theorem 2.31. For this reason, we shall give other criteria of asymptotical smoothness. The following result is due to Ceron and Lopes (see [29]). We recall that a pseudometric p (., .) is precompact (with respect to the norm of X), if any bounded sequence in X has a subsequence which is a Cauchy sequence with respect to p. PROPOSITION 2.34. Let X be a Banach space and S(t), t E G +, be a semigroup defined on a closed positively invariant subset M o f X. Assume that, f o r any bounded set B C M, there exists r0(B) in G +, such that, f o r any ul, u2 in B and f o r any t in G +, t ~ r0(B),
IIs
- s
k(t,
IIBIIx)llul -u211x + P,,lIBtlx(u , u2),
(2.20)
where k: (s, r) E [0, +cx~) x [0, +cx~) w-~ [0, + c ~ ) is a function such that k(s, r) --+ 0 as s --+ +c~ and Pt,IIBIIx is a precompactpseudometric, f o r t in G +, t > r0(B). Then S(t) is asymptotically smooth.
The proof of Proposition 2.34 is very similar to the one of Theorem 2.31.
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We now give a third criterium of asymptotical smoothness, which deals with functionals and has first been introduced by Ball [14] and then applied to dispersive equations by Abounouh [1], Ghidaglia [82] etc. PROPOSITION 2.35. Let Y be a topological space and X be a uniformly convex Banach space such that X is continuously embedded into Y. We consider a semigroup S(t), t ~ O, on X satisfying the following properties: (i) f o r any t >~ O, the mapping S(t) is continuous on the bounded subsets of X, f o r the topology o f Y; (ii) f o r any bounded set B of X such that y + (B) is bounded in X f o r some r ~ O, every sequence S ( t j ) b j , where bj E B and tj --+.i~oc +ec, is relatively compact in Y; (iii) f o r any xo 6 X and t ~ O, we have f ' ( S ( t ) x o ) -- exp(-yt).T'(xo) +
e x p ( - y (t
-
-
S)).~'I ( S ( s ) x o ) d s ,
(2.21)
f0 p where g > 0, .)t-(x) -- Ilxllx + ~'0(x), p > 0 and ~o, 37! are continuousfunctionals on the bounded sets of X f o r the topology of Y and are bounded on the bounded sets of X. Then the semigroup S(t) is asymptotically smooth in X. PROOF. We recall that any uniformly convex Banach space is reflexive. Let B be a bounded set in X such that v + ( B ) is bounded in X for some r ~> 0 and let S ( t j ) b j be a sequence such that bj E B and tj ---->j--+oc +(30. Since X is reflexive, there exists a subsequence, still labelled by j, such that S ( t j ) b j ~ 2 weakly in X, where z 6 B0 ----c--d(y+(B)), the closed convex hull of y + ( B ) . Due to (ii), we can also suppose that S ( t j ) b j ~ z in Y as j --+ + e c . We want to show that S(t})bj, --+ z in X, where j ' is a subsequence of j, j~ ~ +oo. Since X is uniformly convex, it suffices to show that limj,__,+~ IIS(tj,)bj'llx - Ilzllx, for some subsequence j ' of j. As S ( t j ) b j converges weakly to z, we already know that Ilzllx ~< liminfj__,+~ IIS(tj)bjllx. Thus, it remains to show that l i m s u p j , _ , + ~ IIS(tj')bj, llx <<. Ilzllx, for some subsequence j ' of j. For each n 6 N, there exists a subsequence j " such that S(tj,, - n)bj,, ~ zn weakly in X and S(tj,, - n)bj,, --> z, in Y. By a diagonalization argument, we can construct a subsequence j ' such that S(tj,
-
n)bj,
~
z,,,
S ( t j , - n)bj, ~ z,, E Y,
weakly in X,
and
Vn 6 N.
(2.22)
We infer from (i) that, for any t >1 O, S(tj, - n -k- t)bj, --+ S(t)z,, in Y. In particular, S ( n ) z , - z. We consider now the equality (2.21) for t -- n and xo - S(tj, - n)bj,, when tj, - n > / r . From (2.21), (2.22) and the dominated convergence theorem of Lebesgue, we deduce that
p l i m s u p ( l l s ( t j ' ) b j ' l l x ) + ~o(z) <~ e x p ( - y n ) sup (]~'(x)l) j'-+ +~x) x e Bo +
f0"
e x p ( - y (n - s ) ) f ' , ( S ( s ) z , ) d s ,
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G. Raugel
or also, since 9t'(z) = exp(-yn).Y'(Zn) 4- fo e x p ( - y (n - s)).Tl(S(s)zn)ds, P) ~< 2 e x p ( - y n ) sup ([.T(x)l)+ [Iz[[Px limsup( [S(tj,)bj , IIx j ~--+ + cx~
x E Bo
Letting n go to +c~, we obtain that
limsuPllS(tj')bj'[[ x ~ Ilzllx. The proposition is thus proved. REMARKS
IS]
2.36.
(i) In the applications, the space Y is often the space X endowed with the weak topology of X. Then the condition (ii) is always satisfied. In this case, we can also assume that X is only a reflexive Banach space and replace the conditions on the functional f" by the hypothesis: (H. 1) Z ' : X ~ I~+ is continuous, bounded on the bounded sets of X and the properties S(tj)bj ~ z weakly in X, where bj is bounded in X, tj '-+j--++cx~ -Jr-00, and l i m s u p j ~ + ~ U(S(tj)bj) ~ U(z) imply that S(tj)bj converges strongly to z in Z, where Z is a Banach space such that X C Z, with continuous injection. Then, one shows like in the proof of Proposition 2.35 that, if B is a bounded set in X such that y + ( B ) is bounded in X for some r ~> 0 and if S(tj)bj is a sequence such that b j E B and t j ----~j--+cx~ -Jr-00, then there exists a subsequence j ' , such that S ( t j , ) b j , converges strongly in Z to some element z E B0 =-- ~ x (9/+ (B)). (ii) Moise et al. [166] consider more general functionals on Y, in the case where Y is the space X endowed with the weak topology. (iii) A similar result holds for discrete semigroups S provided that the equality (2.21) is replaced by
U(S"xo)
=
exp(-yn)U(x0) + ~
e x p ( - y ( n - m)lU, (Smxo)as,
Vn E N.
(2.23)
In z 0
Further example of asymptotically smooth semigroups: u-contracting and condensing semigroups. Another class of examples of asymptotically smooth semigroups is given by the u-contracting and condensing semigroups. Let now X be a Banach space and 13 be the set of its bounded subsets. The mapping ot :B ~ [0, +cxz), defined by or(B) = inf{l > 0 I B admits a finite cover by sets of diameter ~< l}, is called the (Kuratowski)-measure of noncompactness or, shorter, the ol-measure of noncompactness. It has the following properties (see [52], for example): (a) or(B) - - 0 if and only if B is compact; (b) or(.) is a seminorm, i.e., ot(kB) = ]klot(B) and ot(Bl + B2) ~ or(B1) 4- ot(B2);
Global attractors in partial differential equations
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(c) Bl C B2 implies a ( B l ) ~< a(B2); a ( B l U B2) = m a x ( a ( B l ) , a(B2)); (d) a ( B ) = a ( c o ( B ) ) ; (e) a is continuous with respect to the Hausdorff distance Hdistx. A continuous map S : X --+ X is a conditional a-contraction of order k, 0 ~< k < 1, with respect to the measure a if a (S (B)) ~< ka (B), for all bounded sets B C X for which S (B) is bounded. The map S is an a-contraction oforder k if it is a conditional a-contraction of order k and a bounded map. A continuous map S : X -+ X is a conditional a-condensing map, with respect to the measure a if a (S(B)) < a (B), for all bounded sets B C X for which S(B) is bounded and a ( B ) > 0. The map S is a-condensing if it is conditional a-condensing and bounded. Every bounded linear operator S can be written in the form S -- U + V, where the linear map U is compact and the spectral radius of V is the same as the radius of the essential spectrum of S. Also there exists an equivalent norm on X such that IISIIL(x,x) r(Cress(S)), with respect to this new norm. From a result of Nussbaum [171] stating that r(O'ess(S)) = l i m , , - , + ~ ( a ( S " ) ) I/'', it follows that S is an a-contraction with respect to a norm equivalent to the one of X if and only if r (O'ess(S)) < 1. The prototype of a-contraction is given by the nonlinear map S = U + V, where U is a nonlinear compact map and V is a globally Lipschitz mapping with Lipschitz constant k, 0 <~ k < 1. In this case, S is an a-contraction of order k. If S is a mapping satisfying the condition (2.20), for any bounded set B C X, where 0 <~ k < 1 is independent of B, S is an a-contraction of order k. Conditional a-condensing maps are asymptotically smooth (for a proof, see [150,94]). One should notice that asymptotically smooth maps are not necessarily conditional a condensing, as it is shown by an example in [38, Proposition 1.1]. This fact is somehow expected since the definition of a-condensing involves the metric whereas the definition of asymptotically smooth only involves the topology. So we can wonder if for any asymptotically smooth map S on a Banach space, there exists another equivalent norm for which S is conditional a-condensing. The following interesting property also holds: THEOREM 2.37. Let X be a Banach space. If S : X -+ X is an a-condensing and compact dissipative mapping, then S has a fixed point. This theorem was discovered independently by Nussbaum [ 172] and by Hale and Lopes [102]. Similar definitions and properties hold for continuous semigroups S(t), t E [0, +cx~). A semigroup S(t) on X is a conditional a-contraction if there exists a continuous function k:[0, + ~ ) --~ [0, +cx~) such that k(t) --~ 0 as t --+ +cx~ and, for each t > 0 and each bounded set B C X for which S ( t ) B is bounded, one has a ( S ( t ) B ) <~k ( t ) a ( B ) . The function k(t) is called the contracting function of S(t). The semigroup S(t) is an a-contraction if it is a conditional a-contraction and, for each t > 0, the set S ( t ) B is bounded if B is bounded. Likewise, the semigroup S(t), t >~ O, is a-condensing if, for any bounded set B in X and for any t > 0, the set S (t) B is bounded and a (S (t) B) < a (B) if a (B) > 0. It is shown in [29] that if S(t) is a semigroup satisfying the assumptions of Proposition 2.34 with M = X and the function k is independent of [IBllx, then S(t) is an
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a-contraction. In particular, if S(t) is a semigroup satisfying the assumptions of Theorem 2.31, where M = X and V (t) is a globally Lipschitz function with Lipschitz constant k(t) --+t~+~ O, then S(t) is an c~-contraction. Conditional a-contractions S(t) are asymptotically smooth (see [94]). Theorem 2.37 and Theorem 2.26 imply the following result (for a proof, see [94, Section 3.4]): THEOREM 2.38. Let X be a Banach space and S(t), t >~O, be a continuous semigroup on X. If moreover S(t), t >~O, is an u-contraction with contracting function k(t) ~ [0, 1), is point dissipative and if for any bounded set B C X, there exists r ~ 0 such that y+ (B) is bounded, then S(t) has (at least) an equilibrium point.
2.5. Minimal global B-attractors In some applications, it happens that there exists an unbounded invariant set that attracts all bounded sets. For this reason, we recall here the more general definition of a minimal global attractor. Let X be a metric space and S be a semigroup on X. Following Ladyzenskaya [ 137], we say that a set A C X is a minimal global B-attractor if it is a nonempty, closed, set that attracts all bounded sets of X and is minimal with respect to these properties. The following result was noted in [ 110]. PROPOSITION 2.39. The semigroup S on X admits a minimal global B-attractor .Ax on X if S is asymptotically smooth and if for any bounded set B C X, there exists r ~ G + such that V+ (B) is bounded. Moreover, A x is invariant and
As--Clx(U{CO(B)IB
boundedsubsetofX}).
PROOF. It is a direct consequence of Proposition 2.13.
5
If, under the assumptions of Proposition 2.39, the union of the co-limit sets of all the bounded sets is bounded, then A x is the compact global attractor. It was asserted in [ 110] that, under the hypotheses of Proposition 2.39, the minimal global B-attractor is always locally compact. However, this is not the case as has been shown with an example by Valero (see [38]). Consider the flow S(t) of the linear ODE k = Bx where B is a n • n matrix, which, for example, has one positive eigenvalue and n - 1 negative eigenvalues, then the onedimensional unstable manifold of the origin is an unbounded minimal B-attractor. Another simple example is the flow S(t)" ]R 2 ~ ]R2 of the ODE .f - 0, ~ - - y . The minimal global B-attractor for S(t) is the x-axis. This equation has a first integral ~ ( x , y) = x. On every level set {Pc = {(x, y) I q0(x, y) = c}, S(t) has a compact global attractor (c, 0). This example is a special case of an evolutionary equation on a space X, which has a continuous first integral r It is often the case that on each level set r the associated semigroup S(t) admits a compact global attractor Ac. Then the minimal global B-attractor .Ax is given by A x = Clx(Uc~R Ac). Other examples of such systems with first integrals are studied in [ 111, Section 6].
Global attractors in partial differential equations
913
Examples of minimal global B-attractors, that are not necessarily global attractors, arise in the study of damped wave equations with local damping (see [ 110]). An example of an unbounded minimal global B-attractor is also given in [ 186]. Different notions of global attractors involving two different spaces are considered in [13,159,60,98,158], for instance. Often, the semigroups S(t) generated by evolutionary partial differential equations on unbounded domains are no longer asymptotically smooth on function spaces, which are large enough to contain all the interesting dynamics. One way to overcome this difficulty is to introduce two different adequate topologies on the space X, so that, on the bounded sets for the first topology, S(t) is asymptotically smooth for the second topology (see [158], for details).
2.6. Periodic systems In this paragraph, we very briefly indicate that the notion of global attractor can be extended to evolutionary equations, which are nonautonomous. Let X be a Banach space. We consider, for instance, the nonautonomous evolutionary equation db/ - - ( t ) = f (t, u), dt
u(s) = uo e X,
(2.24)
where f is a continuous map from IR x X into X and is Lipschitz-continuous in u on the bounded sets of X. Then, through each point (s, u0) of R x X, there exists a unique local solution u (t, s, u0, f ) of (2.24). Under appropriate hypotheses on f , this solution is global and we set u(t, s, uo, f ) = S(t, s)uo. The operator S(t, s) : X --+ X satisfies the relations S(s, s) = Id,
Vs ~ R,
S(t,s)=S(t,r)S(r,s),
for any t ~> r ~> s,
(2.25)
and has also has the following properties S ( t , s ) ~ C~
g s e l R , gt>~s, (2.26)
s(t, s)uo ~ c ~
+,c). x),
Vs ~ R, Vu0 ~ x.
For later use, we introduce a subset 9c C C~ x X, X), which consists of functions g(t, u) satisfying the above conditions. If there exists co > 0 such that f (t + co, u) = f (t, u), for any (t, u) E R x X, then S(t + co, s + co) = S(t, s),
for any t >~ s.
(2.27)
More generally, let (X, d) be a metric space and let us consider a family of operators S(t, s) : X --~ X, s e IR, t >~ s, satisfying the conditions (2.25), (2.26) and (2.27). Since S(t, s) is periodic, it is meaningful to introduce the associated period map To = S(co, 0) and study the existence of a global attractor for To. We begin with a lemma, which shows
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G. Raugel
that for period maps, there is an equivalent form of point dissipativness which is easier to verify (see [181] in the finite-dimensional case and [99] in the general case). LEMMA 2.40. Let S(t, s) : X --+ X, s 9 IR, t ~ s, satisfying the conditions (2.25), (2.26) and (2.27). Assume that, f o r any bounded set B C X, there exists no = no(B) such that Un>no T~ (Uo~<s
limsupl[s(t,s)uoll x <~ R,
(2.28)
t---+ + o c
is equivalent to the existence of a positive number r such that, f o r any (so, u0) 9 [0, + o o ) x X, there exists a time r > so such that
IIs< ,so>uoll
< r.
(2.29)
As a consequence of Lemma 2.40 and Theorem 2.26, we obtain the following result: THEOREM 2.41. Let S ( t , s ) : X --+ X, s 9 R, t ~ s, satisfying the conditions (2.25), (2.26) and (2.27). Assume that the property (2.29) holds, that To is asymptotically smooth and that, f o r any bounded set B C X, there exists no = no(B) such that Un>~no TO~(UO<<.s<~oS(~ s) B) is bounded. Then, To admits a compact global attractor Ao. Moreover, f o r any 0 <. cr < w, T~ = S(cr + co, ~r) has a compact global attractor A~ = s(,r, O)Ao. REMARK. If S(t, s) : X -+ X, s 9 R, t >~ s, is a family of operators satisfying the conditions (2.25), (2.26) and (2.27), we obtain a particular case of a periodic process by setting U (r, s)uo = S(r + s, s)uo. We recall that a family of operators U (r, s) : X -+ X for r ~> 0, s 9 i s a p r o c e s s o n X , i f U ( O , s ) = Id, U ( t , a + s ) U ( a , s ) = U ( o - + t, s), for any s 9 and for any ~ > 0, t > 0, U (t, s) 9 C ~ X) and U (t, s)u 9 C ~ + ~ ) , X), for any u 9 X and s 9 N. The process is periodic if there exists a positive number w such that U (t, s + w) -- U (t, s), for any t >~ 0, s 9 IR. Let us remark that processes are natural extensions of the notion of continuous semigroups. Indeed, if one defines the operators r ( t ) :[s, u] 9 R x X ~-+ [s + t, U(t, s)u] 9 IR x X, Z'(t), t ~> 0, is a continuous semigroup under the additional hypothesis that U (t, s)u is jointly continuous in (s, u). Unfortunately, since the time variable does not belong to a compact set, the semigroup s will never have a compact global attractor. Thus, when U(r, s) is not periodic, one needs to find another way to generalize the notion of compact attractor to nonautonomous systems. Let us come back to evolutionary equation (2.24). For any t/> 0, we introduce the translation c r ( t ) ( f ) defined by c r ( t ) ( f ) ( s , x) -- f ( t + s, x). We suppose now that the set .T" is a metric space, with the property that f e .T" implies that the translation c r ( t ) ( f ) belongs to .T'. For any t ~> 0, we define re(t) :X x .T" --+ X x ~" by re(t)(uo, f ) -- (u(t, O, uo, f ) , c r ( t ) ( f ) ) .
(2.30)
Global attractors in partial differential equations
915
One easily shows that re(0) = ld and that :r(t + s)(u0, f ) = rc(t)rc(s)(uo, f ) . If the family 9t- is chosen so that zr satisfies the continuity properties required in Definition 2.1, we have thus defined a continuous semigroup on X • .T', which is called the skew-product flow of S(t, s) or the skew-product flow of the associated process U(r, s). Under appropriate compactness hypotheses on ~ , one can thus study the existence of global attractors for the continuous semigroup 7r(t). Skew-product flows had been first exploited by Miller [160] and Sell [ 196] (see also [ 197]) in the frame of ordinary differential equations. Skew-product flows have also been associated to more general processes U(r, s) (see [49]). For further study of global attractors associated to nonautonomous systems, we refer to [34,35,161,94, 118,198].
3. General properties of global attractors In the previous section, only a few properties of global attractors have been given. In general, the invariance and attractivity of the global attractors, combined with some additional hypotheses, imply interesting robustsness and regularity properties. Often also, the flow restricted to the global attractor shows finite dimensional behaviour. In this section, we are going to briefly describe such additional properties for compact global attractors.
3.1. Dependence on parameters One of the basic problems in dynamical systems is to compare the flows defined by different semigroups. In the study of semigroups restricted to a finite dimensional compact manifold (with or without boundary), this comparison is made very often through the notion of topological equivalence (for the definition, see below). If the semigroups are defined on a finite or infinite dimensional vector space for example, then considerable care must be taken in order to discuss the behaviour of orbits at infinity. If each of the semigroups has a compact global attractor, one can hope to consider the topological equivalence of the flows restricted to the global attractors. This is the strongest type of comparison of flows that can be expected in the sense that it uses the very detailed properties of the flows. In particular, it requests the knowledge of transversality properties, that are very difficult to show in the infinite-dimensional case. For this reason, we begin with much weaker concepts of comparison, like estimates of the Hausdorff distance between the global attractors. We shall mainly give general comparison results and refer the reader to Section 4 and to [ 13,94,97, 101,105,106,108,111,130,187,201,204] for applications to (singularly) perturbed systems and discretised equations. In this paragraph, (X, d) still denotes a metric space and we consider a family of semigroups S~ (t), t E G +, depending on a parameter ~, E A, where A = (A, d~) is a metric space. For sake of clarity, we assume that all the semigroups S~ are defined on the same space X, although in many applications, each S~ may be defined on a different space X~. Then one has to determine first how to relate these spaces in order to have a concept of convergence of the semigroups S~, which replaces the hypotheses (H. 1a), (H. lb) or (H. 1c) given below. Such situations arise in the discretisation of partial differential equations, in problems on thin domains etc. (for more details see [ 101,105,111,112]).
916
G. Raugel
We assume that each semigroup S)~ has a global attractor A)~ and, if 1.o is a nonisolated point of A, we are interested in the behavior of A)~ when 1. --+ 1.0. We say that the sets A), are upper semicontinuous (respectively lower semicontinuous) on A at 1. = 1.0 if lim 6 x ( A z , AZo) = 0, XEA--+Xo
(respectively
lim
XcA--~Xo
8x(Azo, Az) = 0).
(3.1)
We say that the sets A)~ are continuous at 1.o if they are both upper and lower semicontinuous at 1.o. Due to the strong attractivity property and the invariance of the attractors, the upper semicontinuity property holds if the dependence in the parameter is not too bad. To get upper semicontinuity, one often assumes that either (H. 1a) There exist ~ > 0, ro 9 G + with ro > 0 and a compact set K C X such that
U
Az C K,
(3.2)
)~ENA(XO,rl)
and if 1.k --+ 1.o, xk 9 Azk, for k 7~ 0 and xk ~ xo, then Szk (ro)xk --+ Szo (ro)xo;
or (H. lb) There exist 7/> 0, to 9 G + with to > 0 and a bounded set Bo C X such that
U
Az C Bo,
(3.3)
)~ENA(X0,t]) and, for any e > 0, any r 9 G +, r ~> to, there exists 0 < 0 = 0(e, r) < ~ such that
SX (S~('t')x),, Szo(r)x)~) <~ e,
Vx)~ 9 A)~, V1. 9 NA(1.o, 0).
(3.4)
In most of the cases, the hypotheses (H. la) or (H. lb) are rather easy to check. Actually, in the case of evolutionary partial differential equations, the compactness condition 3.2 is often proved by showing that, due to an asymptotic smoothing effect, the attractors .A), are uniformly bounded with respect to 1. in a metric space X2 which is compactly embedded in the space X. Often also, the stronger convergence property: (H. lc) there exists to E G + with to > 0 such that Sz(t)x --+ Szo(t)x uniformly for (t,x) in bounded sets of G + x X2, as A. E A ~ 1.0, holds. Hypotheses (H. 1a) or (H. 1b) imply upper semicontinuity of the attractors at 1. = 1.0 (see [94, Theorem 2.5.2]). PROPOSITION 3.1. Let 1.0 be a nonisolated point of A. If the hypothesis (H.la) or (H.lb) holds, the global attractors A z are upper semicontinuous on A at 1. = 1.o; that is, lim)~eA~)~0 6X(A)~, A)~o) = 0. PROOF. (1) We give only the proof in the case when G + = [0, +oo). Under the hypothesis (H. 1a), the global attractors Az for 1. E NA (1.0, 7/) are compact. We remark that .Az is also the compact global attractor of the discrete semigroup Sz (r). To prove upper semicontinuity, it suffices to show that, for any sequences 1.k E NA (1.0, rl), k >~ 1, xk E .Azk, k >~ 1,
Global attractors in partial differential equations
917
such that )~k -+ )~0, xk -+ x0, the limit x0 E Az 0. Since A;~k is invariant under Sz k (to), there exists x k E Azk such that xk -- S~,k (r0)(x2). Without loss of generality, due to the compactness of K, we can assume that the sequence x~ converges to some element x~. The hypothesis (H.la) implies that x~ -- Sz k (r0)x0. Using a recursion argument, one thus 9
.
.
obtains an infinite sequence x~ e K, j --~ +oo, where SzJk(ro)x~ -- xo. Clearly, the complete orbit V(xo) - {S '1 (ro)xo In E Z} is bounded in X which implies that xo E .Az o. Lk (2) Let e > 0 be fixed. Since .Azo is the global attractor of Szo(t), there exists a time r~ ~> to such that
Szo(t)Bo C Nx(Azo, e/2),
'v't ~> rc.
(3.5)
By hypothesis (H. lb), there exists 0 > 0, such that, for )~ E NA ()~0, 0),
6x(Sx(r~)x~,, Sx0(r~)xx ) ~< e/2,
'v'x;~E A;~,
which, together with (3.5), implies that Sz(r,)A~, C Nx(Azo,e). Since Az is invariant, A~ c Nx(A~o, e). D In general, the lower semicontinuity property does not hold, as shown by the simple ODE ~f - - (1 - x ) ( x 2 - )~),
(3.6)
where ~ E [--1, 1]. Here, A0 = [0, 1], Ax = 1, for )~ < 0 and A~, = [ - q ' ~ , 1], for )~ > 0 and a bifurcation phenomenon occurs. In general, lower semicontinuity at a given point ~.0 is obtained only by imposing additional conditions on the flow. It is mainly known to hold in the case of gradient like systems, when all the equilibrium points are hyperbolic (see Section 4 below). However, as pointed out in [7], the lower semicontinuity property is generic, under simple compactness assumptions. We recall that a subset Q of a topological space A is residual if Q contains a countable intersection of open dense sets in A. We say that a property (P) of elements of A is generic if the set {)~ E A I~. satisfies (P)} is residual. Let K be a compact metric space and K c be the set of compact subsets of K. It is well known (see [132]), that, if A is a topological space, and f : A -+ K C is upper semicontinuous at any ,k E A, then there exists a residual subset A0 C A such that f is continuous at every )~0 E A0. Here we apply this property to compact global attractors A~,, satisfying the hypothesis (H. la), with ,~ E NA (X0, 17) replaced by )~ E A. COROLLARY 3.2. If the hypothesis (H.la) holds, with )~ E NA ()~0, 17) replaced by )~ E A, there exists a residual subset Ao of A such that the sets .A~ are continuous at any )~o E Ao. The parameter ~. can be the domain f2 C IR'l on which one defines a partial differential equation. In [7], Babin and Pilyugin have applied Corollary 3.2 to prove generic continuity of global attractors of nonlinear heat equations with respect to the domain S-2, thus recovering some of the earlier results of Henry [ 122,124].
918
G. Raugel
The hypotheses (H. 1a) or (H. 1b) do not allow to estimate the semidistance 6x (.Az, .Azo). It becomes possible, if one imposes stronger attractivity properties on AZo (see [ 101 ], [ 13, Chapter 8]): PROPOSITION 3.3. Assume that ~o is a nonisolatedpoint of A and that Hypothesis (H. lb) holds. Suppose also that there exist positive constants oto, go, Yo, co and c~ such that
6x(Szo(t)Bo, AZo) <. coexp(-otot),
Vt ~> to, t ~ G +,
(3.7)
and, for any )~ ~ NA ()~0, ~), for any x ~ Bo, 6x(Sz(t)x, Szo(t)x)
Vt ~to, t ~ G +,
(3.8)
then, there exist c > 0 and Ol <~ rl such that, for any )~ ~ NA (~o, 771), we have, 8x(Az, Azo) <~C3A ()~, ~o) ~~215176176162176
(3.9)
PROOF. We introduce the time tl--ln(~o60~,)~o)~~215176176176 We remark that tl /> to, if r/l > 0 is small enough. From the estimates (3.7) and (3.8), we deduce that 6x(Sz(tl)Az, Az o) <~ CaA()~, ~0) ~~215176176176 which implies (3.9) by invariance of .Az. V] The property (3.7) is difficult to verify. However, we shall prove it below for gradient systems, whose equilibria are all hyperbolic. REMARK 3.4. If the conditions (3.7) and (3.8) hold for every ~ ~ A, with constants oto, go,/o, co and c l independent of ~, we obtain the estimate Hdistx (Ax, Ax o) <~CaA (~, )~o)~~215176176176 .
(3.10)
In particular, the attractors Az are continuous at every )~ 6 A. Other concepts of comparison of the attractors Az, which are weaker than continuity, have been introduced in [111 ] and may be more appropriate when the limiting system Sz o is not necessarily dissipative. For sake of clarity, we assume now that G + - [0, +oo); obviously the results below also hold for maps. DEFINITION 3.5. Let X0 E / , , where L is a subset of A. The co-limit set ~L(A., X0) of the family of sets Ax, X ~ NA (X0, ~7) f-) L, r/> 0, is defined by
~,~(A.,z0)- ~ clx 0<~
U
Az.
(3.11)
ZCNA (XO,~)fqL
Several properties of the set ~/~ (A.,)~0) are given in [ 111 ]. If )~0 6 L and, for each ,k NA ()~o, 7) f') L, S~ has a compact global attractor, then the upper semicontinuity (respectively the lower semicontinuity) of the attractors at )~ - ,k0 implies that ~c (A.,)~o) c .A~0
Global attractors in partial differential equations
919
C1x(ULcLANA()~o,,1)r
(respectively A~0 c ~L(.A., ~0)). If is compact, it follows from the inclusion ~/~ (A., ~.0) c A~ 0 that the attractors are upper semicontinuous at ~0. We remark that the inclusion A~ 0 c ~L (A., ~0) does not imply lower semicontinuity of the attractors A~. Indeed, consider the ODE ~ -- - x ( ( - 1)'1~,, + (x - 1)2) with ,k,, -- 1/ n, that is L -- {1, 1/2 . . . . . 1/n . . . . }. There is no continuity of the attractors at )~ - 0" however, ~L (A., 0) - A0 - [0, 1]. We notice that ~L (A., ~.0) does not involve directly the semigroup S~0 . In particular, S~0 could be conservative. The following question then arises: how much information can we obtain about a conservative system by considering the limit of dissipative systems, when the dissipation goes to zero? We cannot hope to obtain too many specific properties of the dynamics of the limit system in this way, but one should be able to obtain some information about the manner in which the orbits of the dissipative systems wander over the level sets of the energy of the limit system. Consider the ODE it - v, iJ - f ( u ) - fly, where fl ~> 0 is a constant, f 9 C2(IR, I~) has only simple zeros and f ( u ) is dissipative (i.e., l i m s u p l ~ l ~ + ~ f(~') ~ ~< ot < 0). The energy functional is q~ (u, v) -- (1/2)v 2 - f0 ~f (s) ds. For fl > 0, the ODE is a gradient system and has a global attractor A/~. Let {s j, j -- 1, 2 . . . . . M} be the set of the saddle equilibrium points of the system. If 9 (s j) ~ cb (sk), for j =/=k, j, k = 1,2 . . . . . M, then, for any interval (o, r
o~L (A., 0) - {(u v) ~
I 9 (u, v) <~ cM },
where CM = max{q~(sj) j = 1,2 . . . . . M} (for details, see [111]). The limit ~L (A., ~0) only uses information about the attractors. As a consequence, the transient behaviour of the semigroups Sx for initial data not on the attractors is completely ignored. To gain some information about this transient behaviour, one can consider the following concept of co-limit set: DEFINITION 3.6. Let k0 9 [,, where L is a subset of A and let S~(t) be a semigroups on the metric space X. For a given subset B of X, the co-limit set respect to the f a m i l y o f semigroups Sx (t), ~ 9 L A NA(Z0, r/), r / > 0, is denoted and is defined in the following way: a point y 9 GL (B) if and only if there are )~,~ 9 L A NA (Z0, r/), Z,, --~ )~0, tn ~ +r and x,, 9 B such that S~,, (t,,)x,, --+ y.
family of o f B with by ~ c (B) sequences
One remarks that the definition of Gc (B) treats ~ as if it were also a time parameter; it does not prescribe an order in the limits. The notion of continuity of global attractors or the definition of ~L (A., ~.0) prescribes the limit t --~ + e ~ before the limit ,k -+ ~0. However, in many practical situations, it is not clear that an order in the limits should be imposed (see, for instance, the discussion in [ 157]). Suppose that )~0 9 A and that Hypothesis (H. lc) holds. If B is a bounded set such that the co-limit set co~0(B) of B with respect to S~0 exists, is nonempty, compact and attracts B, and if, either co)~o (B) C B or coxo(B) attracts a neighbourhood of B, then, co)~o(B) -- ~L (B). In particular, if the semigroup S)~o has a compact global attractor .A~o c B, then the equality ~A (B) -- .Axo holds. Applications of this property to situations, where the limit system
G. Raugel
920
Sz 0 has a first integral, are given in [ 111 ]. For example, consider the retarded differential difference equation ,f = - ( 1 -t- e ) f ( x ( t ) ) -4- f ( x ( t -
1)),
(3.12)
where e ~> 0 is a parameter, f 6 C1 (/I~, ]I~) satisfies f ( 0 ) = 0 and f ' ( x ) ~> 6 > 0, for all x 6 R. For any e ~> 0, one defines a semigroup Se(t) on the space X = C ~ IR) by the relation (Se(t)qg)(O) = x ( t + O, qg), 0 E [ - 1 , 0], where x(t, qg) is the unique solution of (3.12) with initial data qg. It is shown [111] that, for e > 0, the global attractor Ae reduces to {0}, which implies that ~(0,e0l(,A., 0) = {0}, for any positive number e0. On the other hand, for e = 0, the function 9 (r
-- 99(0) +
s
f(cp(s)) ds 1
is a first integral. On each level set {]b-1 (C), there is a unique equilibrium point e(c) of (3.12), for e = 0, and the co-limit set co0({/,-l(c)) with respect to the semigroup So(t) reduces to {e(c)}. It is then proved that, if B is an arbitrary closed, bounded set in X, we have ~(0,E0](B) -- co0(B) -- I ( B ) , where I ( B ) -- {e(c~o) I q9 6 B} and c~o -- q} (qg). We next want to study how the flows-restricted to the compact global attractors- vary with the perturbation parameters. If the semigroups are defined on an infinite-dimensional vector space, we are lead to make severe restrictions on the flows, that we consider. For this reason, we restrict our discussion to Morse-Smale systems. Since general comparison results are mainly available in the frame of discrete Morse-Smale systems, we shall restrict our study to this class. M o r s e - S m a l e maps. As we have already explained, in the infinite dimensional case, the strongest expected comparison of the dynamics of two different semigroups is the topological equivalence of the flows restricted to the compact global attractors. In the case of discrete semigroups, the notion of topological equivalence is replaced by the conjugacy of the trajectories. In [ 177,178], it has been proved that any Morse-Smale Cr-diffeomorphism S, defined on a compact manifold M is stable, that is, there exists a neighbourhood N r (S) of S in the set Diff r (M) of all Cr-diffeomorphisms, r >~ 1, such that, for each T ~ N r (S), there exists a homeomorphism h -- h ( T ) : M --+ M and h o T = T o h holds on M. This important stability property has been generalized to the Morse-Smale maps defined on a Banach manifold by Oliva (see [175,103]). For sake of simplicity, we describe this result only in the case of a Banach space X. We begin with some definitions and notations. Let S ~ C r (U, X), r >/1, and U be an open subset of X. For any fixed point x0 of S, we introduce the stable and unstable sets of S at x0 by W s (x0, S) -
{y E X [ S" (y) -+ x0 as n --+ + e c },
W u (xo, S) -
{y ~ X I there exists a negative orbit Uy of S such that uy(O) - y and u y ( - n ) -+ xo as n -+ +cx~}.
(3.13)
A fixed point x0 of S is hyperbolic if the spectrum a (DS(xo)) does not intersect the unit circle {z ~ C IIzl = 1} in C with center 0.
Global attractors in partial differential equations
921
REMARK 3.7. If x0 is a hyperbolic fixed point of S 6 C" (U, X), r >~ 1, then there exists a neighbourhood V of x0 in U such that the sets Wio c(xo, S) -- W s (xo, S, V) - { y ~ W s (xo, S) l S'' (y) ~ V, n >~ O},
W~"o~(X0,S) -- W" (xo, S, V) = {y 6 W" (x0, S) I S-" (y) exists and S-" (y) 6 V, n >~ 0}
(3.14)
are embedded C"-submanifolds of X. These sets are called the local stable and local unstable manifolds of x0. The manifold Wioc (xo, S) is positively invariant, whereas W{'oc(x0, S) is negatively invariant. Moreover W{'oc(x0, S) is locally positively invariant. If the part of the spectrum cr (DS(xo)) lying outside the unit circle is composed of a finite set of m eigenvalues, then Wl"oc(x0, S) (respectively Wi~oc(xo, S)) is an embedded C"-submanifold of dimension m of X (respectively of codimension m of X). If S and the derivative DS(y) are injective at each point y of U,7~>0(S'1(W~oc(X0, S)), then W"(xo, S) = U,~>0(s" (W{~oc(X0,s)) is an injectively immersed Cr-submanifold of X, of the same dimension as W{'oc(xo, S), and is invariant. If Wi~oc(xo, S) is of finite codimension m, if S is injective and the derivative DS(y) has dense range at each point y of U,,>~o(S-"(Wi~oc(XO, S)), then W~(xo, S) = ~,,>~o(S-'l(Wioc(XO, S)) is an injectively immersed C"-submanifold of codimension m of X (see [121, Theorem 6.1.9]). Moreover, W s (xo, S) is invariant under S. For further details, see also Section 4.1. A point x0 is a periodic point of period p if SP(xo) = xo, S" (xo) 7~ xo, for 0 < n ~< p - 1. A periodic point x0 of period p is hyperbolic if the (finite) orbit O(xo) -{x0, S(xo) . . . . . S/'-j (x0)} of x0 is hyperbolic, that is, if every point y E O(xo) is a hyperbolic fixed point of S p. As above, one introduces the sets Wi~oc(y, S) and W~'oc(y, S) and W"(y, S) = U,,>~o(S"P(W~'oc(Y, S)), for every y E O(xo). These stable and unstable sets have the properties mentioned in Remark 3.7. Hereafter, we denote by Per(S) the set of periodic points of S. Let S 6 C" (X, X), r >~ 1. The nonwandering set I2(S) of S is the set of all points x 6 J ( S ) (where J ( S ) is the maximal bounded invariant set of S) such that, given a neighbourhood V of x in J ( S ) and any integer no, there exists n >~ no with S" (V) A V -r 0. If S-2(S) is finite, then I2 (S) = Per(S). One also notices that, if J ( S ) is compact and S is injective on J ( S ) , then I2 (S) is compact and invariant. Following [ 175,103], we introduce a topological subspace 1CCr (X, X) of C~ (X, X), r >~ 1, with the following properties: (KC 1) S c/CC' (X, X) implies that ff (S) is compact; (KC2) the sets if(S) are uppersemicontinuous on E c r ( x , X), that is, for any S E /CC" (X, X), given a neighbourhood U of if(S) in X, there exists a neighbourhood V(S) of S in ICCr(X, X) such that i f ( T ) C U, for any T E V(S); (KC3) for any S E EC" (X, X), S and D S are injective at each point of J ( S ) . EXAMPLE. Let S)~ 6 C~ (X, X), r ~> 1, be a family of maps depending on a parameter k A, where A is a metric space. Assume that each map Sx admits a compact global attractor ,Ax and satisfies the hypothesis (H. la) or (H. lb) at every point k0 c A. Then, J(S)~) - r
G. Raugel
922
and the sets r are uppersemicontinuous in )~. If moreover the above condition (KC3) holds, the family Sz, )~ 6 A, can be chosen as a EC, r (X, X)-space. Finally, we introduce the class of Morse-Smale maps: DEFINITION 3.8. A map S 6 C~ (X, X), r/> 1, is a Morse-Smale map if the above conditions (KC 1) and (KC3) as well as the following conditions are satisfied: (i) S2 (S) is finite (hence ~ ( S ) = Per(S)); (ii) every periodic point x0 of S is hyperbolic and dim W" (x0, S) is finite; (iii) W" (x0, S) is transversal to Wi~oc(x l, S), for any periodic points x0 and Xl of S. If S is a Morse-Smale map, then
,7(s) -
~_J
w ~(x0, s).
x0EPer(S)
The Morse-Smale maps have a remarkable property, namely they are J-stable. DEFINITION 3.9. A map S 6 K~Cr (X, X) is J-stable or simply stable if there exists a neighbourhood V(S) of S in ]CCr (X, X), such that each T E V(S) is conjugate to S, that is, there exists a homeomorphism h = h (T) : J (S) --+ ff (T) satisfying the conjugacy condition h o S = T o h on ,,7"(S). Adapting the arguments used in [ 177] and in [ 178], Oliva showed, mutatis mutandis, the following basic result (see [ 175,103]): THEOREM 3.10. Let a subspace ]~C r (X, X) of C~(X, X), r ~> 1, be given. The set of all r-differentiable Morse-Smale maps is open in 1CCr (X, X). Moreover, every Morse-Smale map S in 1CCr (X, X) is J-stable. This result has important applications in the study of partial differential equations depending on various parameters, including time or space discretisations. In Section 4, we shall apply it to gradient systems. If Sz ~ C r (X, X), r ~> 1, is a family of maps depending on a parameter )~ ~ A and Sz0 is a Morse-Smale map, Theorem 3.10 allows to conclude that, for )~ close to )~0, Sz has the same type of connecting orbits. If Sz0 is no longer a Morse-Smale map, this persistence of connecting orbits can still be proved in some cases, with the help of the Conley index (see [46,163,164], for example). If $1 (t) and S2(t) are two continuous semigroups on a Banach space X, we say that $1 (t) is topologically equivalent or simply equivalent to $2 (t), if there exists a homeomorphism h : ff(Sl) ~ J ( S e ) , which preserves the orbits and the sense of orientation in time t. A continuous semigroup Sl (t) is stable if there exists a neighbourhood N(S1 (.) of S1 (.) within a given class of continuous semigroups such that every semigroup T(.) ~ N(S1 (.)) is equivalent to $1 (.). Like above, one can define Morse-Smale continuous semigroups. Very recently, Oliva has given a proof of a stability result for Morse-Smale continuous semigroups in the infinite-dimensional case. Also, stability of certain continuous MorseSmale semigroups Sz (t) generated by evolutionary equations has been proved by reducing
Global attractors in partial differential equations
923
Sz (t) to a Morse-Smale system Sz(t), defined by a finite-dimensional system of ODE's depending smoothly enough on the parameter ,k (see Section 3.4 below on inertial manifolds).
3.2. Dimension of compact global attractors The existence of a (compact) global attractor A C X leads to the question of whether there exists a finite-dimensional dynamical system whose dynamics on its global attractor reproduces the dynamics on A or at least whose attractor has the same topological properties as A. Also, from the computational point of view, one is interested in knowing if the solutions on the attractor can be recovered by solving numerically a large enough system of ODE's and how big should be this system. A first step in this direction consists in showing that the "dimension" of the set ,A is finite and in giving a good estimate of it. Various notions of dimension have been studied in conjunction with global attractors. Among them, the Hausdorff and fractal dimensions have played a primordial role. We will briefly describe both notions and state some results. For an exhaustive study in the Hilbertian framework, we refer to the book of Temam [202]. Let E be a topological space. We say that E has finite topological dimension if there exists an integer n such that, for every covering H of E, there exists another open covering H' refining H so that every point of E belongs to at most n + 1 sets of H'. In this case, the topological dimension dim(E) is defined as the minimal integer n satisfying this property. It is a classical result that, if E is a compact space with dim(E) ~< n, where n is an integer, then it is homeomorphic to a subset of •2,,+1. Moreover, the set of such homeomorphisms is residual in the set of all maps from E into R 2''+1 . However, special properties of such embeddings are not known, and, in the case where E is contained in a Banach space, it could be more convenient to deal with linear projections (simply called projections in what follows). As generalizations of the topological dimension, there are several stronger fractional measures of dimension applicable to sets which have no regular structure. The most commonly used are the Hausdorff and fractal dimensions. Let E C X, where X is a metric space. The Hausdorff dimension is based on approximating the d-dimensional volume of the set E by a covering of a finite number of balls with radius smaller than ~, that is,
r? [ri <~e and E C U Bx(xi, ri) I'
tt(E, or, e) -- inf, Z i
i
I
where Bx(xi, ri) is the ball of center xi and radius ri. The a-dimensional Hausdorff measure of E is then defined as /z ~ (E) = lim # ( E , or, e), g---+0
and the Hausdorff dimension dimH (E) of E is essentially the value of c~ for which/z" (E) is a finite nonzero number, dimH (E) ----inf{ot > 0 [ / z ~ (E) --0}.
G. Raugel
924
It is known [125] that dim(E) ~< dimH (E) and that dim(E) = dimH(E) if E is a submanifold of a Banach space. There are also examples of sets E in •n, for which dim(E) = 0 and dimH(E) = n. The fractal dimension also called limit capacity or box dimension is a stronger measure than the Hausdorff dimension; here all the balls in the covering are required to have the same radius. Given e > 0, let n(e, E) be the minimal number of balls Bx(xi, e) of radius e needed to cover E. One defines the fractal dimension dimF(E) as log n(e, E) dimF(E) = lim sup ~0 log(l/e) It is easily proved that d i m , ( E ) ~ dimF(E) (see [149]). The quantities d i m , ( E ) and dimF (E) can be different (see [ 149] for an example of a compact subset K of 12 with finite Hausdorff dimension and infinite fractal dimension). One of the first estimates of the Hausdorff dimension of compact invariant sets has been given by Mallet-Paret in 1976. In [ 147], he showed that, if K is a compact subset of a separable Hilbert space H and is negatively invariant under a mapping S ~ C l (U, H), where U is a given open neighbourhood of K and where, for any z ~ K, the derivative DS(z), restricted to some linear subspace Y C X of finite codimension, is a strict contraction, then d i m , ( K ) is finite. From this property, he deduced that, if the contraction hypothesis is replaced by the condition that the derivative DS(z) is a compact operator, for any z ~ K, then d i m , ( K ) is finite. The same conclusion holds if the property on DS is replaced by the property that S(U) is relatively compact in H. This implies that the compact global attractor of a large class of delay equations and of parabolic equations, including the twodimensional Navier-Stokes equations, has finite Hausdorff dimension (see [147], for an application to RFDE and the heat equation). The hypothesis "DS(z)Ir is a strict contraction, for any z ~ K" is a so calledflattening condition onto a finite dimensional subspace of X. Using the same ideas as Mallet-Paret and a squeezing property of the semigroup S(t), Foias and Temam [74] showed that any compact invariant subset under the flow generated by the two-dimensional Navier-Stokes equations has finite Hausdorff dimension and gave estimates of the dimension. Assuming flattening conditions similar to [ 147], but on the mapping S itself, Ladyzhenskaya [ 135] has improved the estimate of the Hausdorff dimension for the global attractor of the two-dimensional Navier-Stokes equations given in [74]. Notice that, in [103, Theorem 6.8], there is an interesting result of existence of retarded functional differential equations the attractors of which have infinite Hausdorff dimension. In 1981, Marl6 [149] generalized the abstract result of Mallet-Paret to the case of a Banach space and weakened the "flattening condition" in the following way. Let X be a Banach space and L~(X, X) be the space of bounded linear mappings Z that can be decomposed as r = r l + Z2 where Zl ~ L(X, X) is compact and [Ir2llLCx, x) < )~. One remarks that, for r E Lx/2(X, X), there exists a finite-dimensional subspace Y C X such that, if r r : Y C X -~ X is the map induced by r on Y, then Ilrrllc~x) < ~. This amounts to define the number v ~ ( r ) as the minimal integer n such that there exists a subspace Y C X of dimension n with [iZrllL(x) < )~, when
Z E L;~/2(X, X).
Global attractors in partial differential equations
925
THEOREM 3.1 1. Let X be a Banach space, K be a compact negatively invariant set under a mapping S ~ C l (U, X), where U in an open neighbourhood of K in X. IfDS(x) belongs to the space L l (X, X), for any x ~ K, then dimF(K) < +cx~. Theorem 3.11 is proved in [149]. There, the following explicit bound for dimF(K) is given in the case where DS(x) belongs to the space LI/4(X, X), for any x 6 K, dimF (K) ~<
logXl log( 1/2(1 + e)X)'
where )~l = v2"(1 + (k + X)/0~e)) ~', 0 < X < 1/2, 0 < e < (1/2X) - 1, k-
sup I[DS(x)Ic(x.x, x~K
and
v -- sup v;~(DS(x)). x~K
One then remarks that, if DS(x) belongs to the space L I(X, X) for any x 6 K, there exists an integer n, sufficiently large, such that DS"(x) ~ L I / 4 ( X , X ) , for any x 6 K,, = ~ S - j (K), which implies that dimF (K) -- dimF(K,,) is finite. REMARK. In practice, Theorem 3.11 is very useful to show that global attractors have finite fractal dimension, even if the bounds of the dimension given in the proof are not so accurate. Indeed, in most of the cases where the semigroup satisfies the hypotheses of Theorem 2.31, one can also show that the hypothesis, required on DS in Theorem 3.11, holds. In the case of Hilbert spaces, the notion of m-dimensional volume element and its evolution are easily expressed, which gives much more accurate bounds of the various dimensions of the attractor (see [54,45], where DS(x) is a compact mapping and [85] for the non compact case). In particular, the Lyapunov exponents of the flow on the attractor have become a standard tool in the description of the evolution of volumes under the semigroup S. A result, first proved by Constantin and Foias [44] in the case of the attractor of the twodimensional Navier-Stokes equations, states that, if the sum of the first m global Lyapunov exponents on an invariant compact set K is negative, then the Hausdorff dimension of K is less than m and the fractal dimension of K is finite and estimated, up to the product by a universal constant, by m (see [45, Chapter 3] and, for refinements using the local Lyapunov exponents, [55]). We refer especially the reader to the book of Temam [202], where these topics are well explained and estimates of the dimension of the global attractors of numerous partial differential equations, including the reaction-diffusion equations, the damped wave equations, the Navier-Stokes, Kuramoto-Sivashinsky and Cahn-Hilliard equations, are given in terms of physical parameters. Other various estimates of dimensions of attractors are contained in [13,138,139] (see also [18] in this volume). Finally, we note that Thieullen [203] has given estimates of the Hausdorff dimension of K, involving "Lyapunov exponents" in the frame of Banach spaces. If K is a compact subset of a Banach space X, with dimF(K) < m/2, m ~ N, then, for every subspace Y C X of dimension dim Y ~> m, the set of projections P : X ~ Y
G. Raugel
926
such that PlY is injective on Y is a residual subset of the space of all the (continuous) projections from X onto Y, endowed with the norm topology. This result has been given in [149], where, by an unfortunate mistake, dimF(K) has been replaced by dim/4(K). One notices that the statement is no longer true with the hypothesis dim/4 (K) < m / 2 (see [ 195]). Recently, in the Hilbertian case, the above result has been improved by Foias and Olson, who showed that the inverse (Ply)-1 of most projections P are H61der continuous mappings. In general, these inverse are not Lipschitz-continuous [ 165]. THEOREM 3.12 (Marl6; Foias and Olson). Let X be a Hilbert space and K be a compact subset of X with fractal dimension dimF(K) < m/2, m ~ 1~. Then, for any projection (respectively orthogonal projection) Po onto a subspace Y of X of dimension m and for any e > O, there exist a projection (respectively orthogonal projection) P onto Y and a positive numberO <<.1, such that PIK is injective, liP - eollc(x,Y) <~e and Pig has H61der inverse, i.e.,
IlP- x -
P-lYllL(X,x)
~ CIIx -
yll~
x, y E P(K).
As an application of this theorem, we go back to Example 2.2, in the Hilbertian case, and write Equation (2.1) under the following short form, where we assume that Y -- X, db/
dt
=.T(u),
u(0) - u0 6 X.
(3.15)
We assume that the continuous semigroup S(t) has a compact global attractor ,A of finite fractal dimension and that FI,A is H61der continuous from X into X. Let P be a projection given by the above theorem. On P A, the following dynamical system is well defined, dz
dt
= P F ( P - l z ) = - - F(z),
z ~ P,A,
(3.16)
where, under the above hypothesis, F is H61der continuous from P A into PA. The next step is to extend this system to a system of differential equations defined everywhere in IRm, by using a standard extension theorem like the theorem of Stein. Since F is only H61der continuous, the solutions of this generalized system of ODE's may not be unique and differentiable. However, one can show that the solutions of this extended system exist globally and are attracted by P A (see [56, Chapter 10]). One can also use Theorem 3.12 to show that, under the above hypotheses, it is always possible to reproduce "approximately" the dynamics of (3.15) on the global attractor A by a system of ODE's in 1R3. This is done in [ 190], where it is proved that, if F is bounded and continuous on A, then, for any T > 0 and any e > 0, there exist two functions g" R 3 ~ ~3 and q~" R 3 ~ X, which are Lipschitz and H61der continuous, respectively, such that, for any solution u(t) E A, there exists a solution z(t) of dz
dt
=g(z),
with [[~(z(t)) - u(t)llx ~ ~, for any t ~ [0, T].
Global attractors in partial differential equations
927
If one drops the requirement of obtaining a conjugacy between the flows on the attractors, one can construct a homeomorphism between the attractor A and the one of a finite system of ODE's. More precisely, if A is the compact global attractor of a continuous semigroup generated by an evolutionary equation on a Hilbert space X, with finite Hausdorff dimension, then there exists a finite system of ODE's with a global attractor K, homeomorphic to A. Conversely, if K is the global attractor of a finite system of ODE's, there exists an infinite-dimensional continuous semigroup S(t) generated by an evolutionary equation on X, with a compact global attractor A homeomorphic to K and having finite Hausdorff dimension (see [ 190], where also a review of other related results is given).
3.3. Regularity o f the flow on the attractor and determining modes If S(t) is a continuous semigroup on X, which has a compact global attractor A, the semigroup S(t) restricted to A can have interesting "regularity" properties that are not shared by the semigroup S(t) and are a consequence of the existence of the compact global attractor. For instance, if S(t) is generated by a partial differential equation defined on a domain C IR", A can be composed of functions u(y) which are more regular in the space variable y ~ ~2 than the typical elements of the space X. On the other hand, for semigroups defined by evolutionary equations in the infinite-dimensional case, it is not expected, in general, that, for each x ~ X, S ( t ) x is differentiable in t, for t ~> 0. In many cases, however, the function t w-> S ( t ) x will be differentiable in t if x belongs to a compact global attractor or even to a compact invariant set. Time regularity results in a general setting had been obtained in 1985 by Hale and Scheurle [115] and have recently been generalized in [ 113], through the use of a Galerkin decomposition. We present here these results in the simplified situation of Example 2.2. In the same time, we want to emphasize the relation between regularity and finite-dimensionality character. We recall that X is a Banach space, A is the infinitesimal generator of a C~ E0(t) in X, and that f : Y --+ X is a Lipschitzian mapping on the bounded sets of Y, where either (1) Y -----X, or (2) exp(At) is an analytic linear semigroup on X and Y = X ~ = D ( ( M d - A)~), with ot c [0, 1), X an appropriate real number. In the case (1), we set Y = X ~ for u = 0. Assuming that all of the solutions of the following equation on Y,
dr/ -t d( t )
-- A u ( t ) + f ( u ( t ) ) ,
t > O, u(O) -- uo E Y,
(3.17)
are global, we obtain a continuous semigroup S(t) on Y. If E0(t) is an analytic semigroup, time regularity properties of the mapping S ( t ) x , for x ~ Y, are well known (see [121, Chapter 3]). Actually, if f :u ~ Y ~-> f ( u ) E X is of class C k or analytic, then, for t > 0, the mapping t -~ S ( t ) x , x E Y, is of class C k or analytic. Time regularity properties have been especially addressed in the case of the Navier-Stokes equations and other parabolic systems (see [74,75,185], for example). For this reason, in
G. Raugel
928
the next theorem, we consider only the case where Y = X. We suppose that there exists a positive number 0 such that the radius of the essential spectrum r(Oess(exp At)) satisfies r(Cress(expAt)) <~e x p ( - 0 t ) ,
Yt ~> 0.
(3.18)
The following regularity result is proved in [ 115] and [94]. THEOREM 3.13. Suppose that A satisfies the condition (3.18), f 6 Ck (X, X), 1 <<.k < +oo, (respectively f is analytic from X to X) and that S(t) has a compact global attractor A. Then, there exists a positive number rl depending on r such that, if
IlOf (u)llL(X,X)
o,
for any u in a neighbourhoodof A,
(3.19)
the mapping t : R w-~ S(t)u ~ X is of class C k (respectively analytic), for any u ~ A. The proof of Theorem 3.13 is rather long. The most important steps of the proof will be described below, when we indicate how the above theorem can be generalized. The above time regularity property in Theorem 3.13 cannot be true if f is not at least a C l-function, as illustrated by the counterexample given in [94, p. 57]. The smallness condition (3.19) is rather restrictive and limits the applications of this result. However, if one exploits further the ideas contained in [ 115] and uses a Galerkin decomposition, one can generalize Theorem 3.13 in a significant way. For example, if X is a Hilbert space and the eigenfunctions of the operator A form a complete orthonormal system, we can decompose any solution u of (3.17) into the sum of two functions Vn and Wn, with Vn being the sum of the first n terms in the expansion of u and w,, being the remainder. Under additional natural conditions on A and f , one shows that there exists an integer N1 such that each solution on the compact global attractor can be represented in the form
u(t) = I)NI (t) + lION! (t) = UNI (t) + w* (UNI)(t),
(3.20)
where the function w* (UNI)(t) depends on UNI (S), S ~ t, and is as smooth in I)N! as the vector field f . Equation (3.17) is thus reduced to a finite-dimensional system of N1 (nonautonomous) functional differential equations with infinite delay. It is very natural to refer to the coefficients of the projection vu~ (t) of u(t) onto the first Nl eigenfunctions as the determining modes of Equation (3.17) on the attractor A. Furthermore, wu~ (t) = w* (VN~ )(t) is a solution of an equation similar to (3.17), where, if Nl is large enough, the nonlinearity satisfies the smallness condition (3.19). This property, together with the reduction of (3.17) to a finite-dimensional system of FDE's, will imply the regularity in time of the solutions on the attractor A. We now state these results more precisely. We keep the assumptions of Example 2.2, which have been recalled above and assume further that (H 1) f : Y --+ X is a Ck function, k ~> 1; (H2) The continuous semigroup S(t) has a compact invariant set if; (H3) For any n ~> 1, n 6 1% there is a continuous linear map P,, :Z ~ Z, where Z - X or Y, such that APn = Pn A on D(A), and the following additional properties hold:
Global attractors in partial differential equations
929
(i) P,, converges strongly to the identity in Z as n goes to infinity; (ii) there exists a positive constant K0/> 1 such that, if Q,, - Id - P,,, then,
I P,,llL(z.z) <~ Ko,
II Q,,llc(z.z)~< K0,
gn c N;
(3.21)
(iii) there exist an integer n l, two positive constants 61 and K1 ~> 1 such that, for n >~ n l, t > 0, we have,
HeAtw Z <~ Kle-a'tllwllz'
re,,
I eA'
re,,
-< x, e
Hi II
(3.22)
where W,z - Q,, Z, for Z - X or Y. We also set V,z - P,, z . In addition, our main result will depend upon either: (H4) There are a positive constant K2 ~> 1, independent of n E N, and a sequence of positive numbers 6,,, 6,, --+ ,,__,+oo + o c , such that, for t > 0,
IeAtw Z <~ K2e-a"tllwilz' eAt 11olaY ~ K2 e -a''' (t-~ + a,~')ll w IIx,
w,5-
(3.23)
or
(H5) The set { D f ( u l ) u 2 I U l E J ' ,
Ilu2 II Y ~ 1 } is relatively compact in X.
REMARK 3.14. In several concrete situations, there exists a neighbourhood U of ,7 in Y such that f ' U ---> X is completely continuous. If X is a reflexive Banach space, one shows, by arguing as in [147, p. 339] that this property implies the hypothesis (H5). We recall that, for any k > 0, B z (0, k) denotes the open ball in Z of center 0 and radius k. Let r be a positive constant such that Ilullv ~< r / K o , for any u E ,.7". In what follows, we denote by C~,,(By(O, 4r)" X) the subset of Ck(By(O, 4r); X) of the mappings g whose derivatives D j g(u), j <~ k, are bounded for u c By(0, 4r) and the derivative D k g(u) " u c By(0, 4r) ~ D kg(u) ~ L k (Y, X) is uniformly continuous. To prove that the restriction of the flow S(t) to ,.7 is of class C k, k ~> 1, we shall assume that, f c C k (Y, X) belongs to the space C~,(By(O, 4r)" X). If u(t) is a mild solution of (3.17) contained in ,.g and
u(t) - P , u ( t ) + Q,,u(t) = v(t) + w(t),
(3.24)
then (v(t), w(t)) is a mild solution of the following system
dl) dt dw dt
= Av + P , , f ( v + w), (3.25)
= A w + Q , , f ( v + w).
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930
If n ~> n l, the property (3.22) implies that
w(t) -
i
eA(t-S) Qn f (V(S) + w(s)) ds.
(3.26)
oo
In what follows, we shall always choose n ~> n l. For d > 0 and n ~> 1, we introduce the following sets,
Up,,y(J,d)
-
UQ,,y(d) -
{v 9 y y I IlvllY < 2r, disty(v, Pnfl) < d}, {w 9 WV~ l llwllY < inf(d, 2r)}
as well as the subsets
c ~ r ( y , d) -
C~
C~. ~ (d) = C~
ue,,r(a,
d)),
p , , y ( J , d ) - c~. (R; u~,,y(J,d)), k,. r (d) - Cb, k , (IR. Uo,, r (d)), CO,,
c k , ll
UO,,~ (d)),
where k ~> 1. The equality (3.26) suggests that, given v 9 C~ tained as a fixed point of the operator Tv (w)'C~
Tv(w) -
f
d), the function w(t) can be oby (d) --+ CoQ,, r (d) defined by
eA(t-S) Qn f (V(S) + w(s)) ds.
(3.27)
oo
The problem consists now in finding d > 0 and No ~> n l, such that, under the above hypotheses, the mapping Tv is indeed a uniform contraction from C~ into itself, for n ~> No. Actually, motivated by the next remark, we shall also prove that Tv is a uniform r'~QnY (d) into itself, for n ~> No. contraction from ,.. REMARK. If ,.7 is a compact invariant set for S(t), the set of the complete orbits u(t) of (3.17) contained in ,.7 is uniformly equicontinuous; that is, for any positive number r/0, there exists a positive number ~/l such that, for any t 9 R, for any complete orbit u(R) C ,.7, ]lu(t + r) - u ( t ) [ i y <~ rio if Iri ~< 7/1. If the hypotheses (H4) or (H5) hold, one can prove that, for n large and d small enough, the map Tv is a uniform contraction from C oQ,, y (d) into itself [ 113]. THEOREM 3.15. Assume that the hypotheses (H1), (H2), (H3) and either (H4) or (H5)
hold. Then, there is a positive constant dl such that, for 0 < d <. dl, there exist an integer No(d), and, for n ~ No(d), a (unique) Lipschitz-continuous function w * "v E C O p,,r(J,
d) e--> w * (v) E COQ,,y(d),
which is a mild solution of dw*(v)
~(t) dt
= Aw*(v)(t) + Q n f ( v ( t ) + w*(v)(t)).
(3.28)
Global attractors in partial differential equations
931
The mapping w*(v)(t) depends only upon v(s), s <. t, and the map w* is also Lipschitz9 0 II 9 OII connnuous from Cpily(J" , d) mto CQ, r(d). Moreover, w*(v) is as smooth in v as the vectorfield f " that is, if f ~ ck,(By(O, 4r); X), ty, O,u , k ) 1, then the mapping w* is in Ckbu~...p,,y(J d)" (,O,u ,.,Q,,y(d)). Furthermore, if f ~ ck.(By(O, 4r)" X), k ) 1, then w* is a uniformly continuous map from (,k, u f,k, u ...p,,y(J, d) into ,~O,,y(d). REMARK 3.16. Under the hypotheses of Theorem 3.15, we can choose NI = Nl(dl) No(dl), such that, if u(t) e J is a mild solution of (3.17), then, for n >>.Nl, w(t) = Q,u(t) belongs to CoQ. Y (dl) and thus, by uniqueness of the mild solutions, u(t) must be represented as
u(t) = v(t) + w* (v)(t),
(3.29)
where v(t) -- P,,u(t) is the mild solution of the system of functional differential equations du
d-~ ( t ) - Av(t) + P,,f (v(t) + w*(v)(t)).
(3.30)
r'~ (dl) , respectively. If, for any n, the Moreover, v(t) and w(t) are in /"~ vp, y ( J , dl) and ,~O,,y range of P,7 is of dimension n, the above property leads to say that the flow on the compact
invariant set J is determined by a finite number Nl of modes. More classically, one says that Equation (3.17) has a finite number Nl of determining modes if the range of PN~ is finite-dimensional of dimension Nl and if the condition
]PN, Ul(t) - PN, u2(t)]]y --+t--++~ 0
implies that
The property of finite number of determining modes has been extensively studied for parabolic equations and, especially for equations arising in fluid dynamics, like the NavierStokes equations. We refer to Foias and Prodi [72] and to Ladyzhenskaya [133] for the earliest results on the two-dimensional Navier-Stokes equations. Later, various estimates for the minimal number of determining modes in terms of the Reynolds number or the Grashoff number have been established (see the estimates in [70], which have been improved in [ 128]). A generalisation of the concept of determining modes to the one of determining functionals together with applications to dissipative parabolic equations has been recently given by Cockburn et al. [41 ]. Based on the ideas of [41 ], Chueshov [40] extended their results to other dissipative equations, including the damped wave equation. Here the property of finite number of determining modes is a direct corollary of Theorem 3.15, if the compact invariant set J is moreover the global attractor [ 113]. In the next theorem, we do not need the assumption that PN~ has finite-dimensional range. Of course, in view of the applications, the case when PN~ has finite-dimensional range is more interesting.
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G. Raugel
THEOREM 3.17. Assume that the compact invariant set ,7 is the global attractor of Equation (3.17) and that the hypotheses (H1), (H2), (H3) and either (H4) or (H5) hold. If ul (t) and uz(t) are two solutions of(3.17) satisfying (3.31)
[[PN, U ~(t) - PN, U2(t) ll Y ~ ,_++~ o,
where the integer Nl has been defined in Remark 3.16, then (3.32)
[[Ul (t) -- u2(t)[ g ----~t~+~ 0.
We now go back to the regularity in time of the complete orbits contained in J . Arguing as in [ 115, p. 154] by considering the auxiliary differential equation
du
ds = PN, Av + PN, f (v + w* (v)),
v(0) =
v0,
with vo given in the Banach space cO, (R; PN~ Y) and using Theorem 3.15 together with Remark 3.16, we obtain [ 113]: THEOREM 3.18. Assume that the hypotheses (H1), (H2), (H3) and either (H4) or (H5) hold and that, for any n >~ 1, A PIz is a linear bounded mapping from Y into Y. If f belongs to Ckbu(By(O, 4r); X), k ~> 1, then, for any mild solution uo(t) 9 J of (3.17), the mapping t 9 N w-~ uo(t) 9 Y belongs to Cbku(R; Y) and uo(t) is a classical solution of (3.17). Moreover, there exists a positive constant Kk,j such that, for any uo(t) 9 J ,
dJuo
Kk,j
sup --d~(t) tEN
'r
1 <<,j <~k.
(3.33)
Y
To show that the restriction of the flow S(t) to ff is analytic when f is analytic, we complexify the spaces X and Y, the operators A, P,,, Qn etc. and assume, as in [ 115], that: (H6) there exists a real number/9 > 0 such that f has an holomorphic extension from Dy(4r,/9) = {Ul + iu2 l Ul 9 By(O, 4r), u2 9 By(O, p)} into the complexified space X and f is bounded on the bounded sets of Dr (4r,/9). We also complexify the time variable. Given a small positive number 0, we introduce the complex strip Do = {t 9 C I IImtl < 0} and the Banach space Co(Z), defined by Co (Z) -
{ u " Do ~ Z lu is continuous, bounded in Do, holomorphic in Do, and u(t) 9 R, gt 9 R},
and equipped with the norm [llulllz - suPtEb-S0Ilu(t)llz. Showing first an analytic analog of Theorem 3.15 and arguing as in the proof of Theorem 3.18 yield: THEO R E M 3.19. Assume that the hypotheses (HI), (H2), (H3), (H6) and either (H4) or
(H5) hold and that, for any n ~ 1, A Pn is a linear bounded mapping from Y into Y. Then, there exists 0 > O, such that any solution uo(t) C J of (3.17) belongs to Co(Y). In particular, t E N ~-~ uo(t) E Y is an analytic function.
Global attractors in partial differential equations
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Theorems 3.18 and 3.19 can be applied to several evolutionary PDEs, including the heat and damped wave equations, and even to PDEs with delay. In these cases, regularity in time (up to analyticity) on the compact attractor is obtained, even if the solutions are not regular in the spatial variable, that is, even if the domain s on which the equation is given, is not regular (see Section 4.5 below, for the time regularity of the flow on the global attractor of the linearly damped wave equation). In [113], Theorems 3.15, 3.18 and 3.19 are proved under more general conditions than (H4) or (H5), which allows applications to weakly dissipative equations like the weakly damped Schr6dinger equation. Also the condition that P,, and Q,, are projectors can be relaxed. When the evolutionary equation (3.17) arises from a partial differential equation defined on a domain s E R", one deduces regularity in the spatial variable from Theorem 3.15 and Theorem 3.18. Indeed, under the hypotheses of Theorem 3.18, one shows that, for 1 ~< j ~< k - 1, for any solution uo(t) ~ J of (3.17), the derivative ~ditto belongs to Cb0 (R; D(A)) and is a classical solution of the equation dJ u dJ f (u) dJ+lu (t) -- A (t) + (t) dtJ + t dtJ dtJ '
t > 0
dJ u '
(0) given in Y.
(3.34)
In the general case, due to the boundary conditions, u E J is not expected to belong to Z)(AJ), for j ~> 2. However, we can obtain higher order regularity results [113], as it is illustrated in the next example where A is the generator of a C~ only, Y = X and ot = 0. We introduce a family of spaces ZI, l 6 1~, with ZI+I C Z1, Zo = X, D(A) C Z1, such that, Au = g ,
u EZ)(A), g 6 Zl,
implies
u 6 Zl+l.
(3.35)
A simple recursion argument using Theorem 3.18 shows the next regularity result. THEOREM 3.20. Assume that Y = X, cr = O, that (3.35) as well as the hypotheses (H1), (H2), (H3) and either (H4) or (H5) hold and that, for any n ~ 1, APiz is a linear bounded mapping from X into X. Suppose that f belongs to C~u(Bx(O, 4r)" X), k ~> 1. If for k >~2, k-j-I
f is in Cb
(Zj" Zj), for 1 <~ j <~ k - 1, then, for any orbit uo(t) C J , the mapping
t E R ~ uo(t) belongs to C~ (R" Z k - j ) , 0 <~ j <~ k. And there exists a positive constant ~kj such that, for any uo(t) C J , ]]UO[[ciI(R:Zk_j) ~< K ~ '
Yj, 0 ~< j ~< k.
(3.36)
We remark that regularity in Gevrey classes can also be deduced from Theorems 3.15, 3.18 and 3.20 (for earlier regularity results in Gevrey classes in the case of dissipative equations, we refer to [75,185,62,176], for example). Finally, it should be noticed that the proofs of time regularity in [74,75,185] use a classical Galerkin procedure. Also, in his proof of the space regularity of the global attractor
934
G. Raugel
of the weakly damped Schr6dinger equation, Goubet has introduced a Galerkin decomposition [87,88], in a spirit different from the above one.
3.4. Inertial manifolds Theorems 3.12 and 3.15 of the previous sections allow to embed the compact global attractor of some classes of systems into the one of a finite-dimensional system of differential equations. In the second case, we obtained functional differential equations with infinite delay, while in the first case, we got ordinary differential equations, the solutions of which may not be unique. It is therefore natural to try to exhibit classes of dissipative systems, for which these ODE's define a flow. This is actually the purpose of the theory of inertial manifolds, introduced by Foias, Sell and Temam [73]. Suppose that we are given a continuous semigroup S(t) generated by an evolutionary equation on a Banach space X, which has a compact global attractor A. One can define an inertial manifold .M of S(t) as a finite-dimensional Lipschitzian submanifold of X, which contains r and is positively invariant under S(t) (i.e., S ( t ) M C All, for any t ~> 0). If the semigroup S(t) is one-to-one on .L4, then the flow restricted to A// is determined by a finite-dimensional system of ODE's with locally Lipschitzian vector field. This finite-dimensional system is called inertial form. As indicated below, in the process of constructing inertial manifolds, one shows that .h4 attracts bounded sets exponentially. For this reason probably, Foias, Sell and Temam have included the exponential attraction property of bounded sets, in their definition. Up to the present time, one of the basic ways to construct inertial manifolds has been to obtain the inertial manifold as a Lipschitzian graph over a finite-dimensional space and to apply the classical methods of center manifold theory. With such methods, the construction of inertial manifolds encounters the same technical problems and the same obstructions as the one of global center manifolds. In order to prove the existence of such global center manifolds, one needs some normal hyperbolicity property of the manifold M , that is, the flow in X towards A/[ must be stronger than the dispersion of the flow on AA. Since A// does not only contain the neighbourhood of equilibrium points (as it is the case for local center manifolds of equilibria) and that .A may be large, the dispersion of the flow on A/[ may be large. In the frame of semigroups generated by partial differential equations, this strong normal hyperbolicity requirement leads to the so-called cone condition [148] and gap condition (see [73,39], for example), that we explain below. Unfortunately, these conditions are shown to be satisfied only by some partial differential equations in one space variable, including the reaction-diffusion equations (see [73,202]), the Ginzburg-Landau equation [202], the Cahn-Hilliard equation, the Kuramoto-Sivashinsky equation, etc. and by few reaction-diffusion equations on some special domains in dimensions 2 and 3 [ 148, 109]. To be more specific, we go back to the evolutionary equation of Example 2.2 with Y = X ~ , ot E [0, 1). We assume that the spectrum of A consists of a sequence of real eigenvalues ~.~, n E N, in increasing order. For any integer n, we introduce the spectral projection P,, ~ L(X, X) onto the space generated by the eigenfunctions associated with the first n eigenvalues and assume that the hypotheses (H1), (H2), (H3) and (H4) of Section 3.3 hold with 8n - )~,,+1 and 2 - A.
Global attractors in partial differential equations
935
In order to obtain a globally Lipschitzian nonlinear function on the right hand side of (2.1), we truncate the function f . Let m" IR --+ [0, 1] be a C~ such that m ( y ) - 1 if y 6 [0, 1], m ( y ) - 0 if y >~ 2. Let r > 0 be chosen so that A C B y ( O , r ) . We then set
fm (u) -- m
"11~)
4r 2
f (u),
and consider the modified equation du(t) dt
= Au(t) + fm (u(t)),
t > O, u(O) -- uo E Y.
(3.37)
Clearly, the function fm is globally Lipschitzian and bounded from Y into X and (3.37) also defines a continuous semigroup on Y, denoted by Sm (t). One may construct an inertial manifold for Sm (t) by using a Galerkin method. Like in Section 3.3, we choose an integer n. Given a solution u(t) of (3.37), we write u(t) = Pnu(t) + Q,,u(t) = v(t) + w(t), where (v, w) is a solution of the system dv dt
= a v + Pn fin (v + w),
dw dt
= a w + Qn fin (v + w).
(3.38)
One way for obtaining an inertial manifold A4 of Sm (t) as a graph over Vf is to solve, for every v0 e Vf the system (3.38) on ( - e ~ , 0], under the condition v(O) - vo,
w 6 C~
0), W,,r).
(3.39)
Due to (3.26), given vo e V f , (v(t), w(t)) is a solution of (3.38) and (3.39) if and only if w(t) is a fixed point of the map Tv0" Cb~ 0), Wf) --+ C ~ 0), Wf) given by
Tvo (w) --
f,
e A(t-s) Qn fm (V(S) -+- W(S)) ds, oo
where v(t) is the solution of du
dt
= A v + P,, fm (v + w),
v(0) -- v0.
(3.40)
If we prove that Tv0" C~((-e~, 0), W,Y) --+ CO( ( - o o , 0), Wf) is a strict contraction, when C~ 0), W,Y) is equipped with the norm [[wl[u - supt<~0(ertt [Iw(t)llY), where # > 0 is well chosen, then Tvo has a unique fixed point Wvo (t). One then shows that the graph of the Lipschitzian mapping tp "v0 6 V f ~+ qJ (vo) -- vo + Wvo (0) E Y defines an inertial manifold A4 of Sm (t). For example, Tv0 is a strict contraction, if ~n+l
o/
-- ~-n ~ C ~ , n + 1 ,
(3.41)
where C is a positive constant depending on the Lipschitz constant of fm and on c~. The condition (3.41) is a gap condition on the eigenvalues of A and is rather restrictive.
G. Raugel
936
Due to the positive invariance of the inertial manifold A4, Equation (3.37) on .M reduces to the finite system of ODE's: dl) d t ( t ) : Av(t) + fin(q/(v(t))) =_ g(v(t)),
t>0,
v(0) given in V,Y.
(3.42)
This system defines a flow on V,~ -- Pn Y and has a compact global attractor An - PnA. The solutions of (3.42) on P,zr are written as v(t) -- P,~Sm(t)~(v(O)), that is, the flow of (3.42) on PnA is conjugate to Sin(t). Suppose now that S(t) -- Sz (t) and also Sin(t) depend on a parameter )~ in a Banach space A and that, for each )~, one can construct an inertial manifold A//z over Vnv and an inertial form du -
dt
-
gz (v(t)),
t > O,
v(O) given in V~,
where gz and Dgz are continuous functions of v, ,k. If the flow defined by the vector field gzo is structurally stable, then we know that each flow Sz (t)IA~ is equivalent to the flow of Szo(t)lAZo, for ~ close to ,k0 (see [109,167]).
4. Gradient systems Until now, we have not described the behaviour of the flow on the global attractor. Even in the finite-dimensional case, this behaviour is often not known. In the case of the gradient systems, a partial description of the flow restricted to the attractor can be given. We first recall some general properties of gradient systems and then present a few examples of gradient systems.
4.1. General properties of gradient systems We recall that the set G + denotes either [0, + o c ) or N. DEFINITION 4.1. Let S(t), t E G +, be a semigroup on X. (1) A function 9 E C~ R) is a Lyapunovfunctional if
o(s(t)u) ~
vt E G +, Vu ~ X.
(4.1)
(2) A Lyapunov functional 9 is a strict Lyapunov functional if, moreover,
~(S(t)u) - ~(u),
Vt E G +, implies that u is an equilibrium point.
(4.2)
(3) A semigroup S(t) is a gradient system if it has a strict Lyapunov functional and if, either G + -- I~l or G + - [0, + o o ) and S(t) is a continuous semigroup. In the later case, S(t) is called a continuous gradient system.
Global attractors in partial differential equations
937
The simplest example of discrete gradient system is a monotone map S on R (for example, Sx <~ Sy if x ~< y). NOTATION. We denote g -- {z 9 X I S(t)z -- z, Vt ~ O} the set of equilibrium points of S(t). Clearly, g is an invariant and closed set. If S is a discrete semigroup, f is simply the set of fixed points Fix(S) of S. The following result, known as the Invariance Principle of LaSalle, plays a basic role in the theory of gradient systems. We set G - - { - g i g E G + }. PROPOSITION 4.2 (Invariance Principle of LaSalle). Let S(t) be a gradient system on X with a Lyapunov functional cb. (1) If z is an element o f X such that y+ (z) is relatively compact in X, then, (i) l = l i m t ~ + ~ @(S(t)z) exists and @(v) = l, f o r any v E co(Z), (ii) co(Z) C g; in particular, s ~ 0, and 6 x ( S ( t ) z , g) --+t---,+~ O. (2) Let u~ E C~ - , X) be a negative orbit through some z E X. If the negative orbit u = ( G - ) is relatively compact, then the set or/,=(z) is nonempty, compact, invariant, consists only o f equilibrium points and 6 x ( u = ( - t ) , or,= (z)) ----~t~+~ 0. Furthermore, oz,_ (z) is connected. .,
PROOF. The function t ~ @(S(t)z) is non increasing and bounded from below, since 4 ( . ) is a continuous function on X and that V+(z) is relatively compact in X, hence l -- l i m t ~ + ~ cb(S(t)z) exists. If v E co(Z), there exists a sequence t,, --+ +cxz such that S(t,,)z --+ v. As qs(.) is continuous on X, cb(S(t,,)z) --+ cb(v) and q~(v) - I. One remarks that property (i) also holds if the Lyapunov functional @ is not a strict Lyapunov functional. Since X+(z) is relatively compact, the semigroup S(t) restricted to y + ( z ) is asymptotically smooth. From Proposition 2.13, we then deduce that co(z) ~ {3 and that 6 x ( S ( t ) z , co(z)) --+t~+~ O. The inclusion S(t)co(z) C co(z) implies that @(S(t)v) - l q~ (v), for any t ~> 0 and v E co(z). Thus, v E g. Using Remark 2.16(ii), one proves, like above, the statement (2). By Remark 2.16(ii), or,= (z) is invariantly connected. Since or,= (z) E g, or,= (Z) is connected. D REMARK 4.3. (1) Assume that the hypotheses of Proposition 4.2 hold. The above assertions (i) and (ii) simply mean that co(z) C s where s {v E s I q~(v) = l}. If g/ is discrete, it follows from Lemma 2.9 that there exists only one element v0 E s such that co(z) -- v0, that is, that S ( t ) z ~ vo as t --+ + e ~ ; and the orbit through z is said to be convergent. In general, if the sets s are not discrete, the positive orbits are not convergent (for examples, see [179] in the finite-dimensional case or [184] in the infinite-dimensional case). However, in the frame of Example 2.2, the positive orbit through z is shown to be convergent if there is v E co(z) such that the spectrum of the linear map Z'v(1) intersects the unit circle in C at most at the point 1 and 1 is a simple eigenvalue [108,26], where Zv(t) is the C~ generated by the linear operator A + D f ( v ) . Below, we shall give some examples of reaction-diffusion equations and damped wave equations, for which the convergence property holds even if the sets s are not discrete (see [183] for a review of convergence results).
G. Raugel
938
(2) Assume that S(t) is an asymptotically smooth gradient system, which has the property that, for any bounded set B C X, there exists r ~> 0 such that y + (B) is bounded. Then, S(t) is point dissipative if and only if ~' is bounded. (3) More generally, let S(t) be a continuous semigroup and ,/~ be a Lyapunov functional associated with the semigroup S(t). Let H -- {x E X I ~ ( S ( t ) x ) -- ~ ( x ) , Vt ~ 0} and M be the maximal invariant subset of H. One shows like in the proof of Proposition 4.2 that, if z E X is such that y + (z) is relatively compact in X, then the co- limit set co(z) is contained in M. (4) The simplest example of a gradient system, from which the term "gradient" actually comes, is the ODE : ~ - V F ( y ) , where F E c l ' l ( R n , ~ ) . The associated strict Lyapunov functional is r - - F ( y ) . If F is an analytic function, the bounded orbits are convergent [ 145]. For a continuous semigroup, the definitions of stable and unstable sets are analogous to those given for a discrete system in (3.13). We recall that
w" (E) - w" (E, s(t)) = {v E X I there exists a negative orbit uv of S(t) such that uv(O) -- v and 6 x ( u v ( - t ) , C) - + t ~ + ~ 0},
W ~ (e) -
W ~ (e, S(t))
= {w E X [ there exists a negative orbit Uw of S(t) such that uw(O) -- w and U w ( - t ) ---->t~+~ e}, W s (e) -- W s (e, S(t))
= {v E X IS(t)v-->,~+oc e}, where e is any element of C. Two important remarks should be made about the sets W" (e, S(t)) and W s (e, S(t)). REMARK 4.4. If S(t) is a continuous semigroup, then, for any to > 0, e is a fixed point of S(to) and we have the equalities
W" (e, S(t)) - W" (e, S(to)),
w s (e, S(,)) - w (e, S(,o)).
REMARK 4.5. Assume that X is a Banach space. We recall that a continuous semigroup S(t) is said to be of class C r, r / > 1, if, for any t E G +, S(t) E C r (X, X). An equilibrium point e of a continuous semigroup S(t), t >~ O, is hyperbolic if the linear map DeS(t)lt=l -DSI (e), where Sl -- S(1) satisfies"
, ( O S l (e)) n {z
C llzl-
- 0.
(4.3)
The condition (4.3) means that e is a hyperbolic fixed point of $1. We set L - DS1 (e). Let or+ -- {z E or(L) I Izl > 1} and or_ - {z E or(L) I Izl < 1}. If o-+ is a finite set of ne elements, then ne is called the index ind(e) of e. Let P+ and P_ be the spectral projections
Global attractors in partial differential equations
939
corresponding to the sets o-+ and or_, let X+ = P+ X and L+ = L P+. There exist two small positive numbers 6+ such that sup{Izl I z ~ o-_} < 1 - 26_ and sup{Izl I z c o-(L+l)} < 1 - 26+. For any u+ ~ X+, u_ c X _ , we set
IIL~_"u+llx
IIu_ Ill - sup
IIu+ Ill - sup ,,~>0 (1 - 6+)I'
and, for u c X, we introduce the norm
IIL" u - IIx
n~>0 (1 - 6_)"
I1" I1~ on X, defined by
Ilu II1 - sup(ll P+u Ill, II e _ u Ill),
(4.4)
which is equivalent to the original norm I1" Ilg. For any R > 0, let OR(e) = {y E X I 4IlY - elll ~< R}, O R -- ON(O) A X+. One shows that there exist a positive n u m b e r R, two functions g+ E C 1(O +, O ~ ) , g_ 6 C j (OR, O + ) , such that the local stable and unstable manifolds of Sl at e are given by W s (e, Sl, OR(e)) -- {u E X lu -- e + P_(u - e) + g _ ( P _ ( u - e)),
P_(u - e) E OR}, (4.5)
W"(e, Sl, OR(e)) -- {u E X ] u -- e + P+(u - e) + g+(P+(u - e)),
e+(.-e) The functions g+ satisfy g+(0) = 0 and Dg+(O) = 0. If the index ind(e) is finite, then W"(e, Sl, OR(e)) is a manifold of dimension ind(e). Furthermore, there exist positive constants Re, Ke and fie such that, if Sll'y ~ ORe (e), for n = 1 . . . . . m, then,
6x(S11' y, W" (e, Sl, ORe (e))) <~ K e e x p ( - - f l e n ) 6 x ( y , W"(e, Sl, ORe(e))),
1 ~< n ~< m,
(4.6)
where fie depends on 6+ (see [210,8,33], for example). A s s u m e now that S(t) is a semigroup of class C 1 on the Banach space X and a gradient system with a Lyapunov functional 45 satisfying, for any t > 0,
o ( s ( t ) x ) < O(x),
v~ ~ x , x ~ c.
(4.7)
Suppose also that e is a hyperbolic equilibrium point of S(t) with finite index ind(e). Then one can show that, for any p > 0, there exists a positive n u m b e r r such that
W II (e, S(t)) A B x ( e , r) C W l` (e, S], 01o(e)) ,
(4.8)
where B x ( e , r) = {x ~ X ]]lx - ellx < r}. Often, it is easier to construct and study invariant manifolds of time-one maps rather than those of the flow S(t). Remarks 4.4 and 4.5 show that, in the case of continuous gradient
940
G. Raugel
systems with a Lyapunov functional satisfying the condition (4.7), the local and global unstable manifolds of S(t) at e can indeed be replaced by those of the map S1. From Theorem 2.26 and Proposition 4.2, one easily deduces the following result: THEOREM 4.6. Let S(t), t E G +, be an asymptotically smooth gradient system, which has the property that, f o r any bounded set B C X, there exists r ~ 0 such that y + (B) is bounded. I f the set o f equilibrium points C is bounded, then S(t) has a compact global attractor A and A = W u (C). Furthermore, if E is a discrete set, C is a finite set {el,e2 . . . . . eno} and A - Uej~g
W u
(ej).
If C is a discrete set and uz e C~ +, A) is a complete orbit in A through z, there exist equilibrium points ej and ek such that au~ (z) -- ej and oJ(z) = ek. If Z is not an equilibrium point, Cb(ek) < Cb(a) <~ ~ ( e j ) . The orbit which joins the points ej and ek is called a heteroclinic orbit. Under the hypotheses of Theorem 4.6, we introduce the m0 distinct values vl > v2 > 9" > Vmo of the set {q~(el), ~(e2) . . . . . ~(e,,0) } and let f j -- {eji E S I a~ (eji) - vj },
j -- 1 . . . . . mo.
The sets C 1, ~2, ...~ ~m0 define a Morse decomposition of the attractor A, i.e. (i) the subsets gJ are compact, invariant and disjoint; (ii) for any a 6 A \ g j and every complete orbit Ua through a, there exist k and l,
Uj
depending on Ua, so that k < l, otu~,(a) 6 g k and co(a) E g/. For 1 ~< k ~< m0, one defines
m0
Ak--U w,(e)le~U U
] ,
j=k
and, for 0 < d < do -= infz<<,k<~mo (Igk- 1 -- P k ) , F k -- {~ ~ X I ~ ( ~ ) ~< ~k-, - - S } ,
where v0 is chosen so that v0 > vl + do. Assume now that the hypotheses of Theorem 4.6 hold and that g is a discrete set. Then, the same arguments as those used to prove Theorem 4.6, show that, for any 0 < d < do and any k, 1 ~< k ~< m0, A k is the (compact) global attractor of S ( t ) / Fk . If X is a Banach space and all the equilibrium points of S(t) are hyperbolic, then, using the above Morse decomposition and Remark 4.5, one shows that the global attractor exponentially attracts a neighbourhood of it. This property plays an important role in the lower semicontinuity of global attractors. Its proof is implicitly contained in [ 105] and can be found in [12] (see also [94,13,130,77]).
Global attractors in partial differential equations
941
THEOREM 4.7. Let X be a Banach space. Assume that the hypotheses of Theorem 4.6 hold and that the Lyapunov functional satisfies (4.7), for any t E G +, t =/=O. Suppose moreover that, either S(t) is a continuous semigroup of class C 1 or S is a Cl-mappingfrom X to X and that, in both cases, all the equilibrium points of S(.) are hyperbolic. Then, there exist a bounded neighbourhood Bl of the global attractor ,4 and positive constants C1, dl < do, 7, such that,
6 x ( S ( t ) ( F J , ) , . A ) <~ C, e x p ( - g t ) ,
Vt 6 G +,
(4.9)
where F~I -- F~ A B l. The number t' > 0 in (4.9) depends on the minimum, over e E C, of the distance of the spectrum of DSI (e) to the unit circle in C. The Cl-regularity hypotheses in Theorem 4.7 and in Remark 4.5 can be weakened and replaced by the following assumption S(1)(y + e) = e + Ly + Q(e, y), where L ~ L(X, X) satisfies the spectral hypothesis (4.3), Q(e, 0) = 0 and Q(e, .) : x --> X is Lipschitz-continuous on the bounded sets of X. We also assume that the Lipschitz constant of Q(e, .) on the balls Bx(O, r) is a continuous, nondecreasing function of r, vanishing at r = 0. In this case, the mappings g+ of Remark 4.5 are only Lipschitzian mappings. The proof of Theorem 4.7 actually shows that, for any u0 E F~I' there exists a finite number of trajectories S(t)uoj, uoj c A, t ~ [tj, tj+l), j = 0 . . . . . k(uo), with to = 0 and tk~,0) = + e c such that ]]S(t)uo - S(t)uojllx <~ Cl e x p ( - v t ) , for any t c [tj, tj+l). The "trajectory"
-
{...j j=0
[,_J s(t),o tj
is called a finite-dimensional combined trajectory by Babin and Vishik in [12]. Actually, under additional conditions, it is shown there, that, for any ~ > 0, one can construct a finite-dimensional combined trajectory ri0(t) 6 .A such that IlS(t)uo - fi0(t)llx ~< C(r/) e x p ( - r / t ) , where C(O) > 0 depends on 7. This trajectory ri0(t) belongs piecewise to invariant manifolds, whose dimension increases when r/decays to zero. The assumption (4.7) implies that, for any hyperbolic equilibrium point e 6 g, there is a neighbourhood Ue of e such that, if xo E Ue\W s (e, S(t)), then there exists to = to(xo) > 0 so that S(t)xo q~ Ue, for t ~> to, that is, S(t)xo eventually leaves Ue never to return. This property plays an important role in the proof of (4.9) and also in the proof of the following theorem, stating that the unstable and stable manifolds of a hyperbolic equilibrium point are embedded submanifolds of X. THEOREM 4.8. Assume that the hypotheses of Theorem 4.7 hold and that S1 = S(1) as well as the linear map DSI (y) are injective at each point y of the global attractor .A, then, for each e ~ C, the unstable set W u (e, S(t)) is an embedded Cl-submanifold of X of finite dimension equal to ind(e), which implies that the Hausdorff dimension dim/4 (.A) is finite and equal to maxecc ind(e). If furthermore, for each e E C, SI is injective and DSI (y) has dense range at each point y of W s (e, S(t)), then the stable set is an embedded C l-submanifold of X of codimension ind(e).
G. Raugel
942
The above theorem is a consequence of Remarks 3.7, 4.4 and [121, Theorems 6.1.9, 6.1.10] (see also [8]). Under the hypotheses of Theorem 4.8, one shows that, for any hyperbolic equilibrium point e, there is a neighbourhood Qe of W s (e, S) such that, if xo ~ Q e \ W S (e, S(t)), then there exists to = to(xo) > 0 so that S(t)xo ~ Qe, for t ~> to. REMARKS.
(1) Assume that the gradient system S(t) is generated by the evolutionary equation (2.1). Under additional hypotheses, Theorem 4.8 implies that the Hausdorff dimension of the global attractor of (2.1) is estimated by the maximum of the number of eigenvalues with positive real part of A + D f (e), e ~ s If in (2.1), f is replaced by k f , one thus obtains asymptotic estimates of dimH(A), when /, --+ +cx~, by using asymptotic estimates of the number of positive eigenvalues of the operators A + k D f ( e ) (see [13, Chapter 10, Section 4]). (2) If S(t) is generated by an evolutionary equation, the injectivity of S1 (respectively DSI ) comes usually from a backward uniqueness result of the solutions of the evolutionary equation (respectively the corresponding linearized equation). And one shows that the range of DS1 is dense by proving that the adjoint map is injective. Backward uniqueness results are known to hold for a large class of parabolic equations (see [19], [121, Chapter 7, Section 6], [79]). The hyperbolicity of the equilibrium points is usually a generic property with respect to the various parameters involved in the definition of the semigroup S(t).
Gradient systems depending on parameters. As in Section 3.1, we consider a family of semigroups S~ (t) : X --+ X, t 6 G +, depending on a parameter ,k 6 A, where A -- (A, dx) is a metric space. Let k0 be a nonisolated point of A. We assume that Sx0 (t) is a gradient system satisfying the hypotheses of Theorem 4.7, the conditions (H. lb) and (3.8) of Section 3.1, then Propositions 3.1, 3.3 and Theorem 4.7 imply that, for k close enough to ,k0, we have,
SX(A)~, A)~o) ~ C~A (~,, )~0)V y ~ 1 7 6
,
(4.10)
for some positive constant c. We now turn to lower semicontinuity results and estimates of the semi-distance 6x(Ax 0, A~). Lower semicontinuity properties have been first proved in a very general setting in [10] and [105] (see also [94, Chapter 4, Section 10]). To show some of the ingredients, which are necessary, we begin with a very simple result. Hereafter, we denote by s the set of equilibria of Sx and assume that X is a Banach space, for sake of simplicity. Besides the condition (H. lb) of Section 3.1, we introduce the following hypotheses: k0 (H.2) The set s is a finite set, say s -- {e~~. . . . . e,,~,0}; (H.3) the global attractor ,A~0 is compact and n ~0
X o- U w,,(e?, j=l
11kO
U w,, (e?, ao); j=l
Global attractors in partial differential equations (H.4) for j - 1 . . . . . nzo, there exists a neighbourhood
Oj of e~ ~ in
943 X such that
lim (~x(W"(@ ~ Szo(1), Oj),A;~) - 0 . )~----~)~o
If the hypotheses (H.lb), (H.2), (H.3) and (H.4) hold, the global attractors A z are lower semicontinuous at 1. = 1.o, i.e., 6 x ( A z o , Ax) --~ 0 as 1. --+ 1.o, which, together with Proposition 3.1, implies that the attractors A z are continuous at 1. = 1.o.
PROPOSITION 4.9.
PROOF. We give the proof only when G + = [0, + o c ) . Without loss of generality, we suppose that to = 1 in the hypothesis (H. l b). Assume that the sets .A)~ are not lower semicontinuous at 1. - 1.o. Then, there exist e > 0 and, for any m E N, an element 1.m E NA(1.0, 1/m) and an element ~o,,, E .Az o such that 3x(q)m, Ax,,,) > e. Since .Az 0 is compact, the sequence q),,, converges to an element ~oo E Azo and
6x(~oo, Az,,,) > el2,
(4.11)
gm E N, m >~ mo,
where mo E N, m0 > 0. If ~oo E W" (ejLo , S~o (1), Oj), for some j -- 1 . . . . . nzo, (4.11) contradicts the hypothesis (H.4) Thus, assume that ~oo ~ W" (e~.~ Szo (1) O j) for any j -- 1
nzo, then there are
no and 7ro E W" (e) ~ Sxo(1), Oi), for some i, so that 99 - Szo(no)gro. Since Az o is compact, for any e > 0, there exists 6 > 0, depending only on Axo, such that, if [Iv - gro IIx ~< 6, then, (4.12) The hypotheses (H.4) and (H.lb) imply that we can choose 0 = 0(e, n~) > 0, such that, for i. E NA (1.0, 0), there exists vz 6 ,Az so that II~o - v~ IIx ~ ~,
(4.13)
]]Sz(n~o)vz - Szo(n~o)vz x <~ e/4. We deduce from (4.12) and (4.13) that, for 1. E NA (1.0, 0),
I
-
IIx
e/2,
which contradicts the property (4.11).
D
REMARK 4.10. Usually the hypothesis (H.4) is shown by proving that, for 1. close enough to 1.o, Sz (t) has, at least, nz o equilibrium points e~.l' j - 1 , . . . , nz o, and that, for each j - 1 . . . . . n Xo, there exists a neighbourhood Ojz of ejz such that
lim ~.--~)~o
6x(W"(e~~ Sz0(1), Oj), W"(e), Sz(1),
O~.))-0.
(4.14)
944
G. Raugel
If, for instance, the equilibrium points e) ~ are all hyperbolic and that the mapping Sz (1 ) converges to Sz0(1) in C 1(B, X), as )~ --~ )~0, where B is a bounded neighbourhood of Az 0, then the property (4.14) holds (see [210] and [10], for example). In the general case, if e) ~ is not hyperbolic, the hypothesis (H.4) is much more difficult to prove. However, in a particular regular perturbation of the Chafee-Infante equation, Kostin [ 131 ] proved this condition. If we want to estimate the semi-distance 6x(.Az o, .Az), we need stronger hypotheses. Actually, general results are only known in the case where Sz0 is a gradient system. The next theorem summarizes the above discussion and gives an estimate of the distance Hdistx (Az, Az0). To simplify, we do not state the optimal hypotheses (for a more general statement and various applications to perturbed problems, see [105,94,130,187]). THEOREM 4.1 1. Let be given a family of semigroups Sz, )~ E A, on a Banach space X, with compact global attractors Az. Assume that the hypothesis (H. lb) holds, that Sz o is a gradient system with a Lyapunov functional satisfying (4.7). Suppose moreover that, either Sz o (t) is a continuous semigroup of class C l or Sz o is a Cl-mappingfrom X to X, and that all the equilibrium points of Sz o (.) are hyperbolic. (1) If Sz (1) converges to Sz o (1) in C ! (Bl, X), where B 1 is a bounded neighbourhood of A z o, then the global attractors A z are continuous at )~ = )~o. (2) If moreover, there are positive constants C and p such that [1Sx0(1)- Sx(1)[IC,(B,,X) <~ C6xO~,)~o) p,
(4.15)
f o r any ~ in some n eighbourhood of )~o, then there exist a neighbourhood NA 0~0, g), two positive constants C and ~, 0 < ~ <<.p, such that, for )~ E NA 0~0, g),
Hdistx (A~, A~ 0) ~< C'6x 0~, )~0)~.
(4.16)
Theorem 4.11 gives good information about the size of the compact global attractor Az for )~ near )~0. Under the condition of hyperbolicity of the equilibria, the size of the global attractor does not change. However, the flows may not stay the same for each )~. As in Section 3.1, one gets more precise information, when Sz0(t) is a Morse-Smale semigroup. In the context of gradient systems, Theorem 3.10 implies the following statement, which is very useful in the applications. THEOREM 4.12. Let be given a family of semigroups Sz, )~ E A, on a Banach space X, with compact global attractors Az. Assume that the hypothesis (H.lb) holds, that, for any )~, Sz is a gradient system with Lyapunov functional satisfying (4.7). Suppose moreover that, either, for any )~, Sz (t) is a continuous semigroup of class C 1 or Sz o is a Cl-mapping from X to X and that Sz (1) converges to Sz o (1) in C ! (B l, X), where B1 is a bounded neighbourhood of A z o. In addition, suppose that: (1) Sz(1) a n d D S z ( 1 ) ( y ) are injective at eachpoint y of Az, )~ E A, (2) every equilibrium point ez o E Cz o is hyperbolic and W u (ez0, l, Sz0 (1)) is transversal to Wi~oc(e~.o,2, S~.o (1)), for any equilibria e~o, l, e~.o,2 E s
Global attractors in partial differential equations
945
then, there exist no E 1~ and a neighbourhood NA ()~0, e) of )~o, such that, f o r any )~ E NA ()~0, e), S~ (t) is a M o r s e - S m a l e system (i.e., Sx (no) is a M o r s e - S m a l e map) and Sx (no) is conjugate to Sx0 (n0). In the remaining parts of this section, we are going to describe classical examples of gradient systems generated by evolution equations.
4.2. Retarded functional differential equations In many problems in physics and biology, the future state of a system depends not only on the present state, but also on past states of the system. The theory of functional differential equations probably started with the work of Volterra [205,206], who, in his study of models in viscoelasticity and population dynamics, introduced some rather general differential equations incorporating the past states of the system. Since then, retarded functional differential equations (RFDEs) play an important role in biology (predator-prey models, spread of infections, circummutation of plants, etc.) and in mechanics. Here, we mainly describe a model RFDE arising in viscoelasticity. For further studies and generalizations to neutral functional differential equations, we refer the reader to the book of Hale and Lunel [ 116] as well as to [94,174]. For a given 6 > 0 and n E 1~ \ {0}, let C = C ~ 0); R") be the space of continuous functions from [ - 3 , 0] into ]t~" equipped with the norm II" II = II" IIc. For any ot ~> 0, for any function x : [ - 3 , or) ~ It~" and any t E [0, c~), we let xt denote the function from [ - 6 , 0] to It~'' defined by xt(O) = x ( t + 0), 0 E [--6, 0]. Suppose that f E C k (C, R'z), k >~ 1, and that f is a bounded map in the sense that f takes bounded sets into bounded sets. An autonomous retarded functional differential equation with finite delay is a relation ~c(t) - f (xt),
(4.17)
where ,f(t) is the right hand derivative of x ( t ) at t. For a given 99 E C, one says that x(t, 99) is a solution o f (4.17) on the interval [0, ot~), ot~ > O, with initial value 99 at t = 0, if x(t, qg) is defined on [ - 6 , c~o), satisfies (4.17) on [0, oqo), xt(., 99) E C for t E [0, ot~) and x0(., 99) = qg. Using the contraction fixed point theorem, one shows that, for any 99 E C, there exists a unique mild solution x (t, 99) defined on a maximal interval [ - 3 , ot~). Moreover, x(t, qg) is continuous in (t, qg), of class C k in ~0 and, for t E (k6, ot~), of class C k in t. If c~0 < +cx~, then Ilxt(', 99)11--+ +cx~ as t --+ ot~0. We assume now that all the solutions of (4.17) are defined for t E [0, +cx~). Then, the one-parameter family of maps S(t), t >~ O, on C defined by S(t)q9 = xt(., ~p) is a continuous semigroup on C. We also introduce the linear semigroup V (t) :C --+ C, t >~ 0, given by (V(t)qg)(O) - ~o(t + O) - cp(O), =0,
t + 0 < O,
(4.18)
t+O~O.
The following theorem states the basic qualitative properties of the semigroup S(t) and can be found in [ 116], for example.
946
G. Raugel
THEOREM 4 . 1 3 . If the positive orbits of bounded sets are bounded, then S(t) is a compact map f o r t ~ 6. Moreover, f o r t >~ O, one has (4.19)
S(t)q9 = U(t)q9 4- V(t)qg,
where U(t) is a compact map from X into X f o r t ~ 0 and V(t) has been defined in (4.18). Furthermore, f o r any fl > O, there is an equivalentnorm l . Ion C so that IS(t)~ol ~< exp(-flt)ltpl, t ~ O, and S(t) is an a-contraction in this norm f o r t ~ O. The fact that S(t) can be written as (4.19) had been remarked by Hale and Lopes [102]. The next result is mainly a consequence of Theorems 2.26, 2.38 and 4.13. The analyticity property in the third statement is due to [ 173]. THEOREM 4.14. If the positive orbits of bounded sets are bounded and if S(t) is point dissipative, then: (i) S(t) has a connected compact global attractor A C C; (ii) there is at least an equilibrium point (a constant solution) of(4.17); (iii) if f E C k (C, Rn), k ~ O, (respectively analytic), then any element u of the attractor r is a Ck+l function (respectively analytic); (iv) if f is analytic, then S(t) is one-to-one on A. We now present an example of a gradient system generated by a RFDE, which arises in viscoelasticity [143]. We now let 6 = 1 and suppose that b is a function in C2([-1, 0], R), such that b ( - 1) = 0, b(s) > O, b' (s) >~O, b" (s) >~0 and that b"(00) > 0
(4.20)
for some 00 6 ( - 1,0).
Let g E C l (IR, R) be such that G(x) =
fo Xg ( s ) d s
--+ +cx~
as ix]--+ +cx~.
(4.21)
We consider the equation
~c(t) -- -
/
b(O)g(x(t + 0)) dO.
(4.22)
1
(4.22) is a special case of (4.17) with f(qg) - - f ~ 1 b(O)g(qg(O)) dO. Let S(t) be the local semigroup on C defined by S(t)q9 = xt(., qg), where x(t, qg) is the local solution of (4.22) through 99 at t = 0. To show that the solutions of (4.22) exist globally, one introduces the functional on C --
'/
b' (O)
(f0~
g(~o(s)) ds
)2
dO.
Global attractors in partial differential equations
947
We set Lo (~P) - f0~ g(~P (s)) ds. A short computation shows that, for t ~> 0,
dt(45(S(t)q))) -- - ~
+
b'(-1)[L_,
b"(O)[Lo(S(t)qg)] 2dO . 1
(4.23) The hypotheses on b and g imply that 45(99) --+ + e c as I1~011~ -+-~ and, due to (4.23), that 45(S(t)q)) ~< 45(q9), for t >~ 0. Therefore, the solutions of (4.22) exist globally and the orbits of bounded sets are bounded. The next theorem summarizes the properties of the semigroup S(t) [94,114,99]. THEOREM 4.1 5. Assume that the conditions (4.20) and (4.2 1) hold. Then, the co-limit set
of any orbit is a single equilibrium point. If moreover, the set g of the zeros of g is bounded, the semigroup S(t) generated by (4.22) is a continuous gradient system and admits a compact connected global attractor Abg in C. If in addition, each element of g is hyperbolic, then dim W" (x0) = 1, for any xo ~ g and Abg = ~xo~g Wu (xo). SKETCH
OF T H E P R O O F .
We first observe that any solution x(t) of (4.22) satisfies
Ji(t)+b(O)(g(x(t)))-b'(-1)L_l(xt)+
f
b"(O)Lo(xt)dO.
(4.24)
1
Suppose that 45(S(t)qg) = 45(99), for t ~> 0, then (4.23) and (4.24) imply that the solution x (t) through (p satisfies
s
+ b(O)(g(x(t))) = O,
together with Ls(xt) = 0 for s in some interval I0 containing 00. It follows that ~ is a constant. The boundedness of x(t) implies that x(t) is a constant and thus q9 is a zero of g. Hence 45 is a strict Lyapunov functional. The existence of the compact global attractor 9A b g is a direct consequence of Theorems 4.6 and 4.14. If g(c) --O, the linear variational equation about c is ~(t) -- - f ~ 1 b(s)g'(c)y(t + s)ds and the eigenvalues ~. of the linear variational equation are given by
)~ -- -
f
b(s)g' (c) exp()~s) ds. 1
It is possible to show that the equilibrium point c is hyperbolic if and only if g' (c) # 0. If g' (c) = 0, ~. = 0 is a simple eigenvalue. Property 1) of Remarks 4.3 then implies that the w-limit set of any positive orbit is a singleton. If g'(c) > 0, then c is stable. Finally, one easily shows that, if g' (c) < 0, then c is unstable with dim W" (c) -- 1. D Suppose now that b is a fixed function and consider the global a t t r a c t o r .Abg as a function of the parameter g. Semicontinuity and continuity results of .Abg with respect to g are
G. Raugel
948
proved in [105] and are actually an application of Proposition 4.9 and Theorem 4.11. We have seen that, for each zero c with g'(c) < 0, c is an unstable equilibrium point with dim W"(c) = 1; this means that there are two distinct complete orbits qge(t) and ~e(t) which approach e as t --+ +c~. Since the w-limit set of qge(t) (respectively 7re(t)) is a single equilibrium point e~0 (respectively e 7,), the next problem is to determine if e~o (respectively e~) is smaller or larger than e. If s = {el, e2, e3} with el < e2 < e3, the flow o n J4bg preserves the natural order since r is connected. The case when C -- {el, e2, e3, e4, e5} with el < e2 < e3 < e4 < e5, has been studied by Hale and Rybakowski [114]. To state their result, it is convenient to use the notation j[k, l] to mean that the unstable point ej is connected to ek and et by a trajectory. If g has five simple zeros, then e2, e4 are unstable, while el, e3, e5 are stable. THEOREM 4.16. Let b be fixed. One can realize each of thefollowingflows on J4bg by an appropriate choice of g with five simple zeros:
(i) 2[1, 31, 4[3, 51; (ii) 2[1,4], 4[3, 5]; (iii) 2[1,5], 4[3, 5]; (iv) 2[1, 31, 4[2, 5]; (v) 211, 3], 4[ 1, 5].
4.3. Scalar parabolic equations The simplest and most studied gradient partial differential equation is the semilinear heat or reaction-diffusion equation, which models several physical phenomena like heat conduction, population dynamics, etc. The heat equation belongs to the class of parabolic equations, where smoothing effects take place in finite positive time. Here, we study this equation under very simplified hypotheses on the nonlinearity. Let S2 be a bounded domain in ~n, with Lipschitzian boundary. We consider the following heat equation 0U
O--f(x, t) -- Au(x, t) + f (u(x, t)) + g(x), u(x,t) = 0 , u (x, 0) = u0 (x),
x 9 $2, t > O, x 9 0S2, t > 0, x ~ S2,
(4.25)
where g is in L2(S'2) and f :R ~ R is a locally Lipschitz continuous function. We introduce the operator A -- --AD with domain D(A) - {v E H~(~2) I - A v E LZ(g2)} and set V - H~ (S-2) =_ D(A)1/2. In the case n ~> 2, we assume that the locally Lipschitz continuous function f also satisfies the following growth condition: (A. 1) there exist positive constants Co and or, with (n - 2)or ~< 2 such that If(Yl) - f(Y2)l ~< Co(1 + lyll ~ + lyel~)ly~ - y2l,
Vyl, y2 E IK.
(4.26)
The restriction (n - 2)or ~< 2 has been made only for sake of simplicity. Most of the results of Section 4.3 also hold if (n - 2)or ~< 4 (see Remark 4.18(1) below). The hypothesis
Global attractors in partial differential equations
949
(A. l) together with the Sobolev embeddings properties, allow to define the mapping u V ~ f ( u ) E L2(X2), by ( f ( u ) ) ( x ) = f ( u ( x ) ) , for almost every x 6 X-2. This mapping is Lipschitzian on the bounded sets of V. With the above definitions of A and V, we can rewrite (4.25) as an abstract evolutionary equation in V:
du
t"d( t )
-- - A u ( t ) 4- f (u(t)) + g,
t > O, u(O) -- uo.
(4.27)
Since A is a sectorial operator and f : V --+ L2(.Q) is Lipschitzian on the bounded sets of V, for any r ~> 0, there exists T ---- T ( r ) > 0 such that, for any u0 6 V, with Ilu0llv ~< r, Equation (4.27) has a unique classical solution u E C~ T], V) A C l ((0, T], Lz(.Q)) 71 C~ T], Z)(A)). Later, when we need more regularity on f , we suppose in addition that (A.2) f c Ctl~(IK, R), f~ is locally HOlder continuous and, if n ~> 2, there exist nonnegative constants C l, ell, ill, such that
If'(Y~) - f ' ( y 2 ) l ~ c~(1 + ly~l ~, + l y 2 1 ~ , ) l y l - y21 ~ , (4.28)
Vyl, Y2 e IR,
where or1 > 0, (or1 + / ~ l ) ( n - 2) ~< 2 if n ~> 2. In this case, f is a C l ' ~ - m a p p i n g from V into L 2 (.Q). REMARK. For sake of simplicity, we have provided the above heat equation with homogeneous Dirichlet boundary conditions. All the assertions of this subsection remain true if we replace them in (4.25) by homogeneous Neumann conditions, in which case V = H1(s Even, much more general boundary conditions may be chosen. Furthermore, we can rei,j=n
0
0
place the Laplacian operator by any second order operator Y-]-i.j=l axi (aij(X)~xj)--[-ao(x), where aij, ao are smooth enough functions of x and the matrix [aij(x)]i,j is symmetric, positive definite, for any x 6 X-2. To obtain global existence of the solutions of (4.27), we need to impose, for example, a dissipation condition. Here we assume that (A.3) there exist constants C2 ~> 0 et # c R such that y f (y) <. C2 + #y2,
F ( y ) ~ C2 +
1
#y2,
'v'y E R,
(4.29)
with # < 1.1,
(4.30)
where ~.1 is the first eigenvalue of the operator A and where F is the primitive F ( y ) = fo' f (s) ds of f . Global existence of solutions of (4.25) already holds under the hypothesis
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(4.29). Condition (4.30) will ensure that all the solutions are uniformly bounded. Indeed, introducing the functional 40 6 C~ R) given by
*o(u)-
s
lvu(x)l
2-
F(u(x)) - g ( x ) u ( x ) ) dx,
(4.31)
one shows that, if u(t) is a classical solution of (4.27) for t ~< T, then ~o(u(t)) CO([0, T]) A C 1((0, T]) and
d CPo(u(t) ) _ _ [lu,
Vt ~ (0, T],
(4.32)
which implies that q~0 is a strict Lyapunov function of (4.27). Moreover, using the assumptions (A.1), (A.3) and the property (4.32), we obtain, for 0 < s < ~1 - #, 1(1 2
--
l Z~.l + e ) l l v u ( t ) l l 2L 2 - C l n l -
~< qSo(u(O))~< C*(1 + [[gllL 2 +
~l Le Ilgll2L 2
q:'o( u(t) )
ttu(O)tt'+ llu(o)ll., H l I
(4.33)
where C, C* are positive constants. This implies that all the solutions of (4.27) are global and the orbits of bounded sets are bounded. We notice that the set s of the equilibrium points of (4.27) is given by
gp -- {u E V IADu + f (u) + g = 0 } . Due to the dissipative condition (A.3), this set is bounded in V. If we let S(t)uo denote the solution u(t) of (4.27), with initial data u0 6 V, we have defined a continuous gradient system S(t) on V. We remark that the map (t, uo) ~-~ S(t)uo belongs to C~ + ~ ) • V, V). Due to a backward uniqueness result of Bardos and Tartar [ 19], the mapping S(t) is injective on V, for any t ~> 0. Furthermore, since for any bounded set B C V, the orbit y + (B) is bounded in V, one shows, by using the smoothing properties of S(t), that the orbit 7+(B) is bounded in D(A), for any r > 0 [121, Theorem 3.5.2], which implies in particular that S(t) is compact for t > 0. Applying Proposition 2.5 and Theorem 4.6, one obtains the existence of a compact global attractor. THEOREM 4.17. Assume that the assumptions (A.1) and (A.3) hold. Then, the semigroup S(t) generated by (4.27) has a compact connected global attractor , 4 - WU(ge) in V.
The semigroup S(t)IA is a continuous group of continuous operators. Moreover, the global attractor At is bounded in 79(A) and thus in H2(a"2), if the domain S-2 is either convex or of class C 1"1. Properties of the compact attractor A. In what follows, we assume that the assumptions (A.1), (A.2) and (A.3) hold. First, we recall that A is often bounded in a higher order Sobolev space. If, for instance, the nonlinearity f belongs to Ck(V, L2(S'2)) (respectively is analytic), then the
Global attractors in partial differential equations
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map (t, u0) 6 [r, + ~ ) • V ~ S(t)uo E V is of class Ck, for r > 0 (respectively is analytic) (see [121, Corollary 3.4.6]). Due to the invariance of A, it follows that S(t)IA :t IR w+ S(t)uo ~ V is of class Ck (respectively analytic). Arguing by recursion on k, one deduces from the regularity in time property that, if f ~ Cbk-J(HJ+l (s H j (S-2)), 1 <<,j <~k, (respectively C~ HJ(s-2)),for j ~> 1) and if moreover s is of class Ck+l'l (respectively C ~ and g c Hk(S2) (respectively C~(s then ,,4 is bounded in Hk+2(S2) (respectively C~163 For regularity in Gevrey spaces, when the Dirichlet boundary conditions are replaced by periodic ones, we refer to [75,185,62] and the references therein. The fractal dimension of ,A is finite. An explicit bound is given in [202, Chapter 6], for example. By Theorem 3.17, the property of finite number of determining modes holds. We remark that e is a hyperbolic equilibrium point of (4.27) if and only if the spectrum a ( - A + D f ( e ) ) of ( - A 4- D f ( e ) ) does not intersect the imaginary axis in C or, equivalently here, if 0 is not an eigenvalue of ( - A + D f (e)). Moreover, the index ind(e) is equal to the number of positive eigenvalues lj(e) o f - A + D f ( e ) . We recall that the hyperbolicity of the equilibria e of (4.25) is a generic property in g E L2(S2) ([8], [13, Chapter 6, Theorem 3.4]). Generic hyperbolicity of the equilibrium points also holds with respect to the domain s [ 124]. In the one-dimensional case, generic hyperbolicity with respect to f has also been proved in [23,200], for example. If all of the equilibrium points (e, 0) of (4.27) are hyperbolic, we deduce from Theorems 4.6 and 4.8 that r - Uej~gp W u (ej) and that the Hausdorff dimension dimn (A) is equal to maxecgp ind(e). The unstable and stable manifolds W" (e, S(t)) and W s (e, S(t)) are embedded C l-submanifolds of V of dimension ind(e) and codimension ind(e), respectively. We shall see later that these stable and unstable manifolds always intersect transversally if s is an interval of IR. This property is not known in higher dimension space. If we allow the function f to depend upon x and assume that this function f (x, .) satisfies the assumptions (A.1), (A.2) and (A.3) uniformly in x, then the resulting semigroup S(t) still admits a compact global attractor. Brunovsk2~ and Polfi6ik have proved that the semigroup defined by (4.27) is a Morse-Smale system, generically in such non linearities f (x, .) (see [25]). Furthermore, for the unit ball in R 2, Polfi6ik has shown that there exists a function f (x, u) for which the transversality of the stable and unstable manifolds does not hold (see [182]). In the one-dimensional case, the eigenvalues )~i of the operator A fulfill the gap condition needed in the construction of inertial manifolds. In this case, (4.27) has an inertial manifold (see [73]). Mallet-Paret and Sell [ 148] have proved that this gap condition can be replaced by a cone condition, which is less restrictive. As a consequence, they showed that (4.27) has an inertial manifold of class C 1, if s is either a rectangular domain (0, 2re~a1) • (0, 27r/a2), where al, a2 are arbitrary positive numbers or s is the cube (0, 27r) 3 and if f : (x, y) 6 1"2 • ~ ~ f (x, y) E R is of class C3. These results are valid for Equation (4.25) supplemented with homogeneous Dirichlet or Neumann boundary conditions or periodic boundary conditions. The existence of an inertial manifold can also be proved in the case of domains in R ''+l , which are thin in n directions [109,187]. If S-2 C R, the positive orbit of every point u0 6 V is convergent [214]. If s is a domain in R n+l , which is thin in n directions, the positive orbits are still convergent [108]. In the case n ~> 1, all the orbits of (4.27) are still convergent if f : R --+ R and g are analytic functions [ 199]. For further details on convergence properties, see [ 108,26,183].
G. Raugel
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REMARKS 4.18. (1) For sake of simplicity in the exposition, we have assumed that the exponent oe in (A. 1) satisfies the condition (n - 2)or ~ 2, when n ~> 3. We can still associate with (4.27) a continuous semigroup S(t) on V, provided (n - 2)or ~< 4 and that the domain I2 is either convex or of class C l, 1. In this case, the proof of the existence of the associated continuous semigroup S(t) is less straightforward and uses a fixed point argument introduced by Fujita and Kato (see [77]). This semigroup admits a compact global attractor ,,4 in V, with the same properties as above. The restriction of the above semigroup S(t) to X2 -- H2(S-2) N H~ ($2) is also a continuous semigroup on X2 and ,A is the global attractor
of S(t)lx2. (2) If we do not want to introduce a limitation on the growth rate of f , we can also consider Equation (4.25) in the space X - Co - {v 6 C (~-) I v - 0 in 0 S2 } and introduce the operator Ac -- --A with domain D(Ac) -- {v 6 X A H~(s I Av 6 X}. If g - - 0 and f (0) = 0, local existence and uniqueness of the solutions of (4.25) are well known. If, moreover, there exists a positive constant x such that,
y f ( y ) <<,O,
Vlyl ~> x,
(4.34)
one shows, by using the maximum principle and truncature arguments, that the solutions of (4.25) are all global, which allows to associate to (4.25) a continuous semigroup Sc(t) on X. One proves under the assumption (4.34), that, for any bounded set B C X, any r > 0 and any 0 < / 3 < 1, the orbit y + ( B ) is bounded in cl,~(I-2) and hence that Sc(t) has a compact global attractor in X (see [119,118] for further details, and also [ 183] for a setting in the spaces Ws~p(s2), p ~> 2, 1 ~< s < 2).
4.4. One-dimensional scalar parabolic equations In the one-dimensional case, detailed properties of the flow on the compact attractor can be obtained by using tools like the Sturm-Liouville theory, the Jordan curve theorem as well as the strong maximum principle. Here, we are going to distinguish two types of boundary conditions. Because of lack of space, we describe only a few results and refer to the fairly complete review of Hale [96] for further results.
The case of separated boundary conditions. For sake of simplicity, we consider the following reaction diffusion equation on S2 = (0, 1), provided with homogeneous Neumann boundary conditions: ut = Uxx + f ( x , u, Ux)
in S2 = (0, 1), Ux(0) = Ux(1) = 0.
(4.35)
We could consider more general separated boundary conditions like
boux(O, t) + fl0u(0, t) = blUx(1, t) nt- fllu(1, t) -- 0,
(4.36)
where b0,/30, b t, fll are normalized so that b 2 +/32 - b 2 +/32 - 1, and also replace Uxx with a(x)Uxx, where a 6 C2(I-2) is a positive function on S2. We assume now that f :[0, 1] x IR2 --+ R is a C2-function satisfying both conditions:
Global attractors in partial differential equations
953
(C1) there exist V, 0 ~< 9/< 2, and a continuous function x:[0, + e c ) ~ [0, +oo) such that
I f ( x , y, ~)1 ~< K(r)(1 +
I~IY), v(x, y, ~) E [0,
11 x [ - r , r] x IR;
(4.37)
(C2) there exists a positive constant K such that
y f (x, y, O) <~O,
V(x, y) 9 [0, 1] x N, lYl > K.
(4.38)
Under these hypotheses, Equation (4.35) generates a continuous semigroup S(t) on the space X s = 7)((--AN -+- l)S), 1/2 ~< s ~< 1, where A N is the Laplacian operator with homogeneous Neumann boundary conditions. We recall that ~D(--A N -Jr- I ) = {u E H2(F2) I ux(O) --
Ux(1) =0}.
The dissipation condition (C2) implies that the orbits of bounded sets are bounded. In the one-dimensional case, the presence of gradient terms in the nonlinearity does not prevent the gradient structure as it has been proved by Zelenyak [214].
PROPOSITION 4.19. The continuous semigroup S(t) is a gradient system on X s, s E [ 1/2, 1]. Moreover, every positive orbit is convergent.
SKETCH OF THE PROOF. The first step of the proof consists in finding a Lyapunov functional 4~0(u) for (4.35). For u E X s, one considers functionals 1
CI)o(u) --
f0
G(x, u, Ux) dx,
where G : (x, y, ~) E [0, 1] x IR2 ~ u(x, t) of (4.35), d __ e , o ( u ( t ) ) dt
-
-
G(x, y,
f01
~)
E
11~2 and one observes that, for any solution
.. . . ) . , 2 dx .
(4.39)
provided the mapping G satisfies
~G~y - f G ~
+ Get - G,,,
V(x, y, s~) E [0, 1] • ]1~2,
ut (0, t)G~ (0, y, 0) = ut (1, t)G~ (1, y, 0) = 0,
(4.40)
Vy E N.
One then shows that there exists a solution G of class C2 of (4.40) such that G ~ ( x , y, ~) > O. Thus, 45o is a strict Lyapunov functional and S(t) is a gradient system. To prove that the co-limit set co(q)) is a singleton, for every q) E X s, we apply the general result of [108] mentioned in Remarks 4.3. Indeed, for any equilibrium point e E s of
954
G. Raugel
(4.35), the eigenvalue problem for the linearization of (4.35) at e (called Sturm-Liouville problem) )~v = Vxx + A,(x, e, ex)v -+- f~ (x, e, ex)Vx,
'v'x 6 (0, 1), (4.41)
vx(O) = Vx(1),
has the following well-known properties. All the eigenvalues )~j(e) are real and algebraically simple. The (normalized)eigenfunction ~oj(e) corresponding to the j t h eigenvalue ~,j (e) has exactly j - 1 zeros. In particular, if 0 is an eigenvalue of (4.41), it must be simple and the general convergence result of [ 108] applies. D Probably, the most important property of scalar Equation (4.35), which is the starting point of the qualitative description of the global attractor .A is the transversality property of the stable and unstable manifolds of the equilibria. This result is due to Henry [ 123] (see also [3] for another proof). THEOREM 4.20. If e and e* are hyperbolic equilibria of (4.35), then W" (e) is transversal to W s (e*). Thus, S(t) is a Morse-Smale system and is structurally stable, when the equilibria are all hyperbolic. Here we can only give an idea of the proof of this result (for more details, see [ 123,3,94]). It involves the following two basic results. For a continuous function v : [0, 1] --+ R, let z(v) denote the number (possibly infinite) of zeros of v in (0, 1). We say that a differentiable function v has a multiple zero at x0 6 [0, 1] if v(xo) = Vx (x0) = 0. An application of the maximum principle in two-dimensional domains together with the Jordan curve theorem yields the following result (see [168,154,32]). LEMMA 4.21. Let v(x, t) ~ C 0([0, +cx~), X s) be a solution of the linear nonautonomous equation vt = Vxx + a(x, t)v + b(x, t)Vx,
x 6 (0, 1),
vx(O) = Vx(1) = 0,
(4.42)
where a, b are functions in LCc((O, 1) x IR). Then, if v is not identically zero, the following properties hold: (i) z(v(., t)) is finite f o r any t > 0; (ii) z(v(., t)) is nonincreasing in t; (iii) if v(xo, to) = Vx(XO, to) = 0 f o r some to > 0, xo 6 (0, 1), then z(v(., t)) drops strictly at t = to.
Notice that the above lemma holds as well for other separated boundary conditions and for periodic boundary conditions. The nonincrease of the zero number together with the above Sturm-Liouville properties imply the following restriction on the connecting orbits. LEMMA 4.22. If e E X s, e* ~ X s are hyperbolic equilibrium points of (4.35) and there is an element uo ~ X s such that oe(uo) = e and co(uo) = e*, then dim W" (e) > dim W" (e*).
G l o b a l a t t r a c t o r s in p a r t i a l d i f f e r e n t i a l e q u a t i o n s
955
The main ingredients of the proof of L e m m a 4.22 are as follows. If u(t) is a solution of (4.35) through u0, then u,(t) satisfies a linear equation of the form (4.42), whose coefficients converge exponentially to those of the linearized equations around e and e*, when t --+ - e c and t --+ 4-oo, respectively. Since no solution of (4.42) approaches zero faster than any exponential, one can show that ut (t) ~ 0 as t ~ -cx~ along the direction of one of the eigenvectors of the operator O2/Ox 2 4- f~.(x, e, ex)I + f ; ( x , e, ex)O/Ox. It follows from the above Sturm-Liouville properties that z(u,(t)) <~ ind(e) - 1 for t close to - e c . Likewise, one shows that z(ut(t)) >~ind(e*) for t close to 4-oo. Then L e m m a 4.21 implies that dim W ~ (e) > dim W ~ (e*). To complete the proof of Theorem 4.20, one assumes that the manifolds are not transversal, uses the characterization of the tangent space T W S (e*) of W S(e*) in terms of the adjoint of the linearized equation around u(t) and argues as in the proof of Lemma 4.22 for this adjoint equation to show that dim W u (e) < dim W u (e*), which contradicts L e m m a 4.22. Lemma 4.22 naturally leads to the problem of connecting orbits, when all of the equilibrium points are hyperbolic. We say that C (e, e*) is an orbit connecting e to e* if, for any point uo ~ C(e, e*), we have or(u0) = e and co(u0) = e*. This problem has been discussed for a long time in the special case of the Chafee-Infante equation: ut--Uxx4-#2(u-u
3)
in(O, 1),
ux (0) = Ux (1) = 0.
(4.43)
It has been shown by Chafee and Infante [30] that the only stable equilibrium points of (4.43) are the constant functions 4-1. Furthermore, for each j -- 1, 2 . . . . two equilibrium points e f of index j bifurcate supercritically from 0 at # j -- j j r . In the interval (0, Jr), there are three equilibrium points 0, + 1; in the interval (/~j, ~ j + l ) , 0 has index j 4- 1 and there are exactly 2 j 4- 3 equilibria 0, 4-1, e k+ , k - 1 , . . . , j . The complete description of the attractor ,An has been given by Henry in [ 123]. For # c ( ~ j , ~ j + l ) , the attractor .An is + -4the closure of W '' (0), and, for each 1 ~< k ~< j , there exists an orbit connecting e k to e t , for 1 ~< l < k, and to 4-1. Before presenting general results on the existence of connecting orbits, we describe another important consequence of the properties of the zero number; that is the existence of an inertial manifold of (4.35) of minimal dimension, when the equilibria are all hyperbolic. More precisely, let N = maxe; ~s ind(ej). Using the zero number, Rocha [191 ] has shown that, for any equilibria e j and ek, with j # k, one has
z(ej - ek) < N. As a consequence, he proves the existence of a Lipschitz continuous inertial manifold. THEOREM 4.23. If the hypotheses (C1) and (C2) hold and the equilibria are all hyperbolic, there exists an (Lipschitzian) inertial manifold of (4.35) of minimal dimension N and it is a graph over the linearized unstable manifold of maximal dimension. If f (x, y, ~) - f (x, y), the inertial manifold is of class C 1. This result had been proved before by Jolly [127] in the case f (x, y, ~) = y - y3 and by Brunovsk~ [22] in the general case f (x, y, ~) - f (y).
G. Raugel
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We now go back to the problem of connecting orbits in the global attractor ,4 of (4.35). We consider here the semigroup S(t) acting on X 1 and assume that all of the equilibria are hyperbolic. The case of a nonlinearity f , depending only on u was solved by Brunovsk# and Fiedler [24]. Recently, the general case of a nonlinearity f ( x , y, ~) has been mainly considered by Fusco and Rocha, Fiedler and Rocha, Wolfrum. To determine the set s of the equilibrium points of (4.35), one solves the ODE
Ux = v,
Vx = - f ( x ,
Since .,4 is compact, the set s so that
u, v),
u(0) = u0,
v(0) = 0.
(4.44)
is a finite set of k elements {el . . . . . ek } that we have ordered
el (0) < e2(0) < ... < ek (0).
(4.45)
By uniqueness of the solutions of (4.44), these values are distinct. At the other boundary point x = 1, this order may have changed. We thus obtain a permutation rrf ~ 7/" o f the set { 1 . . . . . k} given by errf(l)(1) < errf(2)(1) < " "
< err/(kt(1).
(4.46)
This shooting permutation was introduced by Fusco and Rocha [76]. It characterizes the existence of connecting orbits as proved by Fiedler and Rocha [67].
THEOREM 4.24. Let f (x, y, ~) be a function i n (32([0, 1] x IR x IK; N) satisfying the conditions (C1) and (C2). Assume that the corresponding Equation (4.35) has only hyperbolic equilibria. Let rcf be the permutation defined by (4.45) and (4.46). Then Jrf determines, in an explicit constructive process, which equilibria are connected and which are not. In other words, this permutation determines which of the sets C (el, e j) are nonempty. The proof of Theorem 4.24 uses, in a crucial way, the zero number of differences of solutions of equations of type (4.35), the transversality of stable and unstable manifolds, the shooting surface, the above mentioned Sturm-Liouville properties and the Conley index. The proof of Theorem 4.24 relies on constructive lemmas that we state, without comments. The first lemma shows that the existence of a connecting orbit from e to e* implies a particular type of cascading. LEMMA 4.25 (Cascading). Under the assumptions of Theorem 4.24, assume that e, e* are two equilibria with n -- ind(e) - ind(e*) > 0. Then C(e, e*) # 0 if and only if there exists a sequence (cascade) e -
vO,
131,
...,
Vn =
e *
of equilibria such that, for every j, 0 <. j < n, we have: (i) ind(vj+l) = ind(vj) - 1, (ii) C(vj, vj+l) # 0.
Global attractors in partial differential equations
957
Due to the cascading lemma, it suffices to check all of the possible connections from e to e*, when ind(e) - ind(e*) = 1. DEFINITION 4.26. If e, e* are two equilibria of (4.35) with ind(e) - ind(e*) = 1, we say that the connections between e and e* are blocked if one of the following conditions holds: (i) z(e - e*) 5~ ind(e*), (ii) there exists a third equilibrium w with w(0) between e(0) and e*(0) such that z(e - w) = z(e* - w) = z(e - e*). Brunovsk3~ and Fiedler [24] had already shown that blocking prevents connections. In [67], the reverse property is proved. LEMMA 4.27 (Liberalism). Let e, e* be two equilibria of(4.35) with ind(e) - ind(e*) = 1. Then, C (e, e*) ~ 0 if and only if the connections from e to e* are not blocked. Theorem 4.24 raises the question, whether the shooting permutation determines the global attractor. More precisely, let us denote by ,AU and rr.f the global attractor and the shooting permutation of the semigroup S f ( t ) generated by (4.35), corresponding to the nonlinearity f . Fiedler and Rocha [68] have shown that the shooting permutation rrf completely determines the global attractor, up to an orbit preserving homeomorphism. THEOREM 4.28. Let fl (x, y, ~) and f2(x, y, ~) be two functions in C2([0, 1] x ]R x IR; IR) satisfying the conditions (C1) and (C2). Assume that the corresponding equations (4.35) have only hyperbolic equilibria. Then the equality rc.fi = rcf2 implies that the global attractors A f t and A fz are topologically equivalent. Theorems 4.24 and 4.28 have been given in the frame of homogeneous Neumann boundary conditions. Although the global attractor for a given nonlinearity depends on the choice of the boundary conditions, the set of their topological equivalence classes is independent of the boundary conditions in the following sense. Let us consider again the reaction diffusion equation ut=uxx+f(x,u,
ux)
in(0,1).
(4.47)
For r = (r0, rl ) given in [0, 1]2, we provide (4.47) with the boundary conditions: - r 0 u x ( 0 , t) + (1 - r0)u(0, t) = f l u x ( l , t) + (1 - rl)u(1, t) = 0 .
(4.48)
If f ( x , y, ~) is a function in C2([0, 1] • IR x IR; R) satisfying the conditions (C1) and (C2), then Equations (4.47) and (4.48) generate a continuous semigroup S T (t) on X 1 f which admits a compact global attractor r Using a homotopy argument together with the Morse-Smale property of the global attractors, Fiedler [64] has obtained the equality of the sets of topological equivalence.
G. Raugel
958
THEOREM 4.29. Let r = (to, rl) e [0, 1] 2 and O- = (o-o, O'l) E [0, 1] 2 be given. If f (x, y, ~) is a function in C2([0, 1] • • • R; ~) satisfying the conditions (C1) and (C2) and if Srf(t) has only hyperbolic equilibria, then there exists a function g(x, y, ~) e C2([0, 1] • R • R; R) satisfying the conditions (C1) and (C2), such that Sg (t) has only hyperbolic equilibria and that the global attractors Arf and Ag~r are topologically equivalent.
The case of periodic boundary conditions. If we allow periodic boundary conditions for the equation in (4.35), then the structure of the flow can be different. Let us consider the equation ut = Uxx + f (x, u, ux),
Vx e S 1,
(4.49)
where f ' [ 0 , 1] x ]R 2 ~ IR is a C2-function satisfying the conditions (C1). Equation (4.49) defines a local continuous semigroup S(t) on X l = H2(S 1) and, if moreover the condition (C2) holds, S(t) admits a compact global attractor A. In the case of separated boundary conditions, we have seen in Proposition 4.19 that the co-limit set of any ~0 e X l is a singleton. Here we may have closed orbits, as it can be shown in some explicit examples (see [4]). Furthermore, Fiedler and Mallet-Paret [65] have proved the following somehow surprising generalization of the classical Poincar6-Bendixson theorem. THEOREM 4.30. lf the conditions (C1) and (C2) hold, then the co-limit set of any ~o e X 1 satisfies exactly one of the following alternatives: (i) either co(q)) consists in precisely one periodic orbit of minimal period p > O, or (ii) ot(~) C s and co(!lr) C St}, for any ~t 9 co({p). The alternative (ii) means that coop) consists of equilibria and connecting (homoclinic or heteroclinic) orbits. Again, the main tool in the proof of Theorem 4.30 is the zero number (for more details, see [65] and also [183]). If f is independent of x, all periodic orbits are actually rotating waves, i.e., solutions of the form u -- u(x - ct). Independently, Massatt [153] had proved that, in this particular case, either co(~0) is a single rotating wave or a set of equilibria which differ only by shifting x. Matano [155] had shown that, if f ( y , ~ ) = f ( y , - ~ ) , then coop) is a single equilibrium point. In the case where f is analytic in his arguments, Angenent and Fiedler [4] had proved before that, if 7t e co(~0), then co(r and ot(gt) contain a periodic orbit or an equilibrium point and that every periodic orbit is a rotating wave. Furthermore, heteroclinic orbits between rotating waves are constructed in [4].
4.5. A damped hyperbolic equation We now illustrate some of the additional difficulties encountered when one considers partial differential equations which do not smooth in finite time but are still dissipative and have a global attractor. As a model we choose the linearly damped wave equation, which
959
Global attractors in partial differential equations
arises as mathematical model in biology and in physics [213]. The equation with nonlinearity f ( u ) = sinu is called Sine-Gordon equation and is used to model the dynamics of a Josephson junction driven by a current source. The equation with non linearity f (u) = [u 1~u arises in relativistic quantum mechanics.
The equation with constant positive damping.
We begin the analysis with the following
equation with constant positive damping: O2U at 2
Ou
(x t) + y ,
(x t) -- Au(x t) + f (u(x t)) + g(x), -5-[
'
,
u(x,t)
t>0,
x e aS2, t > 0, (4.50)
= o,
u ( x , O) -
xEI2,
,
uo(x),
Ou
- z - ( x , O) - v o ( x ) ,
Ot
xEf2,
where V is a positive constant, S2 is a bounded domain in R n, with Lipschitzian boundary. We assume that g belongs to L 2 ( ~ ) and that f : I K -+ IK is a locally Lipschitz continuous function satisfying the assumption (A.1). As in Section 4.3, we introduce the operator A - --AD with domain 79(A) -- {v E H~(I2) [ - A v E L2(S-2)} and the mapping f ' v E V ~-+ f ( v ) ( x ) E LZ(f2) . We write (4.50) as a system of first order dU t- d( t )
-- BU(t) + f*(U(t)) + G,
t > O, U(O) -- Uo,
(4.51)
where 0 B--
-A
-Yl )
f,(U)_( '
0 f(u)
) '
and introduce the Hilbert space X -- V x L2 (;2) -- H i (;2) x L2 (~), equipped with the norm[[U[[ 2 - [ [ V u [ I 2L2 +[Jut[[ 2L2. Since the operator B0 9(u, v) E 79(Bo) w+ (v , -- Au) 6 X is a skew-adjoint operator on X, where 7 9 ( B o ) - 7 9 ( B ) - 79(A) • H~(I2) and that f * :X --+ X is Lipschitz continuous on the bounded sets of X, for any r > 0, there exists T =_ T(r) > 0 such that, for any U0 6 X, with IlU0llx ~< r, Equation (4.51) has a unique mild (or integral) solution U E C ~ T], X). If moreover U0 E 79(B), then U E C o ( [ - T, T], 79(B)) M C l ( [ - T, T], X) is a classical solution of (4.5 1). We now introduce the functional 4) 6 C~ R) defined by
|
- .((.,
- fs~ ( lv2 (x) + 1
\ lv.(x)l 2 - F(u(x)) - g ( x ) u ( x ) ) dx. /
(4.52)
G. Raugel
960
One easily shows that, if U 6 C~ and
d *(U)- -y dt
T], X) is a solution of (4.51), ~(U(t)) ~ CJ([0, T])
I II2c2'
'v't E [0, T],
(4.53)
which implies that q~ is a strict Lyapunov function for (4.51). One remarks that the set of equilibria $H of (4.51) is given by E'H=ocp • { 0 } - - { ( u , 0 ) ~ X l A D u + f ( u ) + g - - 0 } . If f and F satisfy the assumptions (A.1) and (A.3), we deduce from the property (4.53), by arguing as in (4.33), that all the solutions of (4.51) are global and the orbits of bounded sets are bounded. If, for any U0 E X, we let S(t)Uo denote the solution U(t) of (4.51), we have defined a continuous gradient system S(t) on X. In addition, the mapping (t, U0) w-> S(t)Uo belongs to C~ +oo) x X, X). In 1979, Webb has proved that each positive orbit V+ (U0) is relatively compact in X, by using the variation of constants formula and the arguments leading to the proof of Remark 2.33 (see [207,208]). Actually, under the assumptions (A. 1) and (A.3), the semigroup S(t) has a compact global attractor ,4. In the non critical case, that is, under the additional assumption (n - 2)or < 2 when n ~> 3, the existence of a compact global attractor has been proved, in 1985, independently by Hale [93] and Haraux [ 117] (see also [85]). In his proof, Hale showed that the assumptions of Remark 2.33 are satisfied, whereas Haraux proved that the complete orbits belong to a more regular space than X, when the domain I2 is more regular. One notices that the proof of Hale does not require regularity of the domain and also works for more general operators than the Laplacian, with less regular coefficients. In the critical case (n - 2)or -- 2 when n ~> 3, the existence of a compact global attractor has been first given by Babin and Vishik [ 13] under an additional assumption on f and later by Arrieta et al. [5] in the general case. Another proof using functionals has been outlined by Ball [ 14]. Here, we give a sketch of these proofs and explain their comparative advantages. We begin with two preliminaries remarks. From the assumption (A. 1), we at once deduce that
II/r
,.
ot+l
co(ll, ll , + llull,,,~ + II/r I,.,).
(4.54)
If t"2 is a bounded domain in IR" with Lipschitzian boundary, the embedding from H I(,Q) into L2(~+l)(,f2) is compact, if n = 1, 2 or if n ~> 3 and (n - 2)or < 2, which implies that f * :X ~ X is a compact map. We also recall that the operator B is the infinitesimal generator of a linear C~ e t 8. Using adequate functionals as below or spectral arguments, one shows that there exist positive constants c l and c2 such that
]e'sllL~X,X) ~ c~ e x p ( - c 2 v t ) ,
t ~ 0.
The next theorem of existence of a compact global attractor is fundamental.
(4.55)
Global attractors in partial differential equations
961
THEOREM 4.3 1. Assume that the assumptions (A.1) and (A.3) hold. Then, the semigroup S(t) generated by (4.51) has a compact connected global attractor A c X, given by A =
w"(Cp x {0}).
Furthermore, if the domain U2 is either convex or of class C 1"1, and if the additional assumption (A.2) holds, then fit is compact in X2 = (H2(~2) N V) x V and is the global attractor of S(t) restricted to X2. PROOF. Since S(t) is a gradient system, whose set gH of equilibrium points is bounded in X, and since the orbits of bounded sets are bounded, we may deduce the existence of a compact global attractor from Theorem 4.6, as soon as we have shown that S(t) is asymptotically smooth. We shall present three different proofs of this property. (1) Since S(t)Uo = U(t) is a mild solution of (4.51), one can write
S(t)Uo --etBUo +
e ( ' - s ) B ( f * ( S ( s ) U o ) + G)ds.
(4.56)
If n -- 1, 2 or if n ~> 3 and (n - 2)or < 2, we deduce from (4.54), (4.55) and (4.56) that S(t) satisfies the hypotheses of Remark 2.33. Hence S(t) is asymptotically smooth and has a compact global attractor A in X. As f is actually a bounded map from V into H ~($2), for some positive s, one can either use a "bootstrap" argument (see [ 117]) or apply Theorem 3.18 or Theorem 3.20 to show that, under the additional smoothness assumptions on f and 12, the global attractor A is bounded in X2. (2) In the case n ~> 3 and (n - 2)or -- 2, the mapping f * ' X --+ X is no longer compact and a more complicated argument is needed. We first present the functional argument of [14]. To this end, we introduce the space Y -- L2(S2) x V'. One easily checks that S(t) is continuous on the bounded subsets of X for the topology of Y and that, for any bounded set/3 c X, the orbit 9/+ (13) is relatively compact in Y. We set, for any U c X,
Eo(U) --
/.(',
-~u + uv
)
dx, (4.57)
F ( U ) -- v E o ( U ) + 2 r
= Ilvll2C' + IlVull~, + ~-0(U).
A simple computation and a density argument imply
dF(U(t)) dt
@ y.~(U(t))
(4.58)
-- .)E"l (U (t )),
where U (t) - S (t) U0 and
F I ( U ) --
fa( --2•
+ y
v + g f (u)u - 2g F(u) - ggu
)
dx.
(4.59)
Integrating (4.58), we obtain the equality (2.21) of Proposition 2.35. Clearly, the functionals .To and .TI are bounded on the bounded sets of X and continuous on the bounded sets of X for the topology of Y, not only in the case (n - 2)or <~ 2, but also in the case (n - 2)or < 4.
962
G. Raugel
Indeed, the continuity of these functionals is proved by showing, that, if Un converges to u in L2 (S'2) and is bounded in V, then terms like fs2 lUn]~+l] un - u l dx converge to 0. Using the classical Sobolev inequalities, one gets
/,/J dx ~< Ilu~ jj~+l L2~/~.-z~Ilu--
fs21Unl=+llu.
~<
UnllLq (4.60)
IIun I1%+111 u - un IIm,
where 2 ~< q < 2n/(n - 2) and 0 < s < 1, which implies that fs2 ]u,, [ot+l ]Un - u] dx converges to 0. By Proposition 2.35, S(t) is asymptotically smooth and hence has a global attractor .A in X. The boundedness of ,4 in X2 is a consequence of [ 113]. (3) Unfortunately the previous proof can hardly be generalized to the cases where the damping term yut is replaced by y(x)h(ut) with y(x) ~> 0 and h(.) a nonlinear adequate function. In the critical case, the splitting (4.56) of S(t), which is just the linear variation of constants formula does not directly imply that S(t) is asymptotically smooth. Thus, as in [ 13] and in [5], we introduce another type of splitting, which relies on a non linear variation of constants formula (see [5] for further details). In [5], it was first remarked that, if f satisfies the conditions (A.1), (A.2) with tYl = 1 and (A.3), then f can be written as a sum f = fl + f2 where fl et f2 are two functions in C 1(R, R) with locally Lipschitzian derivatives and satisfy
Ifl (Yl) - f[ (Y2)I Cl (1 -'F" lyll r + ly21~')lyl - y2l ~ ,
f[O) =0, If2(y)l <. c2,
My1, Y2 E ~, (4.61)
If~(y)l <~c2,
Yfl (Y) <~lzlY 2,
VyEN, V y E R , #1 < k l ,
where c l, c2 are positive constants. We now introduce the continuous semigroup St (t) : U0 E X ~ UI (t) e X defined by the equation
dU1 ~(t)dt
BUI(t) + f { ( U l ( t ) ) ,
t > 0, Ul (0) = U0,
(4.62)
where f{(U1) - (0, fl (ul)). The mapping Sl (t) is asymptotically contracting, that is, for any r > 0, there exist positive numbers kl (r) and kz(r) such that, if IlU01lx <~ r, we have, for any t ~> 0,
IIs, (t)Uo IIx
k, (r)exp(-k2(r)t).
(4.63)
The property (4.63) is easily proved by using the functional El (U) -- y Eo(U) + 2q~l (U), where ~1 is nothing else as the functional ~ , in which F(u(x)) + g(x)u(x) has been replaced by FI (u(x)) = fo (x) fl (s)ds. Indeed, using the properties (4.61) of fl, one shows
Global attractors in partial differential equations
963
that
f/v,(,)ll 2x <~ E,(U,(t))<<. (1 +C(r))llu,<,)l 2X' d El (l) ~ - v l y IIVl (t)] 2
dt
x'
where vl = min(1/2, 1 - #1/~1). From these inequalities, we easily deduce (4.63). We next consider the solution Uz(t) = (uz(t), uzt(t)) = S2(t)Uo of the following equation
d2u2
du2
Aou2(t) + f2(u(t)) + g + f , ( u ( t ) ) - f l ( u , ( t ) ) ,
dt 2 ( t ) + V - d T ( t ) -
U2(0) = 0 ,
du2 dt
~(0)
t>0, (4.64)
=0,
Since S(t)Uo = SI (t)Uo + S2(t)Uo and that the semigroup Sl (t) is asymptotically contracting, S(t) will be asymptotically smooth, if we show that, for any bounded set B in X, the set {S2(t)Uo I Uo E B} is relatively compact in X, for t > 0. Classical energy estimates arguments show that there exists 0, with 1/2 < 0 < 1, such that, for t ~> 0,
(t)
A 1/2-~
dt
+ ]A'-~
<~k3(r),
(4.65)
L2
where k3(r) is a positive constant depending only on r and B C Bx(O, r) (for more details, see [13, Chapter 2, Section 6], [5] or [77]). This estimate implies in particular that $2 (t) is a compact mapping for t > 0. It then follows from Theorem 2.31 that S(t) is asymptotically smooth and that S(t) admits a compact global attractor in X. The estimate (4.65), which is independent of t > 0, as well that the invariance of A imply that A is actually bounded in H 2-0 (a,,-2) • H l-~ (s Finally a "bootstrap" argument shows that A is bounded in X2. (4) Under the additional smoothness hypotheses, the semigroup S(t)lx~ is also bounded dissipative in X2 and asymptotically smooth in X2 (for a detailed proof see [ 104] or [ 136]). The asymptotic smoothness of S(t) in X2 is proved like in (1). Indeed, for (n - 2)or = 2, f * : X 2 ~ X2 is a compact map. Thus, S(t) has a compact global attractor A2 in X2. Obviously, A2 c A. On the other hand, A2 attracts the bounded invariant set A and thus, A c A2. The theorem is proved. E3 _
REMARK 4.32. In the part (2) of the proof of Theorem 4.31, we have seen that the critical exponent for the energy estimates is actually given by (n - 2)oe = 4. Unfortunately, local existence of solutions of Equation (4.51), when s is a bounded domain, is not known if 2 < (n - 2)oe < 4. However, due to the Strichartz inequalities, local existence of solutions of the wave equation in the whole space R" is known, if 2 < (n - 2)oe < 4. To obtain local existence of solutions of (4.50), one can use these techniques, if one is able to extend Equation (4.50) to an equation on the whole space in an appropriate way. This can be done in the case of Neumann boundary conditions for special domains and in the case of periodic boundary conditions if s = (0, L)", L > 0, for example.
964
G. Raugel
Here let us consider only the case of periodic boundary conditions, when X2 - (0, L) n, n ~> 3 and assume that the inequality (4.26), with 2 < (n - 2)or < 4, and the property (4.29), with/z < 0, hold. Extending the solutions of (4.50) to the whole space I~n and using the Strichartz inequalities allow to show global existence and uniqueness of the solutions of (4.50) as well as the continuity of the mapping (t, U0) ~ [0, +oo) x X w-~ S(t)Uo E X. One also shows that S(t) is continuous on the bounded sets of X for the topology of Y. Like in the case (n - 2)or <~ 2, S(t) is a gradient system with Lyapunov functional 4~. Actually, the functional introduced in (4.57) allows to prove that S(t) is bounded dissipative in X. The same functional argument as in the part (2) of the proof of Theorem 4.31 implies that S(t) has a compact global attractor r in X (for further details, see [129,59,77,189]; see also [ 146] for earlier results). In the case when f satisfies the additional condition (A.2), Kapitanski [ 129] had proved the existence of the compact global attractor ,A in X by using a splitting method similar to the one used above and by showing that ,A is bounded in a more regular space than X. A "bootstrap" argument finally implies that ,A is bounded in X2. In the case of S2 = IR'7, the existence of a compact global attractor in the case 2 < (n - 2)or < 4 had been proved by Feireisl [59]. Properties o f the compact attractor A. We have seen in Section 3.3 that, under additional conditions, the restriction of the flow to the compact global attractor r is a more regular function of the time variable. In the case of the damped wave equation, Theorems 3.18 and 3.19 apply. Indeed, (H1), (H2) and (H3) are easily proved and the boundedness of A in H s+l (12) x H s (12) for some s > 0 implies the compactness condition (H5). From Theorem 3.20, we deduce that the elements in A are more regular functions of the spatial variable x (see [113] for further details). We recall that, in the case of a smooth domain, such regularity results had been proved by Ghidaglia and Temam, when (n - 2)a < 2. Gevrey regularity results for the orbits contained in A in the case of periodic boundary conditions are also given in [ 113]. For sake of simplicity, we assume in the next theorem that n = 1, 2, 3 (for details, see [ 113]). THEOREM 4.33. Assume that the conditions (A.1), (A.2) and (A.3) hold and that ~2 is a bounded domain of class C 1' l in IRn, n = 1, 2, 3. (1) l f f ~ Cku(IR, IR), k ~ 1, and f(k) is locally HOlder continuous (respectively f "IR --+ IR is a real analytic function), then, f o r any Uo ~ A, t ~ IR w+ S(t)Uo E X is of class C k (respectively analytic). (2) If moreover X2 is of class C k - l ' l and g E Hk-l(S-2), then r is bounded in (Hk+l(S'2) N V) x Hk(I2). Since the assumptions (H1), (H2), (H3) and (H4) hold, Theorem 3.17 implies that Equation (4.51) has a finite number of determining modes. Generalizing the results of [45] to the non compact case, Ghidaglia and Temam have shown that, under the hypotheses (A.1), with (n - 2)or < 2, (A.2) and (A.3), the global attractor A of (4.51) has finite fractal dimension. If (n - 2)c~ = 2 or if 2 < (n - 2)c~ < 4 in the periodic case, the same type of proof shows that A has finite fractal dimension. In what follows, we assume that the three assumptions (A. 1), (A.2) and (A.3) hold.
Global attractors in partial differential equations
965
We remark that (e, 0) is a hyperbolic equilibrium point of (4.51) if and only if e is a hyperbolic equilibrium point of (4.27) and that ind((e, 0)) = ind(e). Indeed, if lj(e), j >~ 1, denote the eigenvalues of the operator - A + D f (e), then the eigenvalues of the operator B + (Df(e))* ~ L(X, X) are given by +
1
if V 2 + 41j(e) ) 0, (4.66)
#j
-~ ( - ? , -i- i
2
if g 2 +
41j (e)
< 0.
Thus, if all the equilibrium points (e, 0) are hyperbolic, Theorem 4.6 and Theorem 4.8 imply that
A - U w~'(ej ), ej cEp
and the Hausdorff dimension dim11 (A) is equal to maxeege ind(e). Moreover, the unstable and stable manifolds W ~ ((e, 0), S(t)) and W ~((e, 0), S(t)) are embedded C l-submanifolds of X of dimension ind(e) and codimension ind(e), respectively. In general, one does not know if the stable and unstable manifolds intersect transversally, even when X-2 is an interval of R. As in the parabolic case, we can replace the function f (.) by a function f (x, .) depending on the spatial variable x ~ s If one assumes that the conditions (A.1), (A.2) and (A.3) hold uniformly in x, then the semigroup S(t) still admits a compact global attractor A. Generalizing the proof of [25] to the damped wave equation, one shows that (4.51) is a Morse-Smale system, generically in the pair of parameters (V, f ( x , .)) (see [27]). Unfortunately, even in the one-dimensional case, the orbit structure on A is not really known. Indeed, unlike the parabolic case, arguments using the zero number are not applicable. At this time, no good tools seem to be available. Till now, we do not know, for instance, if A can be written as a graph, nor if it is contained in a Lipschitzian manifold. Moreover, one does not expect that the stable and unstable manifolds intersect transversally for all the values of g. However, if s C IR, we deduce from the Sturm Liouville theorem and from (4.66) that all the eigenvalues of B + ( D f ( e ) ) * , e c gp, are simple, which, together with Remarks 4.3, implies that all the orbits of (4.51) are convergent [108]. Furthermore, if n = 1 and f ( u ) = #2(au - bu 3) for example, the bifurcation diagram for the global attractor Au with respect to the parameter # is essentially the same as the one given by Chafee-Infante for the corresponding parabolic equation [208]. In the case n ~> 1, all the orbits of (4.51) are still convergent if f : R --+ IR is an analytic function [ 120]. However, one expects that the flow on the global attractor A for g > 0 very large is equivalent to the flow on the global attractor of the corresponding parabolic equation, when this system is Morse-Smale. This is the case, indeed. To prove it, it is easier to consider the rescaled wave equation
02u e Ou ~ Ot2 (x,t) + - ~ ( x , t )
-
A u e ( x , t ) + f ( u e ( x , t ) ) + g(x),
u ~(x, t) = O,
x E ~2, t > 0 , x ~ 0s
Ou C u ~(x, O) -
uo(x),
-7-(x,
ot
O) - v o ( x ) ,
XE,Q,
t > 0, (4.67)
966
G. Raugel
where e = ],,-2 > 0. The formal limit of (4.67) is the parabolic equation (4.25). Hereafter, we denote by Se(t) the continuous semigroup generated on X by (4.67) and by ,Ae the global attractor of Se (t). We let Sp (t) be the semigroup on V, defined by Equation (4.25) and denote by Ap the global attractor of Sp (t) on V. To compare the attractors JtE and A e , we introduce the set A0 - {(u, v) e X l u e A p , v - A u + f (u) + g}, which is bounded in X. If S-2 is either convex or of class C 1, l, A0 is also bounded in X2. Beginning with the papers of Zlamal [215,216] on the telegrapher's equation eutt .q_ U et Uxxe -- O, the dependence in e of (4.67) has been extensively analysed (see [11,104,106,167 130,211,188]). At first glance, (4.67) appears as a singular perturbation of Equation (4.25). Actually, it is not the case, if we compare adequate time-r maps instead of comparing the continuous semigroups. For any (u0, v0) ~ X, we write the solution u e (t) of (4.67) as u e - - u e 1 +- U2' e w h e r e u e1a n d Ue2 are the solutions of 8U ltt -~- U ~It -+- AUel -- O,
(o),
- (o, vo),
(u ~2 (0), U~,) (0) -- (U0, 0).
8u e e Au~ 2tt + u2t + -- f ( ue) + g
(4.68)
Using a priori estimates on (u ~1, Ult)(t) and comparing (u~, u~t)(t ) with (u(t), ut(t)), where u(t) - Sp(t)uo, we obtain the following result [189,164]: LEMMA 4.34. There exist a positive constant eo and, f o r any r > O, a positive number C(r), such that, f o r 0 ~ e <~ eo and, f o r any (uo, vo) E X, satisfying Ilu0ll21(s2) + ellv0[[ 2L2(s2) ~< r 2, we have, f o r t ~ O,
d (tu~ (t) - tu(t)) L2
<<,C(r)e2(1 + II(u0,
+ IIt(u
-
II2H 1
vo)ll2)expC(r)t,
where u ~ and u are the solutions o f (4.67) and (4.25), respectively.
Similar estimates hold for the linearized semigroups DSe(t) and DSo(t). Lemma 4.34 leads to define the semigroup So(t) on X by
So(t)(uo, vO)-
Sp(t)uo,
d (Sp ( t ) u o ) ) , -dt
t>0, (4.69)
S 0 ( 0 ) ( u 0 , v0) - (u0, v0).
Due to the smoothing properties of parabolic equation (4.25), So e C~ +cx~) • X, X) and, for t >~ O, So(t) e C 1(X, X). Clearly, So(t) is a gradient system with Lyapunov functional 9 o(u, v) - fs2(1/21Al/2u12 - F ( u ) - g ( x ) u ) d x and A0 is the global attractor of So(t). For r > 0 a fixed number, we introduce the Cl-mapping Se - Se(r) from X into X, for e ~> 0. Lemma 4.34 and its analogue for the linearized semigroups DSe(t) and DSo(t)
967
Global attractors in partial differential equations
imply that Sc converges to So in a C l-sense, when e goes to 0. In particular, there exists a positive constant Co(r, r) depending only on r and r so that, if II(u0, v0)II x ~< r,
I
So(.o.
LI + IIDS (.o.
<~ Co(r, r)e 1/2.
DSo(.o.
o)1, (4.70)
As a direct consequence of (4.70), Theorems 4.11 and 4.12, we obtain the next result. THEOREM 4.3 5. (i) The global attractors rite are upper semicontinuous at e --O. If all the equilibrium points e o f (4.25) are hyperbolic, the global attractors r are lower semicontinuous at e - 0 and there exist positive constants el, C and x ~ 1/2, such that, f o r 0 <~e <~ 81,
8 x ( A o , A~) + ,~x(A~, Ao) <~ Ce ~.
(ii) Assume that the continuous semigroup Sp (t) is a Morse-Smale system. Then, there exist positive numbers r and 82, such that, f o r 0 <<,e <~ 82, Se (t) is a Morse-Smale system and there is a homeomorphism he :A0 --+ r satisfying the conjugacy condition he o So(r) -- S~(r) o he, f o r any e > O. This example illustrates well the relevance of replacing the comparison of continuous semigroups by the one of maps. For more details, we refer to [188,189,164]. Assertions (i) and (ii) had been proved earlier in [104,106] respectively (see also [130]). Using the assertion (ii) in the case n = 1, we obtain, for c small enough, the same orbit structure on Ae as in the parabolic case. In the case n = 1, Mora and Solh-Morales [ 167] had proved that, for e small enough, the semigroup Se (t) admits an inertial manifold and that this inertial manifold converges, in a Cl-sense, to the one of So(t), when e goes to 0, which reduces the comparison of (4.67) and (4.25) to a finite-dimensional perturbation problem. This result can also be deduced directly from the general theorems of C l-dependence of inertial manifolds with respect to parameters. REMARKS 4.36. (1) All the above assertions (except Remark 4.32) remain true if we replace the homogeneous Dirichlet boundary conditions in (4.50) by homogeneous Neumann boundary conditions. In this case, V = H 1(S-2). With some small changes, these assertions also hold even if we consider more general boundary conditions. (2) If we replace Equation (4.50) by a system of m damped wave equations, that is not necessarily gradient, one can still show the existence of a compact global attractor in (V • L2($2)) m, under adequate dissipative hypotheses on the non linearity f . In this case, one shows directly that the associated semigroup is bounded dissipative (see [94,202]). du du (3) If, in (4.50), one replaces ?' 77 by v ( - A + / d ) 77, one obtains the so called strongly damped wave equation. The linear operator B : (u, v) ~ D ( A ) • D ( A ) ~ (v, - A u - Av) E X generates an analytic semigroup on X. Under the conditions (A. 1) and (A.3), one shows
G. Raugel
968
that the corresponding nonlinear semigroup S(t) can be written in the form (2.16) and has a compact global attractor in X (see [209,69,152,94]). In the one-dimensional case, all the orbits are convergent [ 108].
The equation with a variable non negative damping. One can now wonder what happens if the damping term V ut in (4.50) is replaced by a function h(ut) or more generally by y(x)h(ut), where V(x) is a non negative function on the spatial variable x. The existence of a compact global attractor .A in X has been proved by Ceron and Lopes [29], under the assumptions (A. 1), with (n - 2)or < 2, and (A.3), in the case where V > 0 is a constant and h ~ C I(R, It~) satisfies h(O)=O,
O
u
(4.71)
where a, b are positive constants. In their proof, they introduced the criterium (2.20) of asymptotic smoothness and applied Proposition 2.34 (for a generalization to the case V(x)h(ut) and (n - 2)or ~< 2, see [61]). We now try to present the difficulties encountered when the positive constant y is replaced by a nonnegative function y(x) 6 C l (~-, [0, +c~)), which is not identically zero on the closure of $2. Assume that the conditions (A.1) and (A.3) are satisfied. In this case, global existence and uniqueness of mild solutions of (4.51) still hold. As before, we denote by S(t) the associated semigroup on X. If q~ is the functional introduced in (4.52) and U E C~ T], X) is a solution of (4.51), the equality (4.53) becomes
- - ~ ( U ( t ) ) -- dt
'
f.
V(X)lU,(t x)l 2 dx ' '
u 6 [0, T],
(4.72)
which implies, as in the case of a constant positive damping, that the orbits of bounded sets are bounded. Unfortunately, without additional conditions on V (x), we cannot deduce from (4.72) that q~ is a strict Lyapunov functional. As a direct consequence of Proposition 2.39, Theorem 2.26 and the part 1 of the proof of Theorem 4.31, we obtain the following result. PROPOSITION 4.37. Assume that the hypotheses (A.1), (A.3) and (n - 2)or < 2 hold. If we suppose that there exist positive constants K and 0 such that IIeB'
Ke-~
Vt >~O,
(4.73)
then the semigroup S(t) is asymptotically smooth and has a minimal global B-attractor A x . If in addition, S(t) is point dissipative, then A x is the compact global attractor of S(t). To apply Proposition 4.37, one must first obtain conditions that will imply that the linear semigroup e Bt satisfies (4.73). If y(xo) > 0 at some point x0 ~ S2, then each solution e Bt Uo approaches zero as t --+ +cx~ (see [126,50]). However, as remarked by Dafermos [50], one can construct examples, for n ~> 2, where the approach to zero is not uniform with respect to initial data in a ball and so (4.73) is not satisfied. Using geometric optics arguments,
969
Global attractors in partial differential equations
Bardos et al. [17] have shown that, if I2 and g are of class C ~ the property (4.73) holds if the following condition is satisfied: (BLR) There exists r > 0 such that every ray o f geometric optics intersects the set s(g) x (0, r), where s(V) is the support o f g. The condition (BLR) is true in particular if s(y) is a neighbourhood of 0S-2. Condition (BLR) gives a very interesting way to verify (4.73). However, the question of characterizing, for a particular domain, the minimal conditions on the damping y for which (BLR) holds, is not easy. If I C 12
are two intervals of R, (4.74)
g (x) ~> O, Vx 9 f2,
V (x) > O, 'v'x 9 I,
then (4.73) holds. It remains to derive conditions on y(x), which will imply that S(t) is point dissipative. From (4.72), we deduce that, for any U0 9 X, co(U0) must be a subset of the bounded complete orbits of the system ut,(x, t) -- A u ( x , t) -- f (u(x, t)) + g ( x ) ,
x
9
I2\s(y), t
9
Ut(X, t) = O,
x E s(y), t EIR,
u ( x , t ) = O,
x E OS-2, t E IR.
R, (4.75)
We now distinguish the cases n = 1 and n ~> 2. If n = 1, using the classical representation formula of the solution of a wave equation, we show that, if the condition (4.74) holds, any bounded complete orbit of (4.75) is an equilibrium point of (4.51). Thus, in this case, Proposition 4.37 implies that (4.51) has a compact global attractor ([ 110]). Moreover, one shows that the orbits of (4.51) are convergent (see [ 110] and [ 187] for further examples of convergence in locally damped wave equations). If n ~> 2, we remark that, for any bounded complete orbit ( u ( t ) , u t ( t ) ) of (4.75), D f ( u ( t ) ) belongs to C~ L"(X2)) and w -- ut 9 C~(R, V') n C~ L2(S-2)) is a solution of the system w , , ( x , t) - A w ( x , t) - D f (u(x, t ) ) w ( x , t) -- O,
x9
t9
(4.76) w ( x , t) = wt (x, t) = O,
x 9 s(v), t 9
If the only solution w(t) 9 C~ (IR, V') O C~ L2(I-2)) of (4.76) is w -- 0, then the co-limit set of any solution of (4.51) is an equilibrium point. We are thus led to the following unique continuation property (u.c.p.): (u.c.p.) Assume that w is a weak L2(s x (0, T)) solution of the equation w t t - A w + b(x,t)w=O in s x (0, T) where T > diamI2 and b 9 L ~ ((O, T), L" (S2)). Then, if w vanishes in some set O x (0, T), O C s w must be identically zero. Ruiz [192] has shown that the (u.c.p.) property holds when O is a neighbourhood of 0S2.
970
G. Raugel
It follows from the above discussion and from Proposition 4.37 that, if OS2 C s(F) and (n - 2)or < 2, then the semigroup S(t) has a compact global attractor in X. Feireisl and Zuazua [61] have generalized this existence result to the critical case (n - 2)or = 2. In their proof, they have used energy functionals arguments to show that S(t) is bounded dissipative and the same splitting as in Part 3 of the proof of Theorem 4.31 to show that S(t) is asymptotically smooth. We thus can state the following result (see also [113]): THEOREM 4.38. Assume that S-2 is a bounded regular domain in ~n. If the hypotheses (A.1), (A.2), (A.3) are satisfied and if either the conditions n = 1 and (4.74), or s(y) is a neighbourhood of 0S-2, then (4.51) has a connected compact global attractor A = W" (s x {0}) in X. Moreover, the time and spatial regularity properties of the complete orbits in A, given in Theorem 4.3.3 still hold and the property of finite number of determining modes remains true. Finally, we note that many other well-known dissipative gradient systems, having a compact global attractor, could have been approached. Among them, we quote the strongly damped wave equation (see Remark 4.36), the Cahn-Hilliard equation (see [202] and the references therein), nonlinear diffusion systems [94] etc.
5. Further topics So far, we have mainly studied equations, which have a gradient structure. The most famous and most studied non gradient dissipative system arising in PDE's is certainly the one generated by the Navier-Stokes equations on a bounded domain in space dimension two or three. As already shown by Ladyzenskaya in 1972 [ 133,134], in space dimension two, this equation has a compact global attractor, which is of finite fractal dimension [ 147,74,135]. The associated semi-flow is a smooth function of the time variable for t > 0 (up to analyticity) and the global attractor is composed of smooth functions in the spatial variable (see [74,75,62,202] and the references therein). Estimates of the fractal and Hausdorff dimensions of the attractor in terms of various physical parameters have been extensively studied (see [8,44,45,137,55,128]). In the two-dimensional case, the Navier-Stokes equations have also the property of finite number of determining modes (see [72,133,138,41]). For further details and study on the Navier-Stokes equations, we refer to [ 18] in this volume. Among the well-known evolutionary partial differential equations, which have smoothing properties in finite positive time and admit a compact global attractor, we should also mention the one-dimensional Kuramoto-Sivashinsky equation (see [169,43] for example), modeling pattern formation in thermohydraulics and also the propagation of a front flame, as well as the complex Ginzburg-Landau equation in space dimensions one or two, describing the finite amplitude evolution of instability waves. The complex Ginzburg-Landau equation is actually a strongly damped Schr6dinger equation; in space dimensions one or two, it admits a compact global attractor of finite fractal dimension [83,202]. To conclude this paper, we present the weakly damped Schr6dinger equation, which is a system generated by a dispersive equation with weak damping.
Global attractors in partial differential equations
971
A weakly damped Schr6dinger equation. In what follows, s denotes, either the whole space IR", n - 1, 2 or 3, or a bounded C2-polygonal domain in R n, when n - 1, 2. For y > 0 a fixed constant, f a function in L 2 (~Q) and g 6 C 1([0, + e c ) , R) a function satisfying the hypotheses (H. 1) and (H.2) below, we consider the weakly damped Schr6dinger equation, which arises in plasma physics or in optical fibers models (see [170], for instance): in s x (0, + o c ) ,
iut + Au + g(lul2)u + igu -- f,
(5.1) in s
u(O) -- uo,
If s 7~ R", we associate homogeneous Dirichlet boundary conditions to (5.1) u -- 0,
on 0s
(5.2)
Of course, we could consider homogeneous Neumann boundary conditions or periodic conditions as well. We assume that g 6 C l ([0, +cx~), R) and G(y) - fo' g(s) ds satisfy the following conditions: (H. 1) there exist two constants Cl > 0 and oeI E [0, 2 / n ) such that
G(y) <~ Cl y(1 -Jr-ya'),
y~>O,
yg(y) -- G(y) <~ Cly(1 + yOt,),
y~>O,
(5.3)
(H.2) in the case when n - 2 or 3, there exist two constants C2 > 0 and oe2 ~> 0, with (n - 2)c~2 < 4, such that, for any (~, ~') E C 2, Ig(]~[2)~ - g(l~']2)~'[ ~< C2(1 + ]s~[a2 +
1 '1 2)1
- ~'1,
(5.4)
(H.3) in the case when s is a bounded subset of I~ 2, there exists a positive constant C3 such that [g'(y)] ~< C3,
y ~> O.
(5.5)
Later we shall also impose the next additional conditions on g, which mainly require that the nonlinearity is subcritical: (H.4) The function g is in C a ([0, + e c ) , R) and there exist two constants C4 > 0 and O~l E (0, 2 / n ) such that
Ylg'(Y)I + [g(Y)l ~< C4y~"
Vy >/O.
(5.6)
Moreover, for k/> 2, the derivatives g(k) are bounded. As before, we denote by A -- - - A B c u the unbounded operator on H -- L 2 ( ~ ) , where A s c is the Laplace operator with the corresponding boundary conditions. We set V 2 = D ( A ) , V - D ( A 1/2) and we denote by V' the dual space of V.
G. Raugel
972
PROPOSITION 5.1. Under the assumptions (H.1), (H.2) and (H.3), for any uo ~ V, there exists a unique solution u(t) E C~ +oo), V) of (5.1) and (5.2). Moreover, u(t) C l ([0, +oo), V') and the mapping S(t)uo -- u(t) defines a continuous semigroup on V. Ifuo E V 2, then u(t) belongs to C~ +co), V 2) n C 1([0, +co), H). Furthermore, for any t ~ O, the mapping S(t) is continuous on the bounded sets of V for the topology of H. PROOF. In the case where S2 = IR", mutatis mutandis, we can follow the proofs of [28, Theorem 7.4.1 and Proposition 7.5.1 ]. These proofs use the well known Strichartz inequalities. The above results are shown in [80], when $2 is a bounded domain in ]R. For ~ a bounded domain in 1R2, the proofs can be found in [1] (see also [2]). The proof of the uniqueness of the solution in V is more delicate than in dimension 1 and requires the hypothesis (H.3). Indeed, in the case of a bounded domain, estimates similar to Strichartz inequalities are not yet known. The continuity of S(t) on the bounded sets of V for the topology of H can be shown by arguing as in [28, Proposition 7.4.2]. [3 Uniqueness of solutions of (5.1) is not known, when S-2 is a bounded domain in IR3. In [80], it was first proved that, if S2 is a bounded domain in 1R, then S(t) has a global weak attractor A1 (respectively A2) in V (respectively D(A)) (see Remark 2.30). Using the functionals given below and applying Proposition 2.35, Abounouh (respectively Lauren~ot) have showed the existence of a compact global attractor ,A in V, when U2 is a bounded domain of R 2 (respectively S2 = JR"). We introduce the functionals 4 , q~0 and q-' defined on V by ~(~)
= IIV~l12 +
(-G(Ivl2)
+ 2Re ( f $ ) ) d x
= IlVvll 2 + q:'o(v), (5.7)
I/1(1)) --
(g(ll)12)ll)l 2 -- G(II)I 2)
Re (f~))dx.
Obviously, due to the hypothesis (H.2), the functionals g'0 and i// are continuous on the bounded sets of V for the topology of H. Taking successively the inner product of (5.1) with K and ~t, one shows (see [2,80,142]) that, for any u0 E HZ(a"2) N V, S(t)uo = u(t) satisfies, for t/> 0, d Ilu(t) II2. a7 d
+
2 2y Ilu
_ 2 fs7 Im (f-a(x, t)) dx,
(5.8)
--*(u(t)) + 2ye(u(t)) - 2ye(u(t)) dt
which implies that, for t ~> 0,
9 (S(t)uo) - e x p ( - 2 y t ) ~ ( u o )
+ 2y
f0t e2•
Vu0 E V. (5.9)
Global attractors in partial differential equations
973
From the hypotheses (H. 1), (H.2) and from the equalities (5.8), one deduces that r > 0 can be chosen so that the ball By (0, r) is positively invariant under S(t) and is an absorbing set for the semigroup S(t). The above properties lead to the following result. THEOREM 5.2. (1) Under the hypotheses (H.1), (H.2) and (H.3), the semigroup S(t) has a connected,
compact global attractor ,4 in V. (2) Suppose that the condition (H.4) holds. If S-2 is either the whole space R n, n -- 1, 2, or a bounded interval of R, the global attractor A of (5.1) and (5.2) is compact in H2($2). Moreover, in the one-dimensional case, for any uo E A, the mapping t E R w-~ S(t)uo is ofclass Ck, for any k ~ 0 (respectively analytic i f g is analytic). l f in addition, f E Hk(l-2), then A is bounded in Hk+2(S2). PROOF. (1) The first statement is a direct consequence of Theorem 2.26, Proposition 2.20, if we show that S(t) is asymptotically smooth in V. Since the mapping S(t), for t ~> 0, and the functionals 450 and q/ are continuous on the bounded sets of V for the topology of H, we apply Proposition 2.35 with X = V, Y -- H, F0 = q~0, F l ( v ) = 2yq/(v) + 2 fs~ Im (fV) dx. If 1-2 is a bounded domain in •", n = 1,2, the condition (ii) of Proposition 2.35 is clearly satisfied, since the Sobolev imbedding HI(S2) ~ L 2 (S2) is compact. When 1-2 = R", n = 1, 2 or 3, the condition (ii) is proved in [142, Lemmas 2 . 6 , 2 . 7 , 2.8], by using a splitting of the solutions like in [58,59]. (2) The first part of the proof does not indicate if the compact global attractor is bounded or compact in a more regular space. When the hypothesis (H.4) holds, the boundedness of A in H2(I2) is shown by Goubet in [87] and [88], by using a splitting of S(t)u into a low wavenumber part Pu(S(t)u) and a high wavenumber part QN(S(t)u). The low wavenumber part is obviously smooth and the high wavenumber part can be approximated asymptotically by the solution of an equation with zero initial data. The compactness in H2(S2) then follows from the fact that A is a bounded invariant set in H2(S-2) and thus in contained in the compact global attractor of S(t) in H 2 ( ~ ) . When S-2 is bounded interval of R, the boundedness of A in Hk+2(I2) is proved in the same way in [87]. The Ck-regularity (respectively the analyticity) of the map t ~ R w+ S(t)uo, for any k ~> 0 and any u0 E A, is shown in [ 113] as a consequence of a generalized version of Theorems 3.18 and 3.19. D REMARKS. (1) In the one-dimensional case, under a relaxed version of the hypothesis (H.4), the system generated by (5.1) and (5.2) has the property of finite number of determining modes (see [ 17 6,77,113 ] ). (2) In the one-dimensional case, the global attractor of Equation (5.1) with periodic boundary conditions is regular in the same Gevrey class as g and f and thus is analytic in the spatial variable (see [176,113]). (3) Goubet [89] has also proved the compactness of the global attractor of Equation (5.1) with periodic boundary conditions in the two-dimensional case. There the proof is more involved and uses spaces introduced by Bourgain.
974
G. Raugel
(4) From [80, Theorem 3.2 and Remark 3.1 ], it follows that in the one-dimensional case, ,t4 has finite fractal dimension. Adapting these proofs, one can certainly show that, in the other cases, r has also finite fractal dimension. Finally, we notice that the existence and regularity of the compact global attractor for other weakly damped dispersive equations like the weakly damped KdV and Zakharov equations are proved by using similar methods (see [91,166,90]).
Acknowledgements I am indebted to Th. Gallay, O. Goubet and E. Titi for their helpful suggestions. I especially express my gratitude to Jack Hale, who introduced me into the field of dynamical systems and over the years did not spare his time to discuss with me and answer my numerous questions. I also thank him for the advices and comments on this manuscript. Finally my thanks go to Bernold Fiedler, who gave me the opportunity to write this text.
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CHAPTER
18
Stability of Travelling Waves Bj6rn Sandstede* Department of Mathematics, Ohio State Universi~, 231 West 18th Avenue, Columbus, OH 43210, USA E-mail: sandstede. 1 @osu.edu
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2. Set-up and examples
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2.1. Set-up
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2.2. Examples
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3. Spectral stability
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3.1. Reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.2. Exponential dichotomies
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3.3. Spectrum and Fredholm properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Fronts, pulses and wave trains
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3.5. Absolute and convective instability
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4. The Evans function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Definition and properties
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4.3. Extension across the essential spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Spectral stability of multi-bump pulses
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4.2. The computation of the Evans function, and applications
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1007 1008 1012 1017
5.1. Spatially-periodic wave trains with long wavelength . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.2. Multi-bump pulses
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5.3. Weak interaction of pulses
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6. Numerical computation of spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1033 1034
6.1. Continuation of travelling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6.2. Computation of spectra of spatially-periodic wave trains . . . . . . . . . . . . . . . . . . . . . . . .
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6.3. Computation of spectra of pulses and fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Nonlinear stability
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8. Equations with additional structure
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9. Modulated, rotating, and travelling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References
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*This work was partially supported by the NSF under grant DMS-9971703. H A N D B O O K OF D Y N A M I C A L S Y S T E M S , VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 983
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Abstract
An overview of various aspects related to the spectral and nonlinear stability of travellingwave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Green's functions) for the linear operator. Among the other topics reviewed in this survey are the nonlinear stability of waves, the stability and interaction of well-separated multi-bump pulses, the numerical computation of spectra, and the Evans function, which is a tool to locate isolated eigenvalues in the point spectrum and near the essential spectrum. Furthermore, methods for the stability of waves in Hamiltonian and monotone equations as well as for singularly perturbed problems are mentioned. Modulated waves, rotating waves on the plane, and travelling waves on cylindrical domains are also discussed briefly.
Stability o f travelling waves
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1. Introduction This survey is devoted to the stability of travelling waves. Travelling waves are solutions to partial differential equations that move with constant speed c while maintaining their shape. In other words, if the solution is written as U (x, t) where x and t denote the spatial and time variable, respectively, then we have U ( x , t) = Q ( x - c t ) for some appropriate function Q(~). Note that c = 0 describes standing waves that do not move at all. In homogeneous media, travelling waves arise as one-parameter families: any translate Q ( x + r - c t ) of the wave Q ( x - c t ) , with r E IR fixed, is also a travelling wave. We can distinguish between various different shapes of travelling waves (see Figure 1): Wave trains are spatially-periodic travelling waves so that Q(~ + L) = Q(~) for all for some L > 0. Homogeneous waves are steady states that do not depend on ~ so that Q(~) = Q0 for all ~. Fronts, backs and pulses are travelling waves that are asymptotically constant, i.e., that converge to homogeneous rest states: l i m ~ • Q(~) = Q+. For fronts and backs, we have Q+ =/= Q_, whereas pulses converge towards the same rest state as --+ 4-oo so that Q+ = Q_. Travelling waves arise in many applied problems. Such waves play an important role in mathematical biology (see, e.g., [121]) where they describe, for instance, the propagation of impulses in nerve fibers. Various different kinds of waves can often be observed in chemical reactions [99,182]; one example are flame fronts that arise in problems in combustion [182]. Another field where waves arise prominently is nonlinear optics (see, e.g., [1 ]): of interest there are models for the transmission and propagation of beams and pulses through optical fibers or waveguides. We refer to [38,43,89] for applications to water waves. Travelling waves also arise as viscous shock profiles in conservation laws that model, for instance, problems in fluid and gas dynamics or magneto-hydrodynamics [ 171 ]. Localized structures in solid mechanics can be modelled by standing waves (see [172174]). We refer to [59] for the existence and stability of patterns on bounded domains. In this article, we focus on the stability of a given travelling wave. That is, we are interested in the fate of solutions whose initial conditions are small perturbations of the travelling wave under consideration. If any such solution stays close to the set of all translates of the travelling wave Q(.) for all positive times, then we say that the travelling wave Q(.) is stable. If there are initial conditions arbitrarily close to the wave such that the associated solutions leave a small neighbourhood of the wave and its translates, then the wave is said to be unstable. In other words, we are interested in orbital stability of travelling waves. There exists an enormous number of different approaches to investigate the stability of waves: which of these is the most appropriate depends, for instance, on whether the
Fig. 1. Travelling waves with various different shapes are plotted: pulses in (a), spatially-periodic wave trains in (b), fronts in (c), and backs in (d). Note that the distinction between fronts and backs is, in general, rather artificial.
B. Sandstede
986
partial differential equation is dissipative or conservative, or whether one can exploit a special structure such as monotonicity or singular perturbations. Given this variety, writing a comprehensive survey is quite difficult: thus, the selection of topics appearing in this survey is a very personal one and, of course, by no means complete. We refer to the recent review [ 184] and to the monograph [ 182] for many other references related to the existence and stability of waves. A natural approach to the study of stability of a given travelling wave Q is to linearize the partial differential equation about the wave. The spectrum of the resulting linear operator s should then provide clues as to the stability of the wave with respect to the full nonlinear equation. As we shall see in Section 3, the spectrum of s is the union of the point spectrum, defined as the set of isolated eigenvalues with finite multiplicity, and its complement, the essential spectrum. Point and essential spectra are also related to Fredholm properties of the operator s - ),. Most of the results presented here are formulated using the first-order operator T()~) that is obtained by casting the eigenvalue operator 13 - )~ as a first-order differential operator. In Section 4, we review the definition and properties of the Evans function, which is a tool to locate and track the point spectrum of s In Section 7, we discuss under what conditions spectral stability of the linearization/2 implies nonlinear stability, i.e., stability of the wave with respect to the full partial differential equation. The stability analysis of a given wave is often facilitated by exploiting the structure of the underlying equation. In Section 8, we provide some pointers to the literature for Hamiltonian and monotone equations as well as for singularly perturbed problems. In many applications, it appears to be difficult to analyze the stability of travelling waves analytically. For this reason, we comment in Section 6 on the numerical computation of the spectra of linearizations about travelling waves. An interesting problem that is relevant for a number of applications is the stability of multi-bump pulses that accompany primary pulses. Recent results in this direction are reviewed in Section 5. Most of the results presented in this survey are also applicable to other waves, for instance, rotating waves such as spiral waves in two space dimensions, modulated waves (waves that are time-periodic in an appropriate moving frame), and travelling waves on cylindrical domains. Some of these extensions are discussed in Section 9.
2. Set-up and examples 2.1. Set-up We consider partial differential equations (PDEs) of the form ut = A ( O x ) u + A ; ( U ) ,
x ~ R, u ~ x .
(2.1)
Here, A(z) is a vector-valued polynomial in z, and 2, is an appropriate Banach space consisting of functions U(x) with x 6 R, so that A(Ox) :,9( __+ ,9( is a closed, densely defined operator. Lastly, A/": 2, --+ 2' denotes a nonlinearity, perhaps not defined on the entire space X, that is defined via pointwise evaluation of U and, possibly, derivatives of U. We refer to [85,133] for more background.
Stability o f travelling w a v e s
987
Travelling waves are solutions to (2.1) of the form U (x, t) = Q ( x - c t ) . Introducing the coordinate ~ - - x - c t , we seek functions U ( ~ , t) - - U ( x - c t , t) that satisfy (2.1). In the (~, t)-coordinates, the PDE (2.1) reads
u, = A(o~)u + cO~U + A;(U),
~ ~ R, U ~ X,
(2.2)
and the travelling wave is then a stationary solution Q (~) that satisfies
o = A(0~)u + cO~ u + A/(u).
(2.3)
The linearization of (2.2) about the steady state Q(~) is given by
u, = A(o~ ) u + cO~ u + OuA;(Q)U.
(2.4)
The right-hand side defines the linear operator s .= A ( ~ ) + cO~ + OuA;(Q). Spectral stability of the wave Q is determined by the spectrum of the operator/2, i.e., by the eigenvalue problem
z u = A(o~)u + cO~U + ~uA;(Q)U = s
(2.5)
that determines whether (2.4) supports solutions of the form U(~, t) = e zt U(r Note that the steady-state equation (2.3) and the eigenvalue problem (2.5) are both ordinary differential equations (ODEs). As such, they can be cast as first-order systems. The steady-state equation, for instance, can be written as !
u =f(u,c),
uE~",
d '------. d~
(2.6)
The travelling wave Q (r corresponds to a bounded solution q (~) of (2.6). The PDE eigenvalue problem (2.5) becomes u'--(O,,f(q(~),c)+~,B)u,
(2.7)
where B is an appropriate n x n matrix that encodes the PDE structure (see Section 2.2 below for examples). An important relation is given by (2.8) which follows from inspecting (2.5). In this survey, we focus on the ODE formulation (2.6) and (2.7). In particular, travelling waves can be sought as bounded solutions of (2.6), and we refer to the textbooks [33,80, 107] for a dynamical-systems approach to constructing such solutions.
988
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2.2. Examples We give a few examples that fit into the framework outlined above. EXAMPLE 1 (Reaction-diffusion systems). Let D be a diagonal N x N matrix with positive entries and F : 1RN --+ IRN be a smooth function. Consider the reaction-diffusion equation Ut = DUxx + F(U),
(2.9)
x E ]~, U E ]1~N ,
posed on the space A" = C uon i f ( ~ , ~ N ) of bounded, uniformly continuous functions. In the moving frame ~ -- x - ct, the system (2.9) is given by Ut = D U ~ + cU~ + F ( U ) ,
(2.10)
~ E ]~, U E ]t~N.
Suppose that U(~, t) = Q(~) is a stationary solution of (2.10) such that DQ~(~)+cQ~(~)+F(Q(~))=O,
(2.11)
~ ER.
The eigenvalue problem associated with the linearization of (2.10) about Q(~) is given by (2.12)
~U = D U ~ + cU~ + Ou F ( Q ) U =:/2U.
This eigenvalue problem can be cast as
V~
=
D -1 ()~U - Ou F ( Q ( ~ ) ) U - c V )
_(
0
id
D -l()~- OuF(O(~)))
- c O -1
)(U)
which we write as u~ = A(~; )~)u = (,4(~) + )~B)u,
u E R n = ~2N,
(2.13)
with u = (U, V) and
_ A(~) -
(
0 _ D _ l Og F ( Q ( ~ ) )
id) - c D -1
'
B=
(o 0) 1 0
"
Bounded solutions to (2.12), namely (/2 - )0U = 0, and (2.13) are then in one-to-one correspondence. In particular, if Q (.) is not a constant function, then )~ = 0 is an eigenvalue of 12 with eigenfunction Q~ (~). This can be seen by taking the derivative of (2.11) with respect to which gives D(Q~)~ +c(Q~)~ + OuF(Q)Q~ =0
Stability of travelling waves
989
so that s = 0 . Hence, u(~) = (Q~ (~), Q ~ (~)) satisfies (2.13) for x = 0 . One important example is the FitzHugh-Nagumo equation (FHN) ut = uxx + f (u) - w, Wt = ~21/3xx + 8(U -- }/tO),
for instance with f (u) = u(1 - u)(u - a). It admits various travelling waves such as pulses, fronts and backs (see, e.g., [91,105,176] for references). The stability of pulses has been studied in [90,185]. Stability results for spatially-periodic wave trains can be found in [53, 156], whereas the stability of concatenated fronts and backs has been studied in [124,147] and [125,154]. Many other results on the stability of waves to reaction-diffusion equations can be found in the literature (see, e.g., [47,60]). One class of such equations that has been studied extensively are monotone systems (see [37,141,182] and Section 8). We also refer to [85, Section 5.4] for instructive examples. EXAMPLE 2 (Phase-sensitive amplification). The dissipative fourth-order equation Ut + (0~ + 2U 2 - ( 2 x - r/2)) (0~ + 2U 2 - ~2)U +4a(3U(O~U) 2 + U20~U) -0
(2.14)
models the transmission of pulses in optical storage loops under phase-sensitive amplification (see [ 106]). This equation with a = 0 admits the explicit solution Q(~) = r/sech(~) for every tc ~> 0. Its stability has been investigated in [6,106]. For a > 0, (2.14) has multibump pulses whose existence and stability has been analyzed in [ 150]. EXAMPLE 3 (Korteweg-de Vries equation). The generalized Korteweg-de Vries equation (KdV) is given by Ut + U~.,-x + U p U,- = 0,
x 6 IR,
where p is a positive parameter. Formulated in the moving frame ~ = x - ct, the generalized Korteweg-de Vries equation reads u, + u~ ~~ - c U~ + u ~'u~ = o,
~ ~ R,
where c denotes the wave speed. This equation admits a family of pulses given by Q ( ~ ) _ [c( p + 1)(p + 2) ] l/p 2
]
sech2J';
2
B. Sandstede
990
for any positive values of c and p. The stability of these solitons was investigated in [ 10], whereas asymptotic stability has been studied in [134,135]. The KdV equation is Hamiltonian for all p > 0 and known to be completely integrable for p = 1, 2. EXAMPLE 4 (Nonlinear Schriidinger equation). The nonlinear Schr6dinger equation (NLS) reads iqot at- qo~ at- 414512~ - 0,
~ ~ IR,
with 4~ 6 C. If we seek solutions of the form qs(~, t) = eic~ equation
iUt -q- U~ - coU -+-4IUI2U : 0,
t), then we obtain the
~ E R,
(2.15)
where U 6 C and co > 0. It is known [ 183] to support stable pulses given by
Q(~)- V~ sech(,f~). The NLS equation is Hamiltonian and, in fact, completely integrable. Of interest is the persistence and stability of these waves upon adding perturbations that model various physical imperfections. An important example is the perturbation to the dissipative complex cubicquintic Ginzburg-Landau equation (CGL) iUt + U~ - c o U + 4lUl2U q- 3otlUl4U
= ie(dl U~ + d2U + d3IUI2U + d4lgl4g)
(2.16)
for small ot 6 1R and e > 0. We refer to [ 177,97,98,1] for various stability and instability results for solitary waves to this equation. Mathematically, the transition from (2.15) to (2.16) is interesting since the perturbation destroys the Hamiltonian nature of (2.15). We refer to [99,121,171,176,182] for problems where reaction-diffusion equations arise naturally. Formal derivations of the KdV, NLS and CGL equation can be found in [ 1] and [38,89] for problems in nonlinear optics and for water waves, respectively.
3. Spectral stability In this section, we review results on the structure of the spectrum of the linearization of a nonlinear PDE about a travelling wave.
Notation. Throughout this survey, we denote the range and the null space of an operator Z; by R(Z;) and N(Z;), respectively. Eigenvalues of operators and matrices are always counted with algebraic multiplicity.
991
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Consider a matrix A ~ C" • We often refer to the eigenvalues of the matrix A as the spatial eigenvalues. We say that A is hyperbolic if all eigenvalues of A have nonzero real part, i.e., if spec(A) AiR = 0. We refer to eigenvalues of A with positive (negative) real part as unstable (stable) eigenvalues. Similarly, the generalized eigenspace of A associated with all eigenvalues with positive (negative) real part is called the unstable (stable) eigenspace of A. The 6-neighbourhood of an element or a subset z of a vector space is denoted by/d~ (z).
3.1. Reformulation As mentioned above, it is often advantageous to write the eigenvalue problem associated with the linearization as a first-order ODE. Therefore, we consider the family T of linear operators defined by 7-(X):7)
~H,
du > ---A(.;X)u,
u~
d~
for X E C. We take either I (R, C" ) , 7) = Cunif
C '' )
or
7 ) - H' (IR, C"),
7-[ = L2(R,
C").
(3.1)
Throughout this survey, we assume that the following hypothesis is met. HYPOTHESIS 3.1. The matrix-valued function A (~; X) ~ C" •
is of the form
A(~'X)--A(~)+XB(~), where A(.) and B(.) are in C~
R'z•
The operators 7-(X) are closed, densely defined operators in 7-/with domain 7). We are interested in the set of X for which TOO is not invertible.
3.2. Exponential dichotomies Spectral properties of T can be classified by using properties of the associated ODE d
---u=A(~;X)u
d~
(3.2)
992
B. Sandstede
with u ~ C n . We denote by 4, (~, () the evolution operator ~ associated with (3.2). Note that q:, (~, () = 4~ (~, (; )~) depends on )~, but we often suppress this dependence in our notation. A particularly useful notion associated with linear ODEs such as (3.2) is exponential dichotomies. Suppose that we consider a linear constant-coefficient equation d
--u
d~
(3.3)
= A ()Ou,
so that A (,k) does not depend on ~. We want to classify solutions to (3.3) according to their asymptotic behaviour as I~l --+ ec. Suppose that the matrix A ()~) is hyperbolic so that the spatial spectrum spec(A (,k)) has no points on the imaginary axis. Consequently, c" -
Eo(2.) G E800 ~
(3.4)
where the two spaces on the right-hand side are the generalized stable and unstable eigenspaces of the matrix A(,k). We denote by P(~(~) the spectral projection of A()0, so that R ( P ~ ( ~ ) ) - E~(~),
N(P~ ( X ) ) - E~(~).
(3.5)
These subspaces are invariant under the evolution 4~(~, () = e A('~)(~-~') of (3.3). Furthermore, solutions u(~) with initial conditions u ( ( ) in E6(~.) decay exponentially for ~ > (, while solutions with initial conditions u ( ( ) in E~)(~.) decay exponentially for ~ < (. We are interested in a similar characterization of solutions to the more general Equation (3.2): DEFINITION 3.1 ( E x p o n e n t i a l d i c h o t o m i e s ) . Let I = N +, IR- or 1R, and fix )~. E C. We say that (3.2), with ~ = ~.. fixed, has an exponential dichotomy on I if constants K > 0 and x s < 0 < Ku exist as well as a family of projections P (~), defined and continuous for E I, such that the following is true for ~, ( E I. 9 With 45s (~, () := 45 (~, () P ((), we have Iqss(~,()]~
~>(,
9 Define qsu(~, () := 45(~, ( ) ( i d - P ( ( ) ) , [ ~ u ( s e , ( ) [ ~ < K e Ku(~-~),
~,(EI. then
se ~ < ( , se , ( E I .
9 The projections commute with the evolution, 45(~, ( ) P ( ( ) = P(~)45(~, (), so that 45s(~, ()uo r R ( P ( ~ ) ) ,
~ ~> ~, s~, ~- E /
qsu(~, ()uo E N ( P ( ~ ) ) ,
~ ~< ~', ~,~" e / .
I I.e., qs(~, ~) = id, 05(~, r)qS(r, () = 45(~, () for all ~, r, ( E IR and u(~) = 45(~, ()u0 satisfies (3.2) for every u0 E Cn .
Stability of travelling waves
993
R(P(~))
N(p(r
~u(r
r
/ Fig. 2. A plot of the stable and unstable spaces associated with an exponential dichotomy. Vectors in the stable space R(P(r are contracted exponentially under the linear evolution 4~s(~, ~') for ~ > ~'. Similarly, vectors in the unstable space N(P (~)) are contracted under the linear evolution 45u(~", ~) for ~ > ~'.
The ~-independent dimension of N ( P ( ~ ) ) is referred to as the Morse index of the exponential dichotomy on I. If (3.2) has exponential dichotomies on R + and on R - , the associated Morse indices are denoted by i+0~,) and i_ (~.,), respectively. Roughly speaking, (3.2) has an exponential dichotomy on an unbounded interval I if each solution to (3.2) on I decays exponentially either in forward time or else in backward time. The set of initial conditions u(~') leading to solutions u(~) that decay for ~ > r with ~, ~" E I, is given by the range R ( P ( r of the projection P(r Similarly, the set of initial conditions u(~') leading to solutions u(~) that decay for ~ < ~', with ~, ~" 6 I, is given by the null space N(P(~')). The spaces R(P(~)) are mapped into each other by the evolution associated with (3.2); this is also true for the spaces N ( P ( ~ ) ) ; see Figure 2 for an illustration. For the constant-coefficient equation (3.3), we have P(~) = P~(~.) due to (3.5). Note that, for constant-coefficient equations, the Morse index of the exponential dichotomy is simply the dimension of the generalized unstable eigenspace. Exponential dichotomies persist under small perturbations of the equation. This result is often referred to as the roughness theorem for exponential dichotomies. If we, for instance, perturb the coefficient matrix A (i.) of the constant-coefficient equation (3.3) by adding a small ~-dependent matrix, we expect that the two subspaces E~(I.) and E~(~.) that appear in (3.4) perturb slightly to two new ~-dependent subspaces that contain all initial conditions that lead to exponentially decaying solutions for forward or backward times. THEOREM 3.1 [36, Chapter 4]. Firstly, let I be R + or R - . C~ C "x") and that the equation d --u=A(~)u d~
Suppose that A(.)
(3.6)
has an exponential dichotomy on I with constants K, K s and x u as in Definition 3.1. There are then positive constants 6, and C such that the following is true. If B(.) ~ C~ C ''•
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B. Sandstede
such that sup~i,l~l~t IB(@)I < ~ / C f o r some 6 < 6, and some L >~ O, then a constant K > 0 exists such that the equation d d U -- (A(~) + B ( ~ ) ) u
(3.7)
has an exponential dichotomy on I with constants K, x s + 6 a n d x u - 8. Moreover, the projections P ( ~ ) and evolutions 45s(~, ~') a n d q~u(~, ~.) associated with (3.7) are 6-close to those a s s o c i a t e d w i t h ( 3 . 6 ) f o r a l l ~, ~ ~ I with I~l, I~l ~ L. Secondly, if l = JR, then the above statement is true with L -- O.
Thus, to get persistence of exponential dichotomies on R + or R - , the coefficient matrices of the perturbed equation need to be close to those of the unperturbed equation only for all sufficiently large values of I~ I. For I = R, the coefficient matrices need to be close for all ~ 6 R to get persistence. Theorem 3.1 can be proved by applying Banach's fixed-point theorem to an appropriate integral equation whose solutions are precisely the evolution operators that appear in Definition 3.1 (see [143,137]). Indeed, suppose that 4~s(~, ~') and 4~u(~, ~') denote exponential dichotomies of (3.6) on I - R +, say. If sup~> 0 IB(~)I is sufficiently small, then the dichotomies q~s(~, ~-) and 4~u(~, ~') associated with (3.7) can be found as the unique solution of the integral equation 0 -- q~s(~, ~) _ 4~s(~, ~) + _ +
f' f0
q:,s(~,r)B(r)q~S(r, ~')dr qss(~, r)B(r)q~U(r, ~')dr,
_
f0'
0 ~< ~" ~< (3.8)
0 -- 4~u(~, ~') - q~u(~, ~.) _
-
qbu(~, r ) B ( r ) ~ s ( r , ~ ) d r
f'
4~u(~,r ) B ( r ) ~ u ( r , ~')dr
4~s(~,r)B(r)q~U(r, ~') dr q~u(~, r)B(r)q~S(r, ~')dr,
0 ~< ~ ~< ~"
(see [ 143,137] for details). We emphasize that exponential dichotomies are not unique: on R +, for instance, the range of P (~) is uniquely determined, whereas the null space of P (0) can be chosen to be any complement of R(P(0)); any such choice then determines the null space of P (~) for every ~ > 0 by the requirement that the projections and the evolution operators commute. The above integral equation fixes such a complement. REMARK 3.1. If the perturbation B(~) in (3.7)converges to zero as I~l ~ ~ with ~ 6 I, then the projections and evolutions of (3.7) converge to those of (3.6) (see, e.g., [ 143,137]).
Stability of travelling waves
995
It is also true that, if (3.2) has an exponential dichotomy for ~. -- ~.,, then the evolutions and projections that appear in Definition 3.1 can be chosen to depend analytically on ~. for ;v close to ~., (see again [143,137]). It is often of interest to distinguish solutions according to the strength of the decay or growth. For instance, for the constant-coefficient system (3.3), we might be interested in distinguishing solutions u l($) that satisfy
[Ul(~) I ~ from solutions
e<+-+>+lu,(o)l,
U2($) that
> 0
satisfy
lu2(+)l ~ e<"++++lu2(O) I,
~<0
for some chosen r/and some small ~ > 0. In other words, rather than separating stable and unstable eigenvalues of A(~.), we divide the spectrum spec(A(~.)) into two disjoint sets according to the presence of a spectral gap at Re v -- r/; see Figure 3(b). This may sound more general than the situation considered above, but, in fact, it is not: the scaling v($) - uC$)e -'1~
(3.9)
transforms (3.3) into the equation d d$ v -
( A ( ) ~ ) - r/)v,
and the two spectral sets associated with the spectral gap at Re v = r/for the matrix A()v) b e c o m e the stable and unstable spectral sets for the matrix A (~.) - JT. Thus, the results stated above are applicable to any two spectral sets of A()0 of the form {v ~ spec(A()0); Re v < 77} and {v E spec(A()~)); Re v > r/} (assuming, of course, that Re v -J= r/ for every v 6 s p e c ( A 0 0 ) ) . Note that the transformation (3.9) changes only the length of vectors but not their direction. In particular, the evolution of subspaces is not changed. In summary, we may wish to replace the condition x s < 0 < x u that appears in Definition 3.1 by the weaker condition x s < x u. Using the transformation (3.9) for an appropriate 77, we see that all the results mentioned above are also true under this weaker condition, i.e., for arbitrary spectral gaps.
(~)
iR "
,
(b)
iR x -~
x x
Reu-
r/
Fig. 3. Two different spectral decompositions of spec(A(~.)) are plotted: in (a) stable and unstable eigenvalues are separated by the imaginary axis, whereas the spectrum in (b) is divided into the two disjoint sets {v E spec(A(~.)); Re v < r/} and {v 6 spec(A00); Re v > ~}, exploiting a spectral gap.
996
B. Sandstede
3.3. S p e c t r u m a n d Fredholm properties We consider the family of operators 7-0~)'79
> 7-[,
u l
>
du
d~
A (. ; ~.) u,
with parameter )~. We are interested in characterizing those )~ for which the operator 7 - O ~ ) ' D --+ 7-[ is not invertible. The set of all such ~, is the spectrum of the linearization s about the travelling wave. We emphasize that the spectrum of the individual operators TOO "79 --+ 7-/, for fixed )~, is of no interest to us. DEFINITION 3.2 (Spectrum). We say that )~ is in the spectrum s of 7- if 7-(X) is not invertible, i.e., if the inverse operator does not exist or is not bounded. We say that ~. 6 s is in the point spectrum s of 7" or, alternatively, that )~ 6 s is an eigenvalue of 7" if 7-(X) is a Fredholm operator with index zero. The complement E \ ~V'pt =" s is called the essential spectrum. The complement of s in C is the resolvent set of 7-. Recall that an operator 12:2' ~ y is said to be a Fredholm operator if R(E) is closed in y , and the dimension of N ( s and the codimension of R(s are both finite. The difference d i m N ( s - codimR(E) is called the Fredholm index of s It is a measure for the solvability of s = y for a given y E Y. Fredholm operators are amenable to a standard perturbation theory using Liapunov-Schmidt reduction. If/2e : 2" --+ y denotes a Fredholm operator that depends continuously on e E R in the operator norm, then Liapunov-Schmidt reduction replaces the equation ~_.e u m 0
by a reduced equation of the form Z2~u -- O,
E"~ "N(s
> RU2o) -J-,
that is valid for e close to zero, where R(s • is a complement of R(s Note that both spaces appearing in the above equation are finite-dimensional. We refer to [ 180, Chapter 3] and [75, Chapters 1.3 and VII] for introductions to Liapunov-Schmidt reduction. For any )~ in the point spectrum of T , we define the multiplicity of )~ as follows. Recall that A(~; )~) is of the form A(~" )~) - A(~) +)~B(~). Suppose that )~ is in the point spectrum of 7-, with 7-(~) -- d d~
,4(~) - )~B(~),
such that N ( T 0 0 ) = span{ul (.)}. We say that )~ has multiplicity g if functions uj be found for j - 2 . . . . . g such that d d~
~
'
'
E
79 can
997
Stability of travelling waves
for j - 2 . . . . . g, but so that there is no solution u 6 D to d d~
u = (X(~) + ~B(~))u + B(~)u~(~),
~ ~ R.
Lastly, we say that an arbitrary eigenvalue )~ of 7- has multiplicity g if the sum of the multiplicities of a maximal set of linearly independent elements in N(7-0~)) is equal to g. EXAMPLE 1 (continued). Recall the operator s -- DO~ + cO~ + Ou F ( Q ) and the associated family 7 - 0 0 7-(~) _
d d~
_
(
A(~) - )~B
with
a(~)
--
0 _D_
I OU F ( Q ( ~ ) )
,d) -cD
-1
. (o 0) 0
'
Suppose that )~ is in the spectrum of/2 and 7-. The Jordan-block structures of the operators / 2 - )~ and 7-0~) are then the same, i.e., geometric and algebraic multiplicities and the length of each maximal Jordan chain are the same whether computed for Z2 - )~ or for 7-0~). This justifies our definition of multiplicity for eigenvalues of 7-. It is also true that the Fredholm properties, and the Fredholm indices, of s - )~ and 7 - 0 0 are the same (see, e.g., [153,157]). REMARK 3.2. The point spectrum is often defined as the set of all isolated eigenvalues with finite multiplicity, i.e., as the set rpt of those )~ for which 7-(~.) is Fredholm with index zero, the null space of 7-0~) is nontrivial, and 7-(~.) is invertible for all 2 in a small neighbourhood of ~. (except, of course, for 2 -- )~). The sets rpt and rpt differ in the following way. The set of )~ for which 7 - 0 0 is Fredholm with index zero is open. Take a connected component C of this set, then the following alternative holds. Either 7 - 0 0 is invertible for all but a discrete set of elements in C, or else 7-0~) has a nontrivial null space for all )~ E C. This follows, for instance, from using the Evans function (see Section 4.1). The following theorem proved by Palmer relates Fredholm properties of the operator TOO to properties pertaining to the existence of dichotomies of (3.2) d - - u -- A(~" )~)u. d~
THEOREM 3.2 [131,132]. Fix )~ ~ C. The following statements are true. 9 k is in the resolvent set o f 7- if and only if (3.2) has an exponential dichotomy on IR.
998
B. Sandstede
9 x is in the point spectrum ~V'pt of'-Jl- if and only if (3.2) has exponential dichotomies on R + and on ]K- with the same Morse index, i+O0 = i - O 0 , and d i m N ( T 0 0 ) > 0. In this case, denote by 1~ (~;)~) the projections o f the exponential dichotomies of(3.2) on R +, then the spaces N(P_(0; X)) f-] R(P+(0; X)) and N(T(X)) are isomorphic via u(O) ~ u(.). 9 )~ is in the essential spectrum 2Tess if (3.2) either does not have exponential dichotomies on R + or on R - , or else if it does, but the Morse indices on R + and on R - differ
As a consequence, eigenfunctions associated with elements in the point spectrum of T decay necessarily exponentially as I~1--> ~ . REMARK 3.3. To summarize the relation between Fredholm properties of T and exponential dichotomies of (3.2), we remark that T is Fredholm if, and only if, (3.2) has exponential dichotomies on R + and on ]K-. The Fredholm index of T is then equal to the difference i - 0 0 - i + 0 0 of the Morse indices of the dichotomies on ]K- and ]K+ (see [131,132]). If TOO is not Fredholm, then typically the range R(T(~)) of T0~) is not closed in 7-/. Suppose that TOO is invertible, and denote by r (~, ~.; )~) and r tial dichotomy of (3.2) on R. The inverse of TOO is then given by u(,~) -- [7()O-lh](~) --
F
qbs(,~, ~'" ,k)h(~') d~" +
~.; ,k) the exponen-
~u(~., ~.. ,k)h(~') d~'.
oo
If T 0 0 is Fredholm with index i, then its range is given as follows. Consider the adjoint equation d d~
(3.10)
~v=-A(~;)~)*v
and the associated adjoint operator T(X)*:D
> ~,
vl
>
dl)
d~
A(.;X)*v
(3.11)
(note that TOO* is the genuine Hilbert-space adjoint of TOO only when posed on the spaces (3.1)). The adjoint operator TOO* is Fredholm with index - i . We have that h E R ( T 0 0 ) if, and only if,
f_~ (7,(se), h (se))dse --0
(3.12)
(3O
for each 7t E N ( T 0 0 * ) , i.e., for each bounded solution ~p(~) of (3.10). In fact, the following remark is true.
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REMARK 3.4. Suppose that the equation d --u=A(~;X)u d~
(3.13)
has an exponential dichotomy on I with projections P(~; X) and evolutions q~s(~, ~';X) and q~u(~, ~-; X), then the equation d --v=-A(~;X)*v d~
(3.14)
also has an exponential dichotomy on I with projections P(~" X) and evolutions q~s(~, ~'" i.) and q~u(~, ~.; i.). The projections and evolutions of (3.13) and (3.14) are related via P(~; X) -- i d - P ( ~ " X)*,
q~(~, ~. z) - ~u(~, ~; ~),,
q,u(~, ~. z) - q~(~, ~. z)*. This is a consequence of Definition 3.1 together with the following observation (see also [157, Lemma 5.1]): if q~(~, ~') denotes the evolution of (3.13), then, upon differentiating the identity q~(~, ~')q~(~', ~) = id with respect to ~, we see that q~(~, ~') -- q~(~', ~)* is the evolution of (3.14). In particular, 7t 6 N(T(X)*) if, and only if, ~(0) e N(P_ (0; X)) A R(P+(0; X)) -- (N(P_ (0; X)) + R(P+(0; X))) -k, where P+(~; X) and P+(~" X) are the projections for (3.13) and (3.14), respectively, on I = R ~. REMARK 3.5. Note that d
for any two solutions u(~) and v(~) of (3.13) and (3.14), respectively. In particular, if u(~) and v(~) are both bounded, and one of them converges to zero as ~ --+ oc or ~ --> - o c , then (u(~), v(~)) = 0 for all ~.
3.4. Fronts, pulses and wave trains In this section, we discuss the consequences of the above results for fronts, pulses and wave trains.
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3.4.1. H o m o g e n e o u s rest states. Suppose that the travelling wave Q(~) is a homogeneous stationary solution, so that Q(~:) - Q0 E ]1~n does not depend on ~. The coefficients of the PDE linearization about Q0 are constant and do not depend on ~. Thus, assume that A (~" ~) -- Ao(~) -- Ao 4- ~Bo does not depend on r and consider (3.2), now given by d
--u
dr
-- Ao(X)u.
This equation has an exponential dichotomy on II~ if, and only if, A0()0 is hyperbolic. In fact, if A000 is hyperbolic, then ~ s ( ~ , (; X) _ eA0(,~)(r162
~ u ( ~ , ( . X) _ eA00~)(r
where P~(,k) and P~(,k) are the spectral projections of A0(~.) associated with the stable and unstable spectral sets, respectively. We have the following alternative: 9 )~ is in the resolvent set of 7- if, and only if, A0(~) is hyperbolic. 9 ~. is in the essential spectrum Z'ess if, and only if, A0(~.) has at least one purely imaginary eigenvalue, i.e., t e s s = {~. E C; spec(A000) A ill~ --/: 0}. In particular, the point spectrum is empty. EXAMPLE 1 (continued). Suppose that Q0 is a homogeneous rest state. Hence,
0 Ao(X)--
D -I(X-0UF(Q0))
id ) -cO -I
'
and X is in the essential spectrum of 7- if, and only if, d0(,~, k) - det[A00~) - ik] -- 0 has a solution k E R. The function d0(X, k) is often referred to as the (linear) dispersion relation. Typically, the essential spectrum consists of the union of curves )~, (k) in the complex plane, where ~., (k) is such that doO~,(k), k) = 0 for k 6 I~. Alternatively, the essential spectrum can be calculated by substituting U(r t) -e'~t+ik~Uo into the linear equation Ut = s An interesting quantity is the group velocity Cgroup = -- d--kImX, (k)
which is the velocity with which wave packets with Fourier spectrum centered near the frequency k evolve with respect to the equation Ut - s We refer to [24, Section 2] for more details regarding the physical interpretation of the group velocity.
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3.4.2. Periodic wave trains. If we consider the linearization about a spatially-periodic travelling wave Q(~) with spatial period L, i.e., about a wave train Q(~) with Q(~ + L) = Q(~) for all ~, then the coefficients of the PDE linearization have period L in ~. Thus, we assume that the matrix A (~;)~) is periodic in ~ with period L > 0, A(~ + L; )~) = A(~; k),
~ ~ IR,
so that d - - u -- A(~" 1.)u -- (,4(~) + k B ( ~ ) ) u d~
(3.15)
has periodic coefficients. By Floquet theory (see, e.g., [83, Chapter IV.6]), the evolution 4~(~, ~'; k) of (3.15) is of the form 9 (se, O; ~.) -- ~per(se" )v)eR()v)~, where R 0 0 E C ''• and qOper( ~ + L; k) = qbper(~; )~) for all ~ 6 IR with ~per(0; )~) = id. Note that it is not clear whether we can choose R(,k) to be analytic in k (though this is always possible locally in k). We have the following alternatives: The point spectrum r p t is empty, and 9 k is in the resolvent set of 7- if, and only if, spec(R00) AiR = 0, i.e.,if q~(L, 0; )0 has no purely imaginary Floquet exponent (or, equivalently, if q~(L, 0; )~) has no spectrum on the unit circle). 9 ~V'ess = {)~ E C; spec(R(k)) n iR 7~ 0} = {)~ 6 C; spec(q~(L, 0; k)) n S 1 r 0}. Consequently, k is in the essential spectrum if, and only if, the boundary-value problem d - - u = A(~; k)u, d$
0 < ~ < L,
(3.16)
u(L) = eiyu(0)
has a solution u(~) for some g c R. This is the case precisely if iv is a purely imaginary Floquet exponent of 45 (L, 0; k). The approach via Floquet theory is also applicable in higher space dimensions [163, 164], then often referred to as decomposition into Bloch waves, and we refer to [23,118, 119] for generalizations and applications to Turing patterns. Suppose that the wave train is found as a periodic solution q (~) to d - - u = f (u, c) d~ and that (3.15) is given by d --u --(0. f(q(~), c)+ kB)u
dse
B. Sandstede
1002 with -
As a consequence, A = 0 is contained in the essential spectrum, since q' (~) satisfies (3.16) for g = 0. Furthermore, spatially-periodic wave trains with period L typically exist for any period L, in a certain range, for a wave speed c(L) that depends on L (see, e.g., [33,80, 107]). It is not hard to verify, using the equations above, that
__dc(L) = -Cgroup---- - d - Imk(g) dL
dg
I•
,
where A(F) denotes the solution to (3.16) that satisfies A(0) -- 0. The group velocity at A = 0 is therefore related to the nonlinear dispersion relation c -- c(L) that relates wave speed and wavelength of the wave trains. We refer to [24,117] for the physical interpretation of the group velocity. We remark that, for each fixed F ~ R, the multiplicity of an eigenvalue A to (3.16) can again be defined as in Section 3.3 by using Jordan chains
d
d~
u = A(~" A)uj + B(~)uj_j j
uj(L) --e'•
'
(see [68]). These eigenvalues, counted with their multiplicity, can be sought as zeros of the Evans function
Oper(Y,A) ~-det[ei•
- ~ ( L , O; X)].
(3.17)
It has been proved in [68] that, for fixed g 6 R, A, is a solution to (3.16) with multiplicity if, and only if, A, is a zero of Dper(Y,A) of order e. 3.4.3. Fronts. Suppose that the travelling wave Q(~) is a front, so that the limits lim
Q(~) -
Q + 6 ]t~ N
exist. The vectors Q+ are homogeneous stationary solutions to the underlying PDE, and we refer to Q+ as the asymptotic rest states. Thus, the coefficients of the underlying PDE linearization have limits as ~ --4 4-oo. We assume that there are n x n matrices A+ and B• such that lim ~-+•
A(~) -- A+,
and define A+(X) -- A-E + XB+.
lim B(~) -- B+ ~+c~
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1003
The existence of exponential dichotomies for the equation d (.4(~) § XB(~))u --u--A(~'k)u-d~
(3.18)
on IR+ is related to the hyperbolicity of the asymptotic matrices A+(A). The next theorem rephrases the statement of Theorem 3.1. THEOREM 3.3 [36, Chapter 6]. Fix X E C. Equation (3.18) has an exponential dichotomy on IR+ if and only if the matrix A+ (k) is hyperbolic. In this case, the Morse index i+ (X) is equal to the dimension dim E~_(X) of the generalized unstable eigenspace E~_ (k) of A+ (k). This statement is also true on 1R- with A+ (X) replaced by A_ (k). Lastly, (3.18) has an exponential dichotomy on IR if and only if it has exponential dichotomies on IR+ and on IR- with projections P+(~; X) such that N(P_(0; X)) G R(P+(0; k)) = C"; this requires in particular that the Morse indices i+(X) and i_(X) are equal. With a slight abuse of notation, we will refer to the number of unstable eigenvalues of a hyperbolic n • n matrix A, counted with multiplicity, as its Morse index. We observe that, using this notation, the Morse indices of the asymptotic matrices A+(A) are equal to the Morse indices i+(A) of the exponential dichotomies on •+ by the above theorem. Note that 7-(A) is Fredholm with index zero if, and only if, the number of linearly independent solutions to (3.18) that decay as ~ ~ -cx~ and the number of solutions that decay as ~ --+ cx~ add up to the dimension n of C 'Z. As a consequence of Theorems 3.2 and 3.3, we have the following options: 9 s is in the resolvent set of 7- if, and only if, A+(A) are both hyperbolic with the same Morse index i+ (s = i_ (X) such that the projections P+ (~; A) of the exponential dichotomies of (3.18)on I = IR• satisfy N(P_ (0; s G R(P+ (0; k)) = C 'z. 9 X is in the point spectrum rpt if, and only if, the asymptotic matrices A• are both hyperbolic with identical Morse index i+(s = i_(A) such that the projections P+(~; A) of the exponential dichotomies of (3.18) on I = R + satisfy N(P_(0; s N R(P+(0; X)) # {0}. 9 X is in the essential spectrum tess if either at least one of the two asymptotic matrices A+(X) is not hyperbolic (so that X is in the essential spectrum of one or both rest states Q+) or else if A+(X) and A_(s are both hyperbolic but their Morse indices differ, so that i+(s ~- i_(X). The reason that the boundary of the essential spectrum depends only on the asymptotic rest states Q+ is related to the fact that the operators T(X) and T(A)-
d d~
AA(9"A)
with
AC~;Z)--
for~ < 0 , { A_CX) A+CX)
for~ >/0,
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differ only by a relatively compact operator (see [85, Appendix to Section 5: Theorem A.1 and Exercise 2]). Typically, the essential spectrum of fronts contains open sets in the complex plane, namely regions where 7-(A) is Fredholm with nonzero index i_(A) - i + ( A ) # 0. Note that A = 0 is always contained in the spectrum with eigenfunction Q' (~). EXAMPLE 1 ( c o n t i n u e d ) . Suppose that Q(~) is a front connecting the asymptotic rest states Q+, so that
0 A+(X)--
id )
D_ I ( X _ O U F ( Q + ) )
-cD
-I
"
Thus, A is in the essential spectrum of T if either A is in the essential spectrum of Q+ or Q_ (see Section 3.4.1) or else if the Morse indices i_(A) and i+(A), i.e., the number of unstable eigenvalues of A+(A), differ. 3.4.4. P u l s e s .
Suppose that the travelling wave Q (~) is a pulse such that
lim Q(~) = Q0 E ]1~N. In other words, a pulse is a front that connects to the same rest state Q0 as ~ -+ -4-oo. Thus, we assume that there are n • n matrices A0 and B0 such that lim A (~) = A0,
lim B(~) -- B0
and define Ao(X) = Ao + XBo. This is a special case of the situation for fronts considered above. The main difference is that the Morse indices at ~ = +cx~ and ~ = -cx~ are always the same. As a consequence, the operator 7-(X) is either not Fredholm or is Fredholm with index zero. We have the following statement. 9 A is in the resolvent set of 7- if, and only if, the asymptotic matrix A0(X) is hyperbolic, and the projections P• X) of the exponential dichotomies of (3.18) on R + satisfy N(P_ (0; A)) @ R(P+(0; A)) = C n. 9 A is in the point spectrum Z'pt if, and only if, A0(X) is hyperbolic, and the projections P+(~; X) of the exponential dichotomies of (3.18) on ~ + satisfy N(P_ (0; X)) fq R(P+(0; X)) :/: {0}. 9 X is in the essential spectrum tess if the asymptotic matrix Ao(X) is not hyperbolic, i.e., if X is in the essential spectrum of the asymptotic rest state Qo. Again, X = 0 is always contained in the spectrum with eigenfunction Q'(~) (see Example 1).
Stability of travelling waves
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3.4.5. Fronts connecting periodic waves. Similar results are true for fronts that connect spatially-periodic waves to each other or to h o m o g e n e o u s rest states. The essential spectrum of such fronts is determined by the essential spectra of the asymptotic wave trains or rest states and their Morse indices: The essential spectra of the asymptotic wave trains or rest states has been computed in Sections 3.4.2 and 3.4.1 above. Exponential dichotomies for the asymptotic linearizations generate exponential dichotomies for the linearization about the front, and vice versa, by T h e o r e m 3.1. We omit the details.
3.5. Absolute and convective instability In this section, we report on absolute and convective instabilities that are related to the essential spectrum of a travelling wave. We refer to [16,24,135,162,155] for further details and more background, and also to [ 140] for instructive examples. Related p h e n o m e n a for matrices (via discretizations of PDE operators) are reviewed in [175]. Suppose that we consider a travelling wave that has essential spectrum in the right halfplane, so that the wave is unstable. Such an instability can manifest itself in different ways. The physics literature distinguishes between two different kinds of instability, namely absolute and convective instabilities. An absolute instability occurs if perturbations grow in time at every fixed point in the domain (see Figure 4(a)). Convective instabilities are characterized by the fact that, even though the overall norm of the perturbation grows in time, perturbations decay locally at every fixed point in the u n b o u n d e d domain; in other words, the growing perturbation is transported, or convected, towards infinity (see Figure 4(b)). 2 We outline how absolute and convective instabilities can be captured mathematically on the u n b o u n d e d domain R. Suppose that the linearization of the PDE about a pulse, say, is given by the operator 12, acting on the space Lz(R) with n o r m ]]. ]]. To describe convective instabilities, it is convenient to introduce exponential weights [ 162]: for a given real number ~7, define the norm I[" 117 by
IIUII2,7 - f ~
le -~ e(~)l 2 d~,
(3.19)
OO
(~)
(b)
T" Fig. 4. The dotted waves are the initial data U0(~) to the linearized equation Ut =/2U, whereas the solid waves represent the solution U(~, t) at a fixed positive time t. In (a) an absolute instability is shown: the solution grows without bounds at each given point ~ in space as t -+ cx~.In (b) a convective instability is shown: the solution U(~, t) grows but also travels towards ~ = +oc; U(~, t) actually decays for each fixed value of ~ as t --~ oc. The operator/2 would then have stable spectrum in the norm II" 11~7for a certain 77> 0. 2 Note that the difference between absolute and convective instabilities depends crucially on the choice of the spatial coordinate system: changing to a moving frame can turn a convective instability into an absolute instability.
1006
B. Sandstede
and denote by L 217(~) ' equipped with the norm I1" I1~ the space of functions U(~) for which IIU I1~ < o~. Note that the norms I1" I1~ for different values of 0 are not equivalent to each other. We consider s as an operator on L~2 (•) and compute its spectrum using the new norm I1" I1~ for appropriate values of 0. The key is that, for 0 > 0, the norm I1" I1~ penalizes perturbations at - c ~ , while it tolerates perturbations (which may in fact grow exponentially with any rate less than r;) at +o<). Thus, if an instability is of transient nature, so that it manifests itself by modes that travel towards +o<), then the essential spectrum calculated in the norm I1" I1~ should move to the left as rl > 0 increases. Indeed, as the perturbations travel towards +oQ, they are multiplied by the weight e - ~ which is small as ~ --+ oe and therefore reduces their growth or even causes them to decay. Exponential weights have been used to study a variety of problems posed on the real line such as reaction-diffusion operators [162], conservative systems such as the KdV equation [135, 136], and generalized Kuramoto-Sivashinsky equations that describe thin films [30,31]. They have also been applied to spiral waves [159] on the plane. Absolute instabilities occur if the spectrum cannot be stabilized by any choice of ~. Conditions for the presence of an absolute instability were derived for homogeneous rest states in [24, Section 2] and for wave trains in [16]. We also refer to [155] for related phenomena. Introducing the weight (3.19) has the effect that the operator T given by
7"(A):D
> 7-/,
ul
>
du
d~
A(.;A)u,
is replaced by the operator
7- ~(A)" 79
> 7-/,
u,
>
du
d~e
[A(." A) - r/]u,
upon using the transformation (3.9). In particular, the essential spectrum of the operator /2 in the weighted space can be computed by applying the theory outlined in the previous sections to the operator T o (A), rather than to the operator T(A). These arguments also apply to fronts instead of pulses: it is then, however, often necessary to consider different exponents for ~ < 0 and for ~ > 0 in the weight function to accommodate the different asymptotic matrices A+(A). We refer to [ 155] for more details and references.
4. The Evans function We have seen that the spectrum of T is the union of the essential spectrum tess and the point spectrum X'pt. For pulses and fronts, the essential spectrum can be calculated by solving the linear dispersion relation of the asymptotic rest states (see Section 3.4). In this section, we review the Evans function which provides a tool for locating the point spectrum [2,134]. The Evans function can also be used to locate the essential spectrum of wave trains [68] (see Section 3.4.2).
Stability of travelling waves
1007
4.1. Definition and properties Consider the eigenvalue problem d --u=A(~;X)u, d~
u9
~9
(4.1)
Since we are interested in locating the point spectrum, we assume that X is not in the essential spectrum Eess of 7- (see, however, Section 4.3). Owing to Theorem 3.2, Equation (4.1) therefore has exponential dichotomies on R + and R - with projections P+(~; X) and P_(~; X), respectively, and the Morse indices dimN(P+ (0; X)) = d i m N ( P _ (0; X)) are the same. Recall from Definition 3.1 that R(P+ (0; X)) contains all initial conditions u(0) whose associated solutions u(~) of (4.1) decay exponentially as ~ --+ ec. Analogously, N(P_ (0; X)) consists of all initial conditions u (0) whose associated solutions u (~) decay exponentially as ~ --+ - e c . In particular, owing to Theorem 3.2, we have that X 9 ~r'pt if, and only if, N(7-(X)) -~ N(P_ (0; X)) N R(P+(0; X)) 7~ {0}. Any eigenfunction u (~) is a bounded solution to the eigenvalue problem (4.1): u (0) should therefore lie in R(P+(0; X)), so that u(~) is bounded for ~ > 0, and in N(P_(0; X)), so that u(~) is bounded for ~ < 0. The Evans function D(X) is designed to locate nontrivial intersections of R(P+(0; X)) and N(P_ (0; X)). Let s be a simply-connected subset of C \ Sess. Note that, in most applications, the essential spectrum Eess will be contained in the left half-plane" otherwise the wave is already unstable. The set s of interest is then the connected component I-2~c of C \ Sess that contains the right half-plane. The Morse index dimN(P_(0; X)) -- dimN(P+(0; X)) is constant for X E I2, see Remark 3.1, and we denote it by k. We choose ordered bases [Ul(X) . . . . . Uk(X)] and [Uk+l(X) . . . . . u,z (X)] of N(P_(0; X)) and R(P+(0" X)), respectively. On account of [100, Chapter II.4.2], we can choose these basis vectors in an analytic fashion, so that uj (X) depends analytically on X E S2 for j -- 1 . . . . . n. We can also choose these basis vectors to be real whenever X is real (recall that we assumed that the matrices A (~) and B(~) are real). DEFINITION 4.1 (The Evans function). The Evans function is defined by D(X) = det[ul (X) . . . . . u,, (X)]. An immediate consequence of this definition is that D(X) = 0 if, and only if, X is an eigenvalue of T. Note that the Evans function depends on the choice of the basis vectors u j (X). Any two Evans functions, however, differ only by a product with an analytic function that never vanishes; this factor is given by the determinants of the transformation matrices that describe the change of bases. Since this ambiguity in the construction is of no consequence, we sometimes use, with an abuse of notation, the shortcut D(X) -- N(P_ (0; X))/x R(P+(0; X))
1008
B. Sandstede
to denote the Evans function associated with the subspaces N(P_ (0;)0) and R(P+ (0; s even though the above construction is not unique. Note that we could make the Evans function unique by, for instance, fixing an orientation for both subspaces N(P_ (0; ,k)) and R (P+ (0;)~)) and by choosing oriented, orthonormal bases. THEOREM 4.1 [54,2,70,134]. The Evans function DO0 is analytic in )~ E I2 and has the following properties. 9 D O 0 E R whenever )~ E R f-) S-2. 9 D()O = 0 if and only if )~ is an eigenvalue o f T . 9 The order of)~, as a zero of the Evans function DOQ is equal to the algebraic multiplicity of )~, as an eigenvalue of 7- (see Section 3.3). Here, the order of )~. as a zero of D(~) is the unique integer m ~> 0 for which dj D dXJ (~'*) = 0
d mD
(for j -- 0 . . . . . m
m
1),
~d)d" (z,)
# 0.
The reason why the Evans function counts the algebraic multiplicity of eigenvalues is related to the fact that, if we denote by 9 (~, (;)~) the evolution of d
d~ then, for each u0 E C n, the derivative 0z,/~ (~, 0; )~)u0 is a particular solution to
--ud _ (A(~) + )~B(~))u + B(~)~(~, O; )Ouo d~
This equation is precisely the equation that determines the algebraic multiplicity of )~ (see Section 3.3). Thus, the Evans function locates eigenvalues of 7- with their algebraic multiplicity. As we have seen in Example 1 and in Section 3.4, the Evans function typically vanishes at 3, = 0, so that D(0) = 0: the derivative Q~(~) of the travelling wave Q(~) generates the eigenvalue )~ --0. Note that, because of analyticity, the Evans function D 0 0 either vanishes identically in I2 or else it has a discrete set of zeros with finite order corresponding to isolated eigenvalues of 7- with finite multiplicity. This proves the statement in Remark 3.2.
4.2. The computation of the Evans function, and applications In general, it is difficult to calculate the Evans function explicitly for a given PDE. One class of PDEs for which the Evans function can often be computed is integrable PDEs for which Inverse Scattering Theory is available. Examples for which Evans functions have been calculated include the Korteweg-de Vries and the modified Korteweg-de
Stability of travelling waves
1009
Vries equation [ 135], Boussinesq-type equations [5,136], the nonlinear Schr6dinger equation [97], and the fourth-order PDE [6] from Example 2. In singularly perturbed reaction-diffusion equations, E'gUt = S2Uxx -+- f (u, v),
(4.2)
vt = 62vxx + g(u, v), with e > 0 small, travelling waves are often constructed by piecing or gluing several singular waves together using, for instance, geometric singular perturbation theory (see [91 ] for a review) or matched asymptotic expansions (see, e.g., [176,112]). These singular waves are travelling waves of (4.2) in the limit e --+ 0 in various different scalings of the spatial variable x or ~. In particular, the singular waves are stationary solutions of certain scalar reaction-diffusion equations, and their stability properties follow immediately from Sturm-Liouville theory (see, e.g., [83, Chapter XI]). The issue is to determine the stability properties of the travelling wave to the full Equation (4.2) for e > 0. This can often be achieved using the Evans function: We refer to [90,185] for results proving the stability of the fast pulses to the FitzHugh-Nagumo equation and to [15,47,70,88,127-129,142] and the references therein for various other results related to the stability of waves to singularly perturbed equations of the form (4.2). One particularly useful argument is provided by the so-called "elephant-trunk lemma" (see [71 ]) that shows that the Evans function of the full Equation (4.2) is often close to the product of the Evans functions for each of the two equations appearing in (4.2), properly scaled and appropriately evaluated (see [2,71 ] for details and applications). The Evans function can be used to test for instability using a parity-type argument. Suppose that the essential spectrum tess is contained in the open left half-plane; otherwise, the wave is unstable. The idea is that the Evans function D(A) is defined and real for all real A >~ 0. Suppose that the Evans function is positive, D(s ~> 3 > 0, for all sufficiently large real A. Note also that D(0) = 0 for any nontrivial travelling wave because of translation invariance. Hence, if D' (0) < 0, then at least one A, > 0 exists with D(A,) --0, and the wave is unstable, since A, is in the point spectrum rpt. The key is that there are computable expressions for the derivatives of the Evans function with respect to A and also with respect to parameters that appear in the underlying PDE (see [ 134,146,95] and Section 4.2.1). Also, the limiting behaviour of the Evans function D(A) as A -+ ~ can often be determined (see Section 4.2.2). The above parity argument has been used to derive instability criteria for solitary waves in Hamiltonian PDEs [134,21,22], for multi-bump pulses to reactiondiffusion equations [3,4,122,187], and for viscous shocks in conservation laws [72]. Lastly, the Evans function is also useful when computing the stability of solitary waves under perturbations. Suppose, for instance, that A = 0 is a zero of D(s with order m > 1. This typically occurs when the underlying PDE has continuous symmetries such as a phase invariance or Galilean invariance. Examples are the nonlinear Schr6dinger equation with m = 4 [183] and the complex Ginzburg-Landau equation with m = 2 [177]. If some of these symmetries are broken upon adding perturbations, then some of the eigenvalues at A = 0 may move away, and it is necessary to locate these additional discrete eigenvalues to determine stability. Using expressions for the derivatives of the Evans function with respect to A and the perturbation parameters as provided by [ 134,146,95], the Evans function can
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B. Sandstede
be expanded in a Taylor series, and its zeros near )~ = 0 can, in principle, be computed. We refer to [94] where this approach has been carried out for the perturbation of the nonlinear Schr6dinger equation to the complex Ginzburg-Landau equation. We emphasize that the perturbed eigenvalues can often be computed more efficiently using Liapunov-Schmidt reduction. The reason is that, near a given isolated eigenvalue )~, in the point spectrum of T, the Evans function D()~) is essentially equal to the determinant of the underlying PDE operator/2 restricted to the generalized eigenspace of)~,. Thus, upon adding a perturbation to the operator/2, the polynomial D 0 0 is perturbed, and we need to compute its zeros: in general, this is a difficult task. It is often far easier to investigate directly the matrix that represents L; restricted to the eigenspace of,k,. In other words, it is often easier to compute the eigenvalues of a matrix directly, using properties of the matrix, rather than finding zeros of the characteristic polynomial which may hide these properties. For instance, symmetries can often be exploited more efficiently. We refer to [ 106,151 ] where Liapunov-Schmidt reduction rather than the Evans function has been utilized. Below, we give an expression for the derivative D ~(0) of the Evans function D 0 0 at )~ = 0 and explore the asymptotic behaviour of D()~) for )~ ~ cx~ in more detail using Example 1. 4.2.1. The derivative D' (0). We assume that )~ = 0 is contained in the point spectrum of T, with geometric multiplicity one. In particular, we assume that T(0) is Fredholm with index zero. We denote the nonzero eigenfunction of ~. -- 0 by q)(~).3 Hence, q)(~) is the unique bounded solution (unique up to constant multiples) of d - - u = A(~; 0)u, d~ and we have span {(p(O) } -- N(P_ (0; 0))N R(P+(O; 0)). We choose the ordered bases [Ul (k) . . . . . uk00] and [Uk+l ()~). . . . . un(k)] of N(P_ (0;)0) and R(P+ (0;)~)), respectively, in the definition of the Evans function in such a fashion that u~ (0) = ~0(0),
uk+~ (0) = ~o(0).
Owing to Remark 3.4 and to the fact that T(0) is Fredholm with index zero, the adjoint equation d - - u = - A ( ~ ; 0)*u d~ also has a unique bounded solution ~r(~). The derivative of the Evans function D 0 0 at k = 0 is then given by D'(0) =
1
17i(0) 12
det[gr(0), U2(0) . . . . . Uk(O), 99(0), U k + 2 ( 0 ) , . . . , Un (0)]
3 If q(~) is the travelling-wave solution, then q)(~) = q1(~).
Stability of travelling waves
•
F
1011
(4.3)
(~ (~), B(~)~o(~)) d~.
This expression can be evaluated once 7t(~) and ~o(~) are known. Note that the right-hand side of (4.3) is a product of two terms: The nonzero determinant measures only the orientation of the basis in the brackets; note that 7t(0) is perpendicular to the vectors u j (0) at )~ = 0 as a consequence of Remark 3.5. The integral decides whether i, = 0 has higher algebraic multiplicity. Indeed, the integral is equal to zero if, and only if, B(-)~o(.) is contained in the range of T(0), see (3.12), which is the case precisely when ~, = 0 has algebraic multiplicity larger than one. This observation illustrates one relation between the statements in Section 3.3 and the Evans function. If the underlying PDE is Hamiltonian, then the derivative D' (0) is typically zero, and an expression for the second-order derivative D" (0) is needed. In [ 134], the quantity D" (0) has been related to the derivative of the momentum functional 4 with respect to the wave speed c. We refer to [ 134] for the details and to [21,22] for extensions that utilize a multisymplectic formulation of the underlying Hamiltonian PDE. 4.2.2. The asymptotic behaviour o f DO~) as )~ --+ co. We illustrate the typical behaviour of the Evans function D()~) in the limit as ~ ~ cc using Example 1. EXAMPLE 1 (continued). Consider the eigenvalue problem 5
(
D - * ()~U - Ou F ( Q ( ~ ) ) U - c V )
o P -1(~.-
,d
OuF(Q(~)))
-cD -l
u,
-
(4.4)
"
Changing variables according to =
,/Izl
v,
Equation (4.4) becomes
(o
0
I~I -I ou F ( Q ( ( / x / ~ ) ) )
_c x/~-
l D- I
(4.5)
In the limit as I)~l ~ ec, we obtain the constant-coefficient equation
(
U~
0
id)
(4.6)
A conserved functional associated with the translation symmetry; see Section 8. 5 Our notation in this example is ambiguous since we denoted the diffusion matrix D of the reaction-diffusion system and the Evans function D(~.) by the same letter.
4
1012
B. Sandstede
The Evans function D0~) for the rescaled equation (4.6) can be computed since there is no dependence on ~. We focus on the case )~ 6 R +, so that arg)~ -- 0" If the diffusion matrix is given by D -- diag(dj), then the eigenvalues of the matrix
(0 ) D-I
are given by l)j - - ~ / ~ and I)j+N = --~//'-dj for j -- 1. . . . . N, and the associated eigenvectors are
fij = (ej, v ~ - l e j ) , where
{tj+N -- (--ej, ~-d~-lej),
j = 1 . . . . . N,
ej denote the canonical basis vectors in ]1~N. In particular,
-
D()~) = det[tTl . . . . .
/~2N] -- det
( id
D_I/2
- i d ) = det[2D - J/2] > 0 D-I/2
for )~ > 0. Since the coefficient matrices in Equations (4.5) and (4.6) are close to each other, uniformly in ~', for )~ sufficiently large, their Evans functions are close uniformly in )~ as a consequence of Theorem 3.1. In particular, the Evans function D 0 0 of (4.4) never vanishes for all real, sufficiently large )~. For the parity argument mentioned above, we would need to compare the Evans function D100 as used in Section 4.2.1 to the Evans function D2 ()~) used in Example 1 above. It is not difficult to see that sign DI (~) = sign D2 (~.) for all large real ~. provided the ordered bases [ut(0) . . . . , U N ( 0 ) ] and [ill . . . . . /gN] of N(P_(0; 0)) have the same orientation, as do the ordered bases [ u u + l ( 0 ) , . . . , U Z N ( 0 ) ] and [tTu+l . . . . . tTZU] of R(P+(0; 0)). Note that, in set-up of Example 1, we have k -- N since we assumed that the essential spectrum is contained in the left half-plane.
4.3. Extension across the essential spectrum We defined the Evans function D()~) for )~ not in the essential spectrum Zess since we were interested in locating the point spectrum. If the essential spectrum is contained in the open left half-plane, then knowing the Evans function for ~ to the right of the essential spectrum, i.e., in the closed right half-plane, is all we need to decide upon stability. For two important classes of PDEs, however, the essential spectrum will always touch the imaginary axis: these are conservation laws on the one hand and integrable PDEs such as the nonlinear Schr6dinger equation
iUt + Uxx + JUJ2U-O,
U ~ C,
Stability of travelling waves
1013
and the Korteweg-de Vries equation
Ut + Uvxx + U Ux - O ,
U 6 IR,
on the other hand. Thus, suppose that we study the stability of a pulse whose essential spectrum lies on the imaginary axis as is the case for the NLS and the KdV equation. Suppose further that we obtained spectral stability of the pulse, so that the spectrum lies in the closed left half-plane. Assume then that the PDE is perturbed in such a fashion that the pulse under consideration persists as a pulse for the perturbed PDE. 6 The issue is then the stability of the pulse to the perturbed PDE. The essential spectrum can again be calculated easily (see Section 3.4), while perturbations of isolated eigenvalues of the unperturbed PDE can be investigated using Liapunov-Schmidt reduction or the Evans function (see Section 4.2). There is, however, an additional mechanism that can create an instability: Recall that we assumed that the essential spectrum resides on the imaginary axis. Upon adding the perturbation, eigenvalues may bifurcate from the essential spectrum leading to additional point spectrum close to the essential spectrum (see Figure 5(c)). These eigenvalues are not regular perturbations of the point spectrum of the unperturbed pulse, but are created in the essential spectrum. Since the new eigenvalues bifurcate from the essential spectrum, it is not possible to use Liapunov-Schmidt reduction or standard perturbation theory as Fredholm properties fail. The Evans function, however, can be used to locate and track such eigenvalues as we shall see below. Consider the equation d - - u -- A(~" A)u
(4.7)
d~
(~)
(b) X X
o[_,
A w X
~,
(c)
I X n
ol ~-~ess
A
.k q[P
.,...
',
',/~o l l l l l l
,
~-]ess
l
Fig. 5. Plotted is the complex A-plane. The insets represent the spatial spectra of the matrix A0 (A) in different regions of the A-plane. Two different spatial eigenvalue configurations near the essential spectrum are plotted: In (a) a single spatial eigenvalue crosses the imaginary axis when A crosses through tess; the dotted line in the inset is given by Re v = 1;. In (b) two spatial eigenvalues cross simultaneously but in opposite directions when A crosses through tess. In (c) we plotted the unperturbed essential spectrum (dashed line) as well as the perturbed essential spectrum (solid line) and an additional eigenvalue that moves out of the essential spectrum upon perturbing the operator. 6 We refer to Section 2.2 for examples of such perturbations for the NLS and the KdV equation.
1014
B. Sandstede
and assume that the limit lim A(~;
X) --
A0(X)
exists. On account of the results stated in Section 3.4.4, we have that )~ is in the essential spectrum of T precisely when A0(~) has at least one spatial eigenvalue on the imaginary axis (see Figure 5). If A000 is hyperbolic, then (4.7) has exponential dichotomies on ]K+ and IK- with projections P+ (~;~) and P_ (~; ,k), respectively. An element ~ e C is in the point spectrum precisely if N(P_(O; X)) A R(P+(O; X)) # {0}. The Evans function D(~) measures intersections of the two subspaces N(P_(0; ,k)) and R(P+ (0;)0). It is defined for any ,k for which A0(,k) is hyperbolic. The key idea is to find an analytic extension of the Evans function D 0 0 for ,k e tess. Zeros of the extended function D(,k) for ,k e Sess correspond to possible bifurcation points of point spectrum: Upon adding perturbations, these zeros may move out of the essential spectrum (see Figure 5(c)). We illustrate the analytic extension of the Evans function in Figure 5. Throughout this section, we assume that )~ is close to the essential spectrum Zess. References for analytic extensions of the Evans function are [ 134,72,97]. First, consider the set-up shown in Figure 5(a). When )~ crosses through Zess from right to left, a spatial eigenvalue v of A000 crosses through ilk from right to left. For ~ to the right of Z'ess, the spectral projection P~ 00 of the matrix A0(~) that projects onto the stable eigenspace along the unstable eigenspace is well defined (see (3.4)-(3.5)). Note that stable and unstable spatial eigenvalues for ~ to the right of Zess correspond to the bullets and crosses, respectively, in Figure 5(a). For )~ on and to the left of Z'ess, we denote by P~ (X) the spectral projection of the matrix A0(~.) associated with the two spectral sets consisting of the bullets and the crosses, respectively, in Figure 5(a). The projection is well defined as long as )~ is close to tess and as long as spatial eigenvalues cross only from right to left through iR as ~ crosses from right to left through ~'ess- In other words, rather than dividing spec(A0(~)) into stable and unstable spatial eigenvalues, we distinguish between spatial eigenvalues v with Re v < 7; and Re v > r/where r/< 0 is close to zero such that the line Re v = r/separates the stable spatial eigenvalues from the formerly unstable spatial eigenvalues that crossed the imaginary axis (see Figure 5(a)). The discussion at the end of Section 3.2 shows that there are projections P+ (~;)~) and P_ (~;)~), defined for ~ e R + and ~ 6 IK- respectively, that are defined and analytic in k for all ~. close to Zess such that
P+(~;z)
~P~(Z), ~ + ~ .
The subspace R(P+(0; )~)) consists of precisely those initial conditions that lead to solutions u(~) of (4.7) that satisfy lu(~)l ~< eO~lu(0)l for ~ ~> 0. The subspace N(P_(0; X)) consists of precisely those initial conditions u(0) for which lu(~)l ~< e ~ lu(0)l for ~ <~ 0; note that these solutions will in general grow exponentially as ~ ---> - o c whenever )~ is to the left of Sess.
Stability of travelling waves
1015
The Evans function D(A) can now be defined as in Section 4.1 via D ( A ) = N(P_(0; A))/x R(P+(0; A)), so that zeros of D(A) correspond to nontrivial intersections of R(P+(0; A)) and N(P_ (0; A)). Note that zeros of D(A) for A to the right of 2Tess correspond to eigenvalues of 7-, but that zeros to the left or on s have no meaning for 7-. The latter zeros are commonly referred to as resonance poles. We conclude that eigenvalues can bifurcate from the essential spectrum only at zeros A, E 1;ess of the extended Evans function D(A). Suppose that this extension is computed, and its zeros determined. Upon adding perturbations to (4.7) that depend on the small perturbation parameter e, a perturbation analysis of the analytic function D(A; e) near each zero then reveals whether the zeros move to the right or to the left of 22ess. This program has been carried out for the generalized Korteweg-de Vries equation U, + U~ ~~ - c U~ + U P U~ -- O,
U ~ R,
by Pego and Weinstein [134,135]: they showed that pulses destabilize at p = 4 where an eigenvalue emerges from the essential spectrum, given by the imaginary axis iN, at A -- 0. An alternative, but equivalent, way of extending the Evans function in the situation shown in Figure 5(a) consists of using the transformation (3.9) v(~) - u(~)e -'1~ which replaces (4.7) by d d~
v-
(A(~" , k ) - ~)v.
(4.8)
For 7; < 0 close to zero, the asymptotic matrix A(A) - r/has imaginary eigenvalues for values of A on a curve Ze~ss that is strictly to the left of tess (see Figure 5(a)). Thus, the Evans function for (4.8) is defined for all A near tess and coincides with the Evans function of (4.7) for A to the right of rest. Next, we consider the case illustrated in Figure 5(b). In this situation, two spatial eigenvalues cross the imaginary axis simultaneously in opposite directions as A crosses through Zess. We are interested in extending the Evans function, defined in the region to the right of Zeus, in an analytic fashion to the left of rest. The procedure outlined above for case (a) seems to fail since there is no spectral gap anymore. Recall that the Evans function D(A) for A to the right of tess is defined by D(A) = det[u I(A) . . . . . u,, (A)], where [ul(A) . . . . . uk(A)] and [uk+l (A) . . . . . u,,(A)] are ordered bases of N(P_(0; A)) and R(P+ (0; A)). One way of obtaining an analytic extension of the Evans function in this setup is to analytically extend the exponential dichotomies ~ _ (~, 0) for ~ ~> 0 and 4~u_(~, 0) for ~ ~< 0; recall that 4~_(0, 0) - P+(0; A) and 4,u(0, 0) - i d - P _ ( 0 " A). For simplicity, we concentrate on ~/> 0.
1016
B. Sandstede
We begin by extending the projections of the asymptotic constant-coefficient equation d
--u
d~
(4.9)
= Ao()~)u.
The stable and unstable spatial eigenvalues of A00,) for )~ to the right of Z:ess correspond to the b u l l e t s and c r o s s e s , respectively, in Figure 5(b). For )~ on and to the left of Z:ess, we denote by P~ 00 the spectral projection of the matrix A0()~) onto the generalized eigenspace associated with the bullets along the eigenspace associated with the crosses. In other words, we assume that we can divide the spatial spectrum of A0()~) into two disjoint spectral sets according to whether the spatial eigenvalues are in the left or right half-plane for ~ to the right of Z:ess; thus, we do not allow that formerly stable and unstable spatial eigenvalues cross through ilR at the same point ik. 7 Using Dunford's integral [ 100, Chapter 1.5.6], we obtain spectral projections, which we still denote by P~ (~.), that depend analytically on )~ for ~ close to the essential spectrum Z:ess. The exponential dichotomies of (4.9) are ea~
and
eA~
(4.10)
for ~. in a, possibly small, neighbourhood of Z:ess, where P~ ()~) = i d - P ~ (~). We can also divide the spatial spectrum of A0(~.) into two different disjoint sets, namely the strong stable and the center-unstable spectrum: the strong stable spectrum consists of all stable eigenvalues that keep a uniform distance from the imaginary axis as we vary % near the essential spectrum, while the center-unstable spectrum is its complement. The strong stable eigenvalues correspond to the leftmost b u l l e t in Figure 5(b), while the center-unstable eigenvalues consist of the rightmost b u l l e t and all c r o s s e s in Figure 5(b). We denote the associated spectral projections by P~s(A.) and p~u(~.) = id_p~s(A.). These projections are again analytic in )~ for )~ close to the essential spectrum. In the next step, we construct the evolution operator @s(~, 0) for the full Equation (4.7). Thus, assume that @s(~, ~) is associated with (4.7) on R + for )~ to the right of the essential spectrum. Roughly speaking, analyticity of these operators with respect to )~ is equivalent to uniqueness. Thus, we shall find a way to extend these operators uniquely to the region to the left of tess. The idea is to seek an extension of these dichotomies by requiring that the extended evolution operators approximate the dichotomies (4.10) of the asymptotic constant-coefficient equation (4.9) as best as possible. Intuitively, such a choice should be possible provided the coefficient matrix A (~;%) converges rapidly enough to the asymptotic coefficient matrix A0(~.). In other words, the solutions of (4.7) should converge rapidly towards solutions of (4.9) provided the coefficient matrix A(~; )~) converges rapidly to A0(%). This is indeed the case as long as the distance, in the real part, between the rightmost formerly stable spatial eigenvalue and the leftmost formerly unstable eigenvalue, i.e., the amount of overlap of the two spectral sets, is smaller than p, where p is the exponential rate with which the coefficient matrices approach each other (see (4.11) below). This result is referred to as the Gap Lemma [72,97]. To illustrate this result, we therefore assume that
Ia( ; x) -
Ao()~)[ ~< Ce -pIll
7 This assumption is actually not necessary, and we refer to [97] for the details.
(4.11)
Stability of travelling waves
1017
as [~] --+ oo for some p > 0 independently of X. To construct the extension of 9 s (~, 0), we define the correction q/s(~, 0) by 0 s (~, O) -- e A~
P(~(Z) --F ~s (~, O)
and seek the correction tps (~, 0) as a solution to the integral equation qjs(~, 0) -- f ~ e A~(~)(~-T)Pocu (X)A(r" X) [eA0(Z)r Pos (X) + tps (r, 0)] dr +
fo
eA~ x [en~
k) -F tps('r, 0)] d'r ,
~ >/0,
(4.12)
where A(~" X ) = A(~" X ) - A o ( X ) - O(e-Pl~l). The above integral equation coincides with the integral equation (3.8), which we encountered in Section 3.2 when we constructed regular exponential dichotomies, upon substituting the above expression for 9 s and setting ~" -- 0 in (3.8). Equation (4.12) is also reminiscent of the integral equation that describes strong stable manifolds; it has a unique solution (possibly after replacing ~ -- 0 by ~ --- L in the second integral for some L >> 1 to make the right-hand side of (4.12) a contraction) that gives the correction qjs. The exponential decay in (4.11) is necessary to compensate for the exponential growth of the solution operators in (4.10). Hence, we can construct analytic extension of the dichotomies for (4.7), and thus of the Evans function D(X), for k to the left of the essential spectrum. Slightly different constructions have been carried out in [72,97]. In [72], the analytically extended Evans function has been used to establish instability criteria of shock waves to conservation laws. In [97,98,96, 110], this approach was used to prove stability and instability of solitary waves to perturbations of the nonlinear Schr6dinger equation. We also refer to [45-47] for applications, and extensions, of the analytic extension of the Evans function across the essential spectrum to the stability of pulses in singularly perturbed reaction-diffusion equations with a strong coupling between the fast and slow subsystems.
5. Spectral stability of multi-bump pulses In this section, we consider the stability of multi-bump pulses. Suppose that we know that a given PDE supports a stable pulse Q(x - cot) that travels with speed co. We refer to this pulse as the primary pulse. Typically, such a pulse is then accompanied by spatiallyperiodic wave trains PL (x -- CLt) that resemble infinitely-many equidistant copies of the pulse (see Figure 6 (a) and (b)). These wave trains have spatial period 2L and wave speed CL; they exist for any sufficiently large spatial period L, and the wave speeds satisfy CL --~
1018
B. Sandstede 2L
2L1
0
,~1
2L2
~2
~a
Fig. 6. Plots of the primary pulse Q(~) in (a), a spatially-periodic wave train with period 2L in (b), and a 3-pulse with distances 2L! and 2L2 in (c).
(a)
(b)
9
9
f.
', O"
0o
Fig. 7. Plots of the spectra of the primary pulse in (a), a spatially-periodic wave train in (b), and a 3-pulse in (c).
co as L --+ ocz. Besides these long-wavelength pulse trains, multi-bump pulses may exist which are travelling waves that consist of several well-separated copies of the primary pulse. Associated with an e-pulse, consisting of e copies of the primary pulse, are the distances 2Lj . . . . . 2 L e - i between consecutive copies and the locations ~l . . . . . ~e of the individual pulses (see Figure 6(c) for a plot of a 3-pulse). Throughout the entire section, we assume that consecutive pulses in a wave train or an e-pulse are well separated, so that L and Lj a r e sufficiently large, say larger than some number L, >> 1. We are interested in the spectra of the wave trains and the e-pulses (if they exist) that accompany the primary pulse. We assume that the primary pulse is spectrally stable, so that its spectrum is contained in the open left half-plane with the exception of a simple eigenvalue at )~ = 0, caused by the translation symmetry (see Figure 7(a)). We shall locate the spectrum of an e-pulse that accompanies the primary pulse: The essential spectrum of an f-pulse is close to the essential spectrum of the primary pulse as it is determined by the asymptotic rest state (see Section 3.4.4). It remains to find the point spectrum. We claim that there are precisely e eigenvalues in the spectrum of an e-pulse near each eigenvalue of the primary pulse (see Figure 7(c)). Recall that, as usual, all eigenvalues are counted with their multiplicity. To make our claim plausible, we argue heuristically and focus on the eigenvalues near )~ -- 0. Consider the 3-pulse plotted in Figure 6(c). Each individual pulse in the 3-pulse resembles the primary pulse, and the individual pulses are well separated. Thus, if we change the position of one of the pulses, the other pulses are not affected. Translating the jth pulse corresponds to adding e Q1( 9 - ~j) to the 3-pulse for e small. Hence, there should be three eigenvalues near )~ = 0, and the associated eigenfunctions should be linear combinations of Q' (. - ~ j ) for j = 1, 2, 3, since adding small multiples of these eigenfunctions to the 3-pulse does not affect the 3-pulse much. The key to this argument is that the pulses are exponentially localized and well separated. Thus, an e-pulse should have e eigenvalues near ~ -- 0 with eigenfunctions of the form Y~'~=1 dj Q' (. - ~ j ) .
Stability of travelling waves
1019
The larger the distances between consecutive pulses in the g-pulse, the closer should these s critical eigenvalues be to )~ = 0. If the primary pulse is stable, then an s is stable provided its s - 1 nontrivial critical eigenvalues near ~ = 0 move into the left half-plane. Similar arguments apply to the wave trains. Since a wave train consists of infinitely many pulses, there should be many eigenvalues near )~ = 0. In fact, for each eigenvalue of the pulse, the wave train has a circle of eigenvalues that is parametrized by the spatial Floquet exponent Y (see Section 3.4.2, in particular (3.16), and Figure 7(b)). To set-up the problem, consider the travelling-wave ODE d
--.u=f(u,c), d~
(5.1)
where c denotes the wave speed. We assume that the rest state is u = 0, so that f (0, c) = 0 for all c. Suppose that q(~) is a pulse to u = 0 for c = co. We assume that 0u f ( 0 , co) is hyperbolic, so that p < min{[Re v]; v ~ spec(O~,f(O, co))} < 3 p / 2 for an appropriate p > 0; it is advantageous to choose p as large as possible as it appears in the estimates for certain remainder terms (see below). We remark that, if the travelling waves are, in fact, standing waves so that c -- 0, then the underlying PDE sometimes features the reflection symmetry x ~ - x that manifests itself as a so-called reversibility of the travelling-wave ODE (5.1). We would then be interested in symmetric waves that are invariant under the reflection x ~ - x . We refer to [ 174,181 ] for more background on reversible ODEs. The PDE eigenvalue problem associated with the pulse q (~) is of the form d
- - u = [Ouf (q(~), co)-+- )~B(se)]u.
d~
We make the following assumptions. HYPOTHESIS 5.1. The only bounded solution to the variational equation
d - - u -- O . f ( q ( ~ ) , co)u d~
(5.2)
is given by q ' (~ ), up to constant scalar multiples. As a consequence, the adjoint variational equation
d - - u = - O . f (q(~), co)*u d~
(5.3)
has a unique, up to constant multiples, bounded nonzero solution which we denote by 7t (~) (see Section 3.3). We refer to Figure 8 for the geometry of the pulse q (~) and the solution
B. Sandstede
1020
W LI
q(-L)
q(L) Fig. 8. The geometry of the pulse q(~) and the solution ~p(~) to the adjoint variational equation. Note that ~(~) _L (Tq(~)WS(0) + Tq(~)WU(0)) for all ~ (see Remark 3.5), i.e., ~(~) is perpendicular to the tangent spaces of both the stable and the unstable manifold 8 of the equilibrium that is approached by the pulse.
HYPOTHESIS
M=
5.2. We assume that
S
(~p(~), Bq'(~))d~ = -
(X)
F
(~P(~),Ocf(q(~),co))d~ :riO
O0
is not zero (recall (2.8)). In the notation used earlier, the above hypotheses say that 0 E ~V'pt, N(T(0)) = span{q'} and D'(0) r 0, i.e., that ~ = 0 is a simple eigenvalue of T (see Sections 3 and 4). In other words, we assume that )~ = 0 is an isolated simple eigenvalue of the pulse q (~). The next hypothesis is not always needed. HYPOTHESIS 5.3. We assume that the point spectrum ~V'pt of the primary pulse is a discrete subset of C. We are interested in the stability of 2L-periodic wave trains PL (~) and e-pulses qe(~) that accompany the primary pulse q (~). Throughout this section, we assume that, for a sufficiently small constant 6 > 0, we have Ic - c0l < ~,
u(~) E Ha({q(~')" ~" E ~ } )
for all ~ 6 R
(5.4)
for any wave u(~) with wave speed c that we consider below. In particular, we assume that the period 2L of any wave train we may consider is sufficiently large. Furthermore, 8 The stable manifold W s (P0) of an equilibrium u = P0 consists of all solutions that converge to that equilibrium as ~ --+ ~ ; analogously, its unstable manifold WU(p0) consists of all solutions that converge to P0 as --+ -cx~ (see [14,33,80,107]).
Stability of travelling waves
1021
we denote by Z ( u ) = ~'pt(U) U tess(U) the various spectra of a wave u(~) with speed c, computed with respect to the operator
Z , 0 0 = --
d~
-
O, f ( u ( . ) ,
c) -
AB(.).
5.1. Spatially-periodic wave trains with long wavelength
Suppose that PL (~) is a 2L-periodic wave-train solution of (5.1) with wave speed CL, so that PL ( - L ) = PL (L). We comment later on the existence of such wave trains (see [11, 111,181]). We recall from Section 3.4.2 that A is in the spectrum of PL (~) if, and only if, the boundary-value problem d d~U -
[ O u f ( P L ( ~ ) , c L ) + AB(~)]u,
I~1 < L,
(5.5)
u(L) = eiyu(-L)
has a solution u (~) for some y 6 R. THEOREM 5.1 [69]. Assume that Hypothesis 5.3 is met. For every sufficiently small e > O, a ~ > 0 exists with the following properties. If PL (~) is a 2L-periodic wave-train solution o f (5.1) such that (5.4) is met, then the following statements are true. 9 2 7 ( P L ) = ress(PL) C b/e(r(q)). 9 For any A, 6 rpt(q) with multiplicity m, and f o r any fixed spatial Floquet exponent y E [0, 27r), (5.5) has precisely m solutions, counted with multiplicity, in the e-neighbourhood o f A,.
Besides the topological proof given in [69], Theorem 5.5 can also be proved using the roughness theorem of exponential dichotomies; we refer to [ 155, Section 4]. REMARK 5.1. It follows from the results in [155, Section 5.2] that the essential spectrum Sess(q) is also approximated by the spectrum S (pL) of the wave trains. Thus, if the spectrum I7 (q) of the primary pulse is contained in the open left half-plane with the exception of a simple eigenvalue at A = 0, then the spectrum 27 (pc) of the wave train is contained in the open left half-plane with the exception of a circle A(V) of simple eigenvalues that is parametrized by y 6 [0, 2rr] with A(0) = 0. To conclude (in)stability of the wave train, it is necessary to locate this circle of critical eigenvalues. THEOREM 5.2 [156].Assume that Hypotheses 5.1 and 5.2 are met. There is a ~ > 0 with the following properties. Assume that PL (~) is a 2L-periodic wave-train solution o f (5.1) such that (5.4) is met. Equation (5.5) has a solution f o r g E IR and A close to zero if and
1022
B. Sandstede
only if 1 M ((eiy - 1)(~(L), q ' ( - L ) ) + ( 1 - e - i • + R(V, L),
(5.6)
where R(y, L) is of the form g(y, L)-
(e iy -- 1)O(e -3pL) + ( 1 - e-i•
The associated solution u (~) of (5.5) is given by u(~s)--eik•
~ 6[(2k-1)L,(2k+
l)L],
kEZ.
Note that the sign of d2
~ Re)~[ dy 2
1
q' ( - L ) ) + ( ~ ( - L ) , q
'(L)))
decides, to leading order, upon stability. Reference [156] contains more general results, with better estimates for R(F, L), that are applicable even if )~ has larger geometric multiplicity due to the presence of additional continuous symmetries. Before we illustrate the theorem by examples, we outline its proof as it provides some insight as to the role of the scalar products appearing in (5.6) (see also Figure 8). 5.1.1. Outline of the proof of Theorem 5.2. Since the ideas for the proof of the stability theorem 5.2 are the same as those that give existence of wave trains, we begin by introducing Lin's method [ 111,143] that can be used to prove existence. Thus, to set the scene, suppose we want to prove the existence of the periodic orbits pc (~) of period 2L near a given homoclinic orbit q (~) to the equation d m u -- f (u, c), d~
u E ]R2,
in the plane, where c denotes the wave speed. Hence, we shall find functions pc (~) and wave speeds cc such that d d~
pc (~) - f (pc (es), CL),
I~1 < L,
PL (0) 6 q(0) + span{~(0) } -- A, PL (--L) = pL (L) and such that cc is close to co and PL (~) is close to q(~) for I~1 < L. Note that A is a line attached to q (0) and perpendicular to the vector field (see Figure 9(a)). To solve this
Stability of travelling waves (b)
(
1023 (
Fig. 9. The homoclinic orbit q(~) and the transverse section A = q(0) + span{~p(0)} that is attached to q(0) are plotted in (a). In (b) the solution u(~, c, L) to (5.7) is plotted for c-r co. In (c) the perturbed stable and unstable manifolds, denoted by u(~; c, ec) are plotted for c =/: co; the distance between the unstable and the stable manifold is, to leading order, given by u(0-; c, co) -u(0+; c, ec) = - M ( c - co). Lastly, in (d) we plotted the unique solution u(~; co, L) to (5.7) for c--co.
problem, we shall seek functions u(~" c, L), defined for all c close to co and all L large, that may have a discontinuity 9 at ~ -- 0 such that d d--}-u(~" c, L) -- f(u(se; c, L), c),
I~l < L,
u(O+;c,L) E q ( 0 ) + s p a n { O ( 0 ) } - - A ,
(5.7)
u ( - L " c, L) -- u(L" c, L) (see Figure 9(b)). Such a solution is the desired periodic wave train if, and only if, 3 (c, L) "-- u ( 0 - " c, L) - u(0+; c, L) -- 0 so that u(~" c, L) is continuous at ~ -- 0. Hence, we focus on solving (5.7). First, we pretend that we can expand the solution u(~" c, L) as a Taylor series in (c, L) centered at (c, L) -(co, e~). In other words, we write u(~" c, L) -- u(~" c, oc) 4- u(~" co, L),
(5.8)
where u($; c, oc) satisfies (5.7) for L = ec (see Figure 9(c)), while u(~; co, L) satisfies (5.7) for c = co (see Figure 9(d)). Note that the solution u(~; c, ec) of (5.7) with L = oc are precisely the stable and unstable manifolds: in other words, u($;c, c~) parametrizes the stable manifold for ~ ~> 0 and the unstable manifold for ~ ~< 0 with a discontinuity at -- 0 (see Figure 9(c)). O w i n g to M e l n i k o v theory [ 111 ], the j u m p at ~ = 0 is given
( ~ ( 0 ) , u(0-" c, ~ )
--(c-co)
- u(0+" c,
~))
(g/(~), Ocf(q(~),co))d~ + O([c - co[ 2) (N3
= -M(~
- co) + O(1~ - ~ol2).
9 If u(~) has a discontinuity at ~ =0, we define u(0-) = lim~s0 u(~) and u(0+) = lim~,,~0 u(~).
(5.9)
B. Sandstede
1024
It remains to find u (~; co, L). Hence, set c = co and write u(~; co, L) = q ( ~ ) + v(~),
(5.~o)
then u(~; co, L) satisfies the ODE in (5.7) if, and only if, v(~) satisfies d d---~v -- f (q(~) + v, co) - f (q(~), co) = Ouf (q(~), co)v + O({v{2), I~l < L .
(5.11)
Omitting the higher-order term O(Iol2), the solution to this equation can be written, for instance, as
v(~) = c I ) ( ~ , - L ) v ( - L ) , v(se) = r
e, L)v(L),
~>0, ~
where 4~(~, () is the evolution of the linear part of (5.11). At ~ = + L , we shall satisfy
q(L) + v(L) = q ( - L ) + v ( - L ) . Thus, we set v(L) = q ( - L ) and v ( - L ) = q(L), and the solution v(~) to (5.11) is therefore given by
v(~) --
{ ( P ( ~ , - L ) v ( - L ) -- ~ ( ~ , - L ) q ( L ) , ~(~, L)v(L) = cp(~, L ) q ( - L ) ,
~ [ - L , 01, ~ [0, L],
upon omitting the terms of higher order in (5.11). In particular,
v(O-)=~(O,-L)q(L),
v(O+) = q:,(O, L ) q ( - L ) .
In summary, using (5.8) and (5.9), we obtain that the discontinuity ,~, (c, L) of u(~; c, L) at ~ = 0 is given by Z(c.
L)
-
( O ( o ) . . ( o - ; c.
-
L) -
u(O+; c.
L)) + (o(o), u(o-;
= ( O ( - L ) , q ( L ) ) - (O(L), q ( - L ) ) -
- u ( o + ; c,
M(c - co)
upon using that ~ ( ~ ) = 4,(o, ~ ) * ~ ( o )
(see Remark 3.4). Thus, a 2L-periodic wave train with wave speed c exists if, and only if,
M(c - co) - - ( O ( - L ) , q(L)) - (O(L), q ( - L ) ) .
Stability of travelling waves
1025
In particular, 2L-periodic wave trains exist under our assumptions for any L sufficiently large, and the nonlinear dispersion relation that relates wave speed and period is given by 1
CL -- co + --~((7r(--L), q(L)) - (gr(L), q ( - L ) ) ) . We refer to [11,111,143,181,144,156] for more details that justify the above heuristic argument. The Stability Theorem 5.2 can be proved in a very similar fashion by applying the approach outlined above to linear Equation (5.5) with the wave-speed parameter c replaced by the eigenvalue parameter )~. Regarding the parametrization (5.10), it is also advantageous to use !
u(~" cL, L) - Pc (~) + v(~) !
instead of u(~; co, L) -- q'(~) + v(~)" this allows us to exploit that Pc (~) is an exact solution to (5.5) for X = 0 and V -- O. We refer to [ 156] for the details. 5.1.2. Discussion of Theorem 5.2. eigenvalues near X = O: X=
((e i • M
Theorem 5.2 gives the location of the circle of critical
1)(O(L)q'(-L))+(1-e-l• ~
(5 12) ,
~
9
where we neglected the remainder term R(y, L). To apply this result, we need to find expressions for the scalar products that involve the solutions ~p(~) to the adjoint Equation (5.3) and q'(~) to the variational Equation (5.2). Note, however, that these solutions are needed only for I~l large. The tails of these solutions are determined by the eigenvalue structure of the matrix 0, f (0, co) (see, for instance, [34, Chapter 3.8] or [83, Chapter X. 13]). We briefly outline the two situations that occur generically. First, we assume that the leading eigenvalue 1~ of 0 , f ( 0 , co) is real and simple. For simplicity, we also assume that its real part is negative (the analysis for a positive real part is completely analogous and can, in fact, be found in [156]). Thus, there is a simple spatial eigenvalue v s e spec(O,f(O, co)) such that IRe v[ > - v s > 0 for every v 6 s p e c ( 0 , f ( 0 , co)) with v -r v s (see Figure 10(a)). It follows from [34, Chapter 3.8] or [83, Chapter X.13] that there is a 6 > 0 and eigenvectors v0 and w0 of 0 , f ( 0 , co) and 0, f ( 0 , co)*, respectively, belonging to the eigenvalue v s such that q'(~) -- e v~ vo + O(e-(Iv'l+~)~),
gr ( - ~ ) -- e ~ wo + O(e -(1r
as ~ --+ oc. Typically, the vectors v0 and w0 are nonzero. Furthermore, we have
Iq'(-~)l + 10 The leading, or principal, eigenvalues of a hyperbolic matrix are those closest to the imaginary axis.
1026
B. Sandstede
(a) A w
A w
(b) A w
=
i ! ',
Fig. 10. The spectrum of Ouf(O, co) and the shape of the tails of the pulse q(~) are plotted for two different cases: in (a) the leading eigenvalue v s is simple and real with negative real part, whereas the leading eigenvalues in (b) are a pair of simple, complex-conjugate eigenvalues with negative real part.
as ~ --+ c~. Upon substituting these expressions and estimates for q' and ~ into (5.12), we obtain (vO, wo} (1 -- e-i• M
2v~L
for 9/ 6 [0, 2zr], upon omitting terms of higher order. In particular, the wave trains are spectrally stable for all large L if (v0, w0)M < 0 and spectrally unstable for all large L if (vo, w o ) M > O.
The second generic case is that the leading eigenvalues of Ouf (0, co) are complex conjugate and simple. We again assume that their real part is negative (see Figure lO(b)). Hence, a pair of simple complex-conjugate eigenvalues v s, ~v E spec(Ouf(O, co)) exists such that [Rev[ > - R e v s > 0 for every v ~ spec(Ouf(O, co)) with v-~ v s, b-~. We assume that Im v s ~ O. Exploiting the expansion q'(~) -- Re[eVS~ v0] q- O(e-(I RevSl+~)~), lp(--~) -- Re[eV~ w0] + O(e -(I RevSl+~)~)
(5.13)
for ~ --+ oe (see [34, Chapter 3.8] or [83, Chapter X.13]), we end up with the expression ~=
a
sin(2L Im v s + b)(1 - e -i•
2L Re us
M for the circle of critical eigenvalues near )~ = 0. Here, a and b are certain real constants. Hence, if a ~ 0, which is equivalent to v0 ~ 0 and w0 ~ 0, then the periodic wave trains change their PDE stability periodically in L regardless of the signs of a and M. This is not too surprising as it is known that, in the case shown in Figure 10(b), the periodic orbits to (5.1) undergo many saddle-node and period-doubling bifurcations as L is varied [ 170,41 ]. At each such bifurcation, the circle of critical eigenvalues crosses through the imaginary axis, and the wave trains either stabilize or destabilize: in the case of a period-doubling, for instance, (5.5) exhibits solutions at )~ - 0 for both y - 0 and y = Jr. Whereas the solution with y - 0 is enforced by the translation symmetry, the eigenvalue with y = zr should cross the imaginary axis upon unfolding the period-doubling bifurcation. Analogous results are true for wave trains near symmetric pulses provided (5.1) is reversible (see [156] for details and applications).
Stability of travelling waves
1027
The above results have been applied in [ 156] to the wave trains that accompany the fast pulse in the FitzHugh-Nagumo equation as well as to wave trains that arise in the fourthorder equation from Example 2.
5.2. Multi-bump pulses In this section, we discuss the stability of g-pulses that consist of g well separated copies of the primary pulse9 Recall that, if the primary pulse is spectrally stable with only one simple eigenvalue at )~ = 0 and with the rest of the spectrum in the open left half-plane, then the spectrum of an g-pulse contains g critical eigenvalues near )~ - - 0 , and the rest of the spectrum is again contained in the open left half-plane, bounded away from the imaginary axis. The next theorem provides a method of computing the location of the g critical eigenvalues. THEOREM 5.3 [146]. Assume that Hypotheses 5.1 and 5.2 are met, then a ~ > 0 with the following properties exists. Assume that qe(~) is an g-pulse with wave speed ce and distances 2L l . . . . . 2 L e - l such that (5.4) is met. The equation d - - u - - [ O . f ( q e ( ~ ) . ce) + )~B(~)]u
d~
has a bounded nonzero solution u(~) f o r I~1 < 6 if and only if
D ( ) ~ ) - det[A - M)~ + R(/k)] - - 0 The g x g matrix A is tridiagonal and given by --al --bl A ~_.
al bl - a2 -b2
a2 b2 - a3
a3
",,
,,
-be-~
be-~
with aj - - ( ~ / ( L j ) . q ' ( - L j ) ) .
bj - - ( g / ( - L j ) , q ' ( L j ) )
f o r j = 1 . . . . . g - 1. The remainder term R()~) is analytic in )~ and satisfies
R0~) -- O(I)~le -pL + e 3PL). where L = minj=l ..... e-l {Lj}.
B. Sandstede
1028
We remark that [ 146] contains a stronger result with better estimates for the remainder term R 0 0 . The proof of Theorem 5.3 can be found in [146] (see also [147] for a less technical proof in R2). The idea of the proof is as outlined in Section 5.1.1 except that we get matching conditions for each individual pulse. The theorem can be generalized to the situation where the eigenvalue )~ = 0 of the primary pulse has higher geometric multiplicity due to additional continuous symmetries (see [148]). Theorem 5.3 states that the critical eigenvalues of the PDE linearization about an g-pulse are given as the eigenvalues of the tridiagonal matrix A, up to the factor M - l , provided we ignore the remainder term R(~). The matrix A can be thought of as the restriction of the PDE operator to the generalized eigenspace belonging to the critical eigenvalues. Note that
Av--O,
v-----(1. . . . . 1)*,
(5.14)
so that the vector (1 . . . . . 1)* corresponds to the translation eigenvalue ,k = 0 of the g-pulse. The entries of the matrix A are again scalar products similar to those encountered for the wave trains in Section 5.1. The only information needed to compute the entries of A are the distances 2Lj between consecutive pulses in the g-pulse. 5.2.1. Strategiesfor using Theorem 5.3. Hence, we first comment on a general approach, namely Lin's method [ 111,143], towards existence of multi-bump pulses that provides us in particular with the distances of bifurcating multi-bump pulses. Afterwards, we illustrate how the eigenvalues of A can be computed using the information gathered from the existence results. It has been proved in [111,143] that an g-pulse with wave speed c and distances {2Lj }j-I .....e-I bifurcates from the primary pulse if, and only if, the equation
M(c-co) = ( ~ ( - L j _ l ) , q ( L j _ l ) ) - ( ~ ( L j ) , q ( - L j ) ) + O ( e -3pL)
(5.15)
is satisfied for j = 1. . . . . g, where we set L0 = Le = oc and L = min{Lj }. We remark that [143,150] contain more general results with better estimates for the remainder term. We refer to Section 5.1.1 for the idea that leads to (5.15). Thus, once the tails of the pulse q (~) and the adjoint solution gt (~) are known, Equation (5.15) can be used to investigate the existence of multi-bump pulses. If such pulses exist, their half-distances L j can be fed into the matrix
A-
--al
al
-bl
bl --ae
ae
-b2
b2 - a 3
",
a3
(5.16)
".o
-be-i
be-1
with
aj =(O(Lj),q'(-Lj)),
bj --(~(-Lj),q'(Lj))
that determines the critical PDE eigenvalues and thus stability of the multi-bump pulses.
Stability of travelling waves
1029
Before we explore examples, we comment on general strategies for solving (5.15) and for computing the eigenvalues of the tridiagonal matrix A in (5.16). Let v s < 0 and v u > 0 denote the real parts of the leading stable and unstable eigenvalues of the matrix 0, f (0, co). Assuming that the leading eigenvalues are semi-simple, we have (gr(L), q ( - L ) )
- O(e-2~uL),
(~p( - L ) , q' ( L ) ) - O(e 2v~c)
(5.17)
(see Section 5.1.2). Typically, we have either ]vS[ > v u or IvS[ < v u so that one of the two scalar products in (5.17) is of higher order. Using appropriate scalings and utilizing the implicit-function theorem, it can be verified that the higher-order scalar product can be dropped from the existence Equation (5.15) and from the matrix A in (5.16) (see [145,146, 150]). The remaining equation is then much easier to analyse. Note that dropping either the entries aj for [vs[ < v u or else the entries bj for [vS[ > v u makes the matrix A either superdiagonal or subdiagonal: in either case, its eigenvalues are given by the entries on the diagonal, and stability of the multi-bump pulses can be determined by inspecting the signs of the scalar products bj or aj (see again [145,146,150]). We emphasize that the stability matrix A is sometimes truly tridiagonal: for instance, if the underlying PDE features the reflection symmetry x ~ - x , so that the travelling-wave ODE is reversible, and the pulse is symmetric, then the stability matrix A is symmetric with aj -- - b j [146]. Owing to the property (5.14), the signs of the eigenvalues of A can still be determined from the signs of the elements aj -- - b j (see [146, Section 5]). Note that reversibility implies in particular that ]vS[ = v u. We illustrate this approach by an example, namely multi-bump pulses for primary pulses q (~) that approach saddle-focus or bifocus equilibria as [~[ --+ cx~. Hence, we assume that the leading spatial eigenvalues of the matrix O, f ( O , co) are a pair of simple, complexconjugate eigenvalues v,, v,. Under this condition, infinitely many g-pulses bifurcate for each fixed g > 1. More precisely, there is a number L, >> 1 with the following property. For each choice of integers k l . . . . . ke-l E N, there is an integer k, E N such that a unique g-pulse with half-distances L j -- L , + (2k + kj)7c/[ Im v,[ exists, for a unique wave speed close to co, for every integer k > k,. The stability properties of the g-pulse described above are determined by the integers kj chosen above: define k_ - #{j; kj is even},
k+ -- #{j; kj is odd},
so that k_ + k+ + 1 = g, then the g-pulse has a simple critical eigenvalue at )~ = 0, k_ stable and k+ unstable critical eigenvalues, or vice versa (whether k+ equals the number of stable or unstable eigenvalues depends on the definition of L,; see [146] for more details). These statements can be proved upon substituting the expansions (5.13) into (5.15) to get M ( c - co) -- a sin(2Lj I Im v,] + b)e -2Ljl Re v*l (with either j = 0 . . . . . g - 1 or j = 1. . . . . g), upon neglecting terms of higher order. The eigenvalues of the stability matrix A can then be computed as outlined above, and we refer to [ 146, Section 6] for the details of the proof. Instead of giving these details, we focus on a fictitious travelling-wave ODE in R 3 and explore the geometric meaning of
1030
B. Sandstede
(a)
W?o~(O)
(b)
~ , , , , , , ,
I q2
Fig. 11. This plot illustrates the existence equation (5.15) and the stability result for 2-pulses that bifurcate from a primary pulse to a saddle-focus equilibrium in R 3 (see the main text).
the existence Equation (5.15) and the stability results mentioned above for 2-pulses (see Figure 11). Thus, suppose that we have two stable spatial eigenvalues v s ~ v s and one unstable spatial eigenvalue v u with 0 < - Re v s < v u. To see why 2-pulses exist for nearby wave speeds, we consider Figure 11 (a). First, we follow the two-dimensional local stable manifold WlSc(0) backward in ~ to get the manifold W~ar(0), which is again close to the equilibrium at u - 0 . Suppose we seek a 2-pulse with distance 2L between the two pulses: We shall follow the primary pulse, starting at ~ -- -cx~, until we reach q(L). We then vary the wave speed slightly such that the solution takes off and leaves WlSoc(0) to get caught by the manifold W~ar(0) near q(-L) (see Figure 1 l(b)). To examine for which values of L this approach works, we observe that the distance between WlSoc(0) and the lower boundary of W~ar(0) at q(-L) is much smaller than the distance between q(L) and u - 0 because of our assumption that IRe vsl < v u. Thus, one should think of the lower boundary of W~ar (0) at q ( - L ) as being extremely close to WlSoc(0) (this is not shown well in Figure 1 l(a)). For fixed L, we can catch the solution that leaves WlSoc(0) at q(L) using W~ar(0) provided q(L) is directly underneath Wfar(0), s i.e. provided q(L) lies on the dotted line in Figure ll(a). This means precisely that (7f(-L),q(L)) --0 since 7 : ( - L ) is perpendicular to Wfar(0) s at q(-L). Thus, if we have q(L) ql or q(L) = q2 (or if q(L) is any other intersection point of the pulse with the dotted line), then we expect that there is a 2-pulse with distance 2L for a slightly perturbed wave speed (see Figure 11 (b)). Next, we discuss the sign of the critical nonzero eigenvalue of the 2-pulse. By Theorem 5.3, the nonzero eigenvalue of the stability matrix A is given by bl -- ( ~ ( - L ) , q: (L)). Hence, if q(L) = ql, then bl < 0, whereas if q(L) -- q2, then bj > 0. This discussion provides some insight as to the geometric meaning of the entries of the stability matrix A. We refer to Section 5.2.4 for references where the approach outlined here has been used to analyse existence and stability of multi-bump pulses.
Stabili~ of travelling waves
1031
5.2.2. An alternative approach using the Evans function. A different approach to determining the critical PDE eigenvalues of multi-bump pulses is to compute the Evans function De()~) of an ~-pulse and to calculate its ~ zeros near ,k : 0. For 2-pulses, the idea is to calculate the derivative D[ (0)" since there is only one nonzero root of D2(X) near )~ - - 0 , we can determine the sign of this root from the sign of the derivative D~,(0) utilizing a parity argument as in Section 4.2. We refer to [3,4,122,123] for further details. Nii [ 124] used topological indices on projective spaces to compute all roots of the Evans function De()~) for g-pulses that bifurcate from doubly-twisted heteroclinic loops; these multi-bump pulses exist for any ~ > 1. (This stability result has also been obtained in [ 147] using Theorem 5.1 ). The approach using indices appears to be restricted to problems where the critical PDE eigenvalues are real. This makes a restriction to real ,k possible and allows the use of indices. We remark that the function D(~.) that appears in Theorem 5.1 is the Evans function De()~) of the ~-pulse. In many cases, it is more convenient to compute the critical eigenvalues of the stability matrix A directly rather than computing the roots of the determinant det[A - M)~]: the matrix A often exhibits a special structure (such as being superdiagonal or symmetric), which simplifies the computation of its eigenvalues, but this structure may not be visible in the determinant. 5.2.3. Fronts and backs. Up to now, we had looked into the stability of multi-bump pulses that bifurcate from a primary pulse. Frequently, multi-bump pulses can also be constructed by gluing fronts and backs together. In other words, they may bifurcate from heteroclinic loops that consist of connections between two different equilibria p l and p2 (see Figure 12). We need to distinguish between two different scenarios. If the equilibria pl and p2 are such that dim WU(pl) = dim WU(p2), which is the geometric configuration shown in Figure 12(a), then Theorem 5.1 and the results reviewed in Sections 5.2.1 and 5.2.2 readily generalize to cover multi-bump pulses
(a)
pl
(b)
p2
pl
P2
Fig. 12. Two heteroclinic loops comprised of a front qf(~) and a back qb(~)- The plotted manifolds are the stable and unstable manifolds of the respective equilibria.
1032
B. Sandstede
that consist of several alternating copies of the front and the back. We refer to [123,124, 147] for theory and applications. If, on the other hand, the geometry is as shown in Figure 12(b), so that dim WU(pj) = dim WU(p2) + 1
or
dim WU(pl) -- dim WU(p2) - 1,
then the situation is more complicated: since the Morse indices of the equilibria pl and p2 differ, one of them, say P2, has essential spectrum in the right half-plane. Thus, on account of the results in Section 3.4, both the front and the back are unstable. The theory outlined above is then no longer applicable. Nevertheless, it has recently been proved that the multibump pulses that converge to the stable equilibrium pl can be stable even though front and back are both unstable (see [125,154]). The reason for this unexpected behaviour is that the essential spectrum behaves rather strangely under matching or gluing [155]. We also refer to Section 6.3.2 for a related phenomenon. 5.2.4. A review of existence and stability results of multi-bump pulses and applications. Over the past decades, bifurcations to multi-bump pulses, and wave trains, have been the subject of numerous articles. Summaries of relevant results can be found in [28,55]. Here, we focus on those bifurcations for which the stability of the bifurcating multi-bump pulses has been analyzed. In Section 5.2.1, we have already mentioned Shilnikov's saddle-focus and bifocus bifurcation. The existence of multi-bump pulses has been studied in [170,73] (see also [41, 111 ]). The stability of the bifurcating 2-pulses, and certain 3-pulses, has been investigated in [3,4,187]. Stability results for e-pulses with arbitrary g > 1 can be found in [146]. Next, suppose that the travelling-wave ODE is reversible, which is the case when the underlying PDE exhibits the reflection symmetry x w-~ - x , and that the primary pulse q (x) is reflection invariant. If the asymptotic equilibrium u - - 0 towards which the pulse converges is a bifocus, i.e., if it has nonreal simple leading eigenvalues, then again infinitely many e-pulses exist for each g > 1 [25,27,42,82]. The stability of these multi-bump pulses has been analyzed in [ 145]. We refer to the theme issue [174] for applications. There are a number of bifurcations that require two parameters (the wave speed c and an additional system parameter) to be encountered and properly unfolded. Among them are the resonant, the inclination-flip and the orbit-flip bifurcation. All of these bifurcations lead, under appropriate conditions, to multi-bump pulses [ 186,32,86,104,143]. The stability of these pulses is studied in [ 126] utilizing Theorem 5.3. As mentioned in Section 5.2.3, doubly-twisted heteroclinic loops also lead to multibump pulses [39]: their stability has been investigated in [ 124,147]. This bifurcation occurs in the FitzHugh-Nagumo equation [40], and the resulting multi-bump pulses where found to be stable in [ 124,147]. Lastly, we again consider reversible travelling-wave ODEs. Bifurcations of codimension one that lead to multi-bump pulses include the reversible orbit-flip [150] and the semisimple bifurcation [ 188]. The stability of e-pulses that bifurcate at reversible orbit-flips has been analyzed in [ 150], and applications to fourth-order equations and parametricallyforced NLS equations that model optical fibers under phase-sensitive amplification can be found in [ 150] and [97], respectively. The instability of e-pulses to coupled NLS equations that admit a semi-simple bifurcation has been investigated in [189,190].
Stability of travelling waves
1033
5.3. Weak interaction of pulses One interesting feature of the stability matrix A that appears in Theorem 5.3 is that this matrix is tridiagonal. Since the j t h row of the matrix A is associated with the translation eigenfunction Q ' ( . - ~j) of the j th pulse in the multi-bump pulse, located at position ~j E R, it appears as if the individual pulses interact only with their nearest neighbours. This is indeed the case: Suppose that we substitute the initial condition
Uo(~) -- ~ Q(~ - ~j) j=l into the PDE, where Cj 6 ]R denotes the position of the j t h pulse (see Figure 13). We call such a function a pulse packet provided the distances between consecutive pulses are large, i.e., provided ~j+l - ~j ) ) 1 for j = 1 . . . . . ~ -- 1. If we solve the PDE with initial condition U0 (~), then it turns out that the shape of each individual pulse in the pulse packet is maintained; the time-dependence of the solution manifests itself only in the movement of the position of each pulse. In other words, the solution U (~, t) is, to leading order, given by
(5.18) j=l where the positions ~j(t) depend upon the time variable t. Using the above ansatz, it is also possible to derive ODEs that govern the evolution of the positions ~j (t) (see [52,117, 130,51,149]). The ODE that describes the interaction of the pulses in the pulse packet can be written as [149] d
1
dt ~j -- ~ ( ( O ( - L j - l ) , q ( L j _ j ) ) -
{O(Lj), q ( - L j ) ) ) + O(e-3pc),
(5.19)
where
Lo--Le=oc,
~l(t)
Lj =
~j+l --~j 2 ,
~(t)
j - - 1 . . . . . g~- 1,
~(t)
Fig. 13. A pulse packet consisting of three identical, well separated pulses. The positions of the pulses are denoted by the time-dependent coordinates ~1, ~2, ~3.
1034
B. Sandstede
and L = m i n j : j .....e-l{Lj }. The interested reader may wish to compare this equation to the existence Equation (5.15) for multi-bump pulses that was derived in [111,143]: the term c - co in (5.15), which is the wave speed of the j t h pulse in an g-pulse (measured relative to the primary pulse), is replaced by the wave speed Ot~j(t) of the j t h pulse in a wave packet (again relative to the speed of the primary pulse). Hence, once we know that the time evolution of pulse packets is, to leading order, given by (5.18), then the interaction Equation (5.19) can be derived by using Lin's method as outlined in Section 5.1.1. Equation (5.18) can be confirmed by proving the existence of a center manifold for the underlying PDE that is formed by pulse packets. This requires to establish normal hyperbolicity (which is a consequence of the stability results in [ 146]) and to utilize a cut-off function that acts only within a finite-dimensional approximation of the center manifold. We refer to [ 149] for details. Equation (5.19) has been derived rigorously in [51 ] using Liapunov-Schmidt reduction for the PDE and, simultaneously and independently, in [ 149] using a center-manifold reduction and subsequent Liapunov-Schmidt reduction for the flow on the center manifold. In fact, [149] contains an improved version of (5.19) that is applicable near homoclinic bifurcations of codimension two. We also refer to [26,62] for earlier results on the interaction of meta-stable patterns in scalar reaction-diffusion equations. Lastly, we mention that there are interesting relations between the interaction equation (5.19) and the nonlinear dispersion relation c(L) that relates the wave speed c and the wavelength L of the wave trains that accompany the pulse (see, e.g., [ 117,130]).
6. Numerical computation of spectra In many applications, it appears to be impossible, or at least very difficult, to investigate the existence and stability of travelling waves by analytical means. In such a situation, numerical computations are often the only way to obtain information about travelling waves. In this section, we summarize some theoretical results in this direction and provide pointers to algorithms and numerical software for the numerical computation of waves and their PDE spectra.
6.1. Continuation of travelling waves Pulses, fronts, and wave trains can be continued numerically as certain system parameters are varied, once a good starting solution is available for one set of parameter values (see the survey [14]). For pulses, the idea is to approximate the condition for having a pulse, namely that the pulse is contained in the unstable and stable manifolds of the equilibrium at u - - 0 (see [ 14,33,80,107]), i.e., u ( - L ) ~ Wl~c(O),
u(L) 9 WlSoc(0),
1035
Stabili~ of travelling waves
by a condition that is posed on a finite interval [ - L , L] to make it computable. For instance, if q (~) denotes the exact pulse, then a numerical approximation can be sought as a solution to the travelling-wave ODE u' = f (u, c)
(6.1)
on the interval ( - L , L) that satisfies the boundary conditions u
u ( - L ) ~ ToWlUoc(O)- E o ,
s
u(C) e ToWoc(O)- e o
(6.2)
together with the phase condition
f
C(q'(~),u(~) - q(~))d~ = 0 ,
(6.3)
L
which breaks the translation invariance and singles out a specific translate. Here, E~ and E~ denote the generalized stable and unstable eigenspaces of O, f (0, co). We remark that the exact pulse q (~) that appears in the phase condition (6.3) can be replaced by any reasonable guess for q (~). We refer to [ 12,61 ] for algorithms related to the continuation of pulses and fronts. Analogously, periodic waves of period 2L can be sought as solutions to (6.1) and (6.3) together with the boundary condition u(L) = u ( - L ) .
These algorithms can be implemented in boundary-value solvers such as AUTO97 (see [44]). In fact, AUTO97 computes the boundary conditions (6.2) for pulses automatically and also detects various homoclinic bifurcations that lead to multi-bump pulses (see [29, 44]). We refer to the survey [ 14] for more details related to the computation, and continuation, of travelling waves and to [8] as a general reference for numerical methods for boundary-value problems.
6.2. Computation o f spectra o f spatially-periodic wave trains Suppose that Q(~) is a wave train with period L so that Q(~ + L) = Q(~) for all ~. We had seen in Section 3.4.2 that ~, is in the spectrum 27 of the linearization about Q(~) if, and only if, the boundary-value problem d ~u d,~
= A(se; )~)u,
0 < ~ < L,
(6.4)
u(L) = eiyu(O)
has a solution u (~) for some 9/6 R. One possible numerical procedure to find all solutions to (6.4) is as follows.
B. Sandstede
1036
First, compute all solutions to (6.4) for y = 0. This can be done by discretizing the operator L; (or 7-) with periodic boundary conditions using, for instance, finite differences or pseudo-spectral methods [8], and to compute the spectrum of the resulting large matrix using eigenvalue-solvers (see, e.g., [7]). Note that, if we restrict to g = 0, Equation (6.4) describes precisely the eigenvalues of the wave train under periodic boundary conditions u(0) = u(L).
Second, once we have calculated all eigenvalues for y = 0, we can utilize continuation codes (e.g., AUTO97 [44]) to compute the solutions to (6.4) for y :/: 0 by using pathfollowing of the solutions for y = 0 in y. We refer to [159] for an example where this procedure has been carried out successfully. The advantage of this approach is that the spectrum is computed with high accuracy. Also, since the most interesting eigenvalues are those close to the imaginary axis or in the right half-plane, one would need to continue only a few relevant eigenvalues in y. We remark that the approach outlined above gives all eigenvalues only if, for each solution (F,, A,) of (6.4), there is a continuous curve (y, A(V)) of solutions to (6.4), parametrized by Y e [0, y,], such that A, = A(y,). If there is an island of solutions that is not connected to any eigenvalue at y = 0, then this island could never be reached by continuation in F- Fortunately, it is possible to prove that, for reaction-diffusion systems, such islands cannot exist: For bounded islands, this is a consequence of winding-number type arguments using the analyticity of the Evans function Dper(Y, A) in (F,A) (see (3.17)). Unbounded islands can be excluded upon using scaling arguments as in Section 4.2.2.
6.3. Computation of spectra of pulses and fronts We consider the operators 7-(A) -- ~ - A (~; A) and the associated eigenvalue problem d ~u d~
-- A(~; A)u
(6.5)
(see Section 3). Suppose that Q(~) is either a front or a pulse, so that there are n x n matrices A+(A) with IA(se" A ) - A+(A)I ~ Ke -pIll as ~ --+ -+-co for certain positive constants K and p that are independent of ~ and A (see Sections 3.4.3 and 3.4.4). As before, we denote by r = ~V'pt U ~r'es s the various spectra associated with 7-. We are interested in computing the spectrum using periodic boundary conditions (for pulses) or separated boundary conditions (for pulses or fronts). 6.3.1. Periodic boundary conditions. sider the operator
'~Lper(A) " Hpler((--L, L),
C")
Suppose that A+(A) = A_(A) for all A. We con-
> L2((-L,L),Cn),
u,
>
du d~
A(.'A)u,
1037
Stability of travelling waves
where the function space
Hler((-L,L),C n)-H I((-L,L),c n)O {u; u(-L)--u(L)}, yTPer the spectrum of ,*L f per and obencodes periodic boundary conditions. We denote by "-'L _per serve that ~L consists entirely of point spectrum (see, e.g., [155]). The next theorem yTPer converges to s as L -+ oo uniformly in bounded subsets of C. states that ~L
THEOREM 6.1. Assume that Hypothesis 5.3 is met. 9 Fix an eigenvalue )~, with multiplicity ~ in r p t . As L --+ ec, there are precisely g. yTPer , counted with multiplicity, close to )~,, and these elements converge elements in "-'L to )~, as L --+ oc. In other words, isolated eigenvalues o f the pulse are approximated yTPer , counting multiplicity [13,155]. by elements in "-'L 9 Fix )~, ~ Z'ess, then, under an additional technical assumption [155, Hypothesis 6], .--.per )~, is approached by infinitely many eigenvalues in z, L as L --+ ec [155]. 9 Fix a bounded domain I2 C C. For any 6 > O, an L , exists such that ( r per A S2) C Lt~(S) f o r a l l L > L , [155]. See, for instance, [ 191 ] for numerical computations that corroborate this statement. 6.3.2. Separated boundary conditions. Recall that we consider a pulse or a front. Separated boundary conditions can be realized by choosing appropriate subspaces E be and Eb_c of C n . We then consider the operator 7";eP (~.) " Hslep ((-- L, L), C n )
> L2((-L, L), C"),
u,
~
du
dse
A(.'2.)u,
where the function space
Hslep((-L, L), C") -- H' ( ( - L , L), C") N {u" u ( - L ) ~ Eb_c and u(L) ~ E+bc }, encodes separated boundary conditions. HYPOTHESIS 6.1. We assume that the following conditions are met. 9 A number p > 0 and an integer i~c ~ N exist such that, f o r all )~ with Re ~ ~ p, the asymptotic matrices A+()O are hyperbolic, and the dimension o f their generalized unstable eigenspaces is equal to iec. 9 The subspaces E bc satisfy dim E bc -- codim Eb+c -- i oc.
In other words, for separated boundary conditions, the integer ioc is singled out as the number of boundary conditions at the right endpoint of the interval ( - L , L); observe that the number of boundary conditions at ~ - + L is equal to the codimension of E be. The integer i~c is also equal to the asymptotic Morse index of the matrices A+ (i.) as Re i. --+ cx~. yTsep ,fsep .--.sep We denote by ._.c the spectrum of./~ and remark that 2,c consists entirely of point .-.sep spectrum (see, e.g., [ 155]). It turns out that the spectrum ~L does not resemble the spectrum • of T but an, in general, entirely different set that we shall describe next.
1038
B. Sandstede
We label eigenvalues of A+(A) according to their real part, and repeated with their multiplicity, Re v~(A) >~-.. >~ Re vi:-,c + (A) >~ Re vi~+1 + (A) ~>-.->~ Re v,~(A) We can now define the so-called absolute spectrum of 7- [155]. DEFINITION 6.1 (Absolute spectrum). We define + 27abs "-- {A 6 C"' Re vlog + (A) = Re vlOG + 2r- l -(A)-|
.
and, analogously, 27gs = {X 6 C; Re vi-~ (A) = Re vi-+ , (A)}. The absolute spectrum 27abs of 7" is the union of 27a;s and 27~s" Next, suppose that A ~ 27abs, SO that there is a gap, in the real part, between the spatial eigenvalues of A+(A) with indices i ~ and ioc + 1, i.e., so that Re v~ 0v) >~-.->~ Re vi~c + (X) > 7• > Re vioc+l • (A) >/--. >/Re v,~00 for some rl+ --rl+(A) (see Figure 3). We denote by E~(A) and E~(A) the generalized eigenspaces of A+(A) associated with the spectral sets {v~-. . . . . v/~} and {V+i e c + l , ' ' ' , vn+ 'J, respectively. Owing to the presence of the spectral gap at Re v = ~+, Equation (6.5) has exponential dichotomies~ on R + for every )v ~ 27abs with pro~ections P+(~" A) such that N(P_(~" A)) --+ Eu_(A) as ~ -+ - c o and R(P+(~" A)) ---, E~_(A) as ~ --, oc (see Section 3.2). We define the analytic functions Dsep(A) : = N(P_(0; A)) A R(P+(0; A)), +
bc
D~c (X):= E bc A ~7s (A),
"u
Dbc(A ) "-- E+ A E+(X) (see Section 4.1 for this notation) that measure eigenvalues or resonance poles of the underlying wave as well as transversality of the boundary conditions with the pseudo-stable and pseudo-unstable eigenspaces that we introduced above. DEFINITION 6.2 (Pseudo-point spectrum). We define 2 7 p t - { A ~ 27abs" ~ s e p q - ~ - - + - ~ + > 0 }
where s and [• denote the order of A as a zero of the functions Dsep and D+ defined above. We call ~ -- ~sep 2r- ~- -+- ~+ the multiplicity of A for A 6 r p t . The next theorem states that the spectrum yTsep "-'L does not approximate the spectrum 27 -~V'pt U ~V'ess of 7- but the set 27pt U ~V'abs.
Stability of travelling waves
1039
THEOREM 6.2 [155].Assume that Hypotheses 5.3 and 6.1 are met. 9 Fix an eigenvalue A, with multiplicity g~ in rpt. As L --+ cx~, there are precisely g~ .-.sep elements in 2.;L , counted with multiplicity, close to A,, and these elements converge to A, as L --+ r 9 Fix A, E ~V'abs, then, under additional technical assumptions [155, Hypotheses 7 yT,sep and 8], A, is approached by infinitely many eigenvalues in "-'L as L --+ cx~. 9 Fix a bounded domain Y2 C C. For any ~ > O, there is an L , such that ( r L ep O ~Q) C /at3 (~V'pt O ~V'abs)for all L > L , . Hence, eigenvalues on large bounded intervals under separated boundary conditions are created via two different mechanisms: First, eigenvalues are created whenever the spaces that encode the boundary conditions are not transverse to the pseudo-stable or pseudounstable eigenspaces which are related to the two spatial spectral sets associated with the number of boundary conditions. Second, eigenvalues arise as zeros of the Evans function Dsep(A) that is again related to the aforementioned pseudo-stable or pseudo-unstable eigenspaces. We emphasize that the sets rpt and ~V'pt coincide to the right of the essential spectrum tess because of Hypothesis 6.1. The absolute spectrum is typically to the left of the essential spectrum. We remark that, if the underlying PDE is reflection invariant, then rpt and tabs are typically equal to rpt and Z:ess, respectively, except possibly for additional eigenvalues that are created on the bounded interval through nontransverse boundary conditions.
7. Nonlinear stability In this section, we consider nonlinear stability of travelling waves. Suppose that Q(.) denotes a travelling wave that is spectrally stable, so that the spectrum of the linearization s of the PDE about the wave Q (-) is contained in the left half-plane. We are then interested in the stability of the wave Q(.) for the full PDE. Since there is an entire family of waves, namely Q(-) together with its translates Q(-4- r), we say that the wave is nonlinearly stable if, for any initial condition U0(-) sufficiently close to Q(.), the associated solution U (., t) stays near the family {Q(. 4- r); r E •} for all t > 0. More precisely, we have the following definition. DEFINITION 7.1 (Nonlinear stability). We say that a travelling wave Q is nonlinearly stable if, for every e > 0, there is a 3 > 0 with the following property: if U0 is an initial condition in U~(Q), then the associated solution U(., t) satisfies U(., t) E Uc({Q(. + r); ~ E ~}) for t > 0. We say that Q is nonlinearly stable with asymptotic phase if, for each U0 as above, a r, exists such that U (-, t) --+ Q (. + r,) as t --+ cx~. Note that we have not yet mentioned any function spaces or norms. Often, one would have to measure the various neighbourhoods that appear in Definition 7.1 in different, not necessarily equivalent, norms.
B. Sandstede
1040 (b)
T
I
(d)
_.Jl
Fig. 14. The spectrum of the linearization /2 about a spectrally stable wave is shown in (a) if/2 is sectorial, and in (b) if/2 generates a C0-semigroup. In (c) the spectrum of the sectorial linearization about a marginally stable wave is plotted, whereas (d) contains the spectrum if/2 -- j ~ t t (Q) comes from a Hamiltonian PDE (see Section 8).
N o n l i n e a r stability properties d e p e n d strongly on the nature of the PDE, in particular, on the properties of the l i n e a r i z a t i o n / 2 about the travelling wave Q (see F i g u r e 14). T h r o u g h out the r e m a i n d e r of this section, we consider a P D E of the form Ut = A U 4- A/'(U).
(7.1)
We a s s u m e that .,4: X --+ X is a d e n s e l y defined, closed operator, w h e r e A" is an appropriate B a n a c h space.
Sectorial operators.
S u p p o s e that the operator A is sectorial: its r e s o l v e n t set contains the sector {~. 6 C; Re ~. > a - b l Im)~l} for s o m e a 6 IK and s o m e b > 0, and the r e s o l v e n t of A satisfies an estimate of the f o r m K
IX-al for )~ in the above sector. We refer to [85] for m o r e b a c k g r o u n d on sectorial operators as well as for the results m e n t i o n e d below. A s s o c i a t e d with the sectorial o p e r a t o r A are its fractional p o w e r spaces X ~ : we have X ~ = X and ,~1 = ~)(,A), and X ~ with ot 6 (0, 1) interpolates b e t w e e n these two spaces. If, for s o m e oe 6 [0, 1), the nonlinearity A f : X ~ X is differentiable, then we can solve (7.1) in X ~ for any initial condition in X ~ (see [85, Section 1]). S u p p o s e that Q(.), together with its translates Q ( - + ~:), is a travelling-wave solution to a P D E that can be cast in the above fashion. 11 We then have the following result [85, Section 5.1 ] that can be briefly stated as spectral stability implies nonlinear stability with asymptotic phase. M o r e precisely, if the s p e c t r u m Z of the o p e r a t o r / 2 = .,4 -4- OuA/'(Q), p o s e d on X , satisfies Z \ {0} C {)~; Re)~ < - 8 } 11 At this point, there is always some restriction by going from the abstract framework (7.1) to concrete applications to travelling waves where A is differential operator posed on a function space A" such as L2(R, RN).
Stability of travelling waves
1041
for some 8 > 0, and if )~ = 0 is a simple eigenvalue of/2, then the travelling wave Q is nonlinearly stable with asymptotic phase (see Definition 7.1 applied to A'~). We refer to [85, Section 5.1] for the proof that uses a center-manifold reduction (see also [56,160]). The above result is, for instance, applicable to the reaction-diffusion system (2.10) provided the diffusion matrix is strictly positive [85]. Next, we consider the situation where the operator .,4 generates only a C~ on A2 (see, e.g., [133, Section 1] for sufficient and necessary conditions on A). Consequently, the PDE (7.1) has mild solutions in R" for each initial condition in R" provided N': R" ~ ,g is differentiable [ 133, Section 6]. Suppose again that Q(.), together with its translates Q(. + r), is a travelling-wave solution to the PDE (7.1). Assume that the linear semigroup e z;t that is associated with the linearization/2 = .,4 + OuN'(Q) and posed on ,g has a simple eigenvalue X = 1, and the rest of its spectrum is contained inside the circle of radius e -~t for t > 0 for some fixed 8 > 0. Under this assumption, the travelling wave Q is nonlinearly stable with asymptotic phase. We refer to [9] for a proof that uses center-manifold reduction. This nonlinearstability result is applicable to the reaction-diffusion system (2.10) if the diffusion matrix D is non-negative; we refer to [54] and [9] for this and other applications. The main difficulty in applying the above nonlinear-stability result is that the spectrum of the linear semigroup e z;t is not necessarily computable using the spectrum of its generator/2: the spectral theorem is not true for generators of C~ (see, e.g., [133, Section 2.2] for counterexamples). If, however, the generator/2 of a C~ satisfies a resolvent estimate of the form C~
] ( / 2 _ X ) - l ] ~ rl for some fixed 7/6 IR and K, then the semigroup satisfies liez;t I] <~ Ce 'Tt owing to a result by Prtiss [ 138, Corollary 4]. This criterion can be used to establish estimates for a semigroup using the spectral information for its generator. We refer to [97] for an application to a perturbed NLS equation; in fact, the result in [97] applies more generally to coupled NLS equation with arbitrary bounded potentials in several space dimensions. Essential spectrum up to iR.
We comment on various problems where the essential spectrum of the relevant operator/2 touches the imaginary axis at X = 0 in a quadratic tangency. This situation occurs naturally when considering spatially-periodic travelling waves (see Section 3.4), shock waves in conservation laws (see [ 192] and references therein) or fronts that connect stable to unstable rest states (see [ 162]). In all these cases, it becomes necessary to introduce polynomial or even exponential weights in the space or time variables. The nonlinear stability of spatially-periodic waves has been investigated for the Ginzburg-Landau equation [17,35], the Swift-Hohenberg equation [166] and reactiondiffusion systems [168]. In higher space dimensions, nonlinear stability has been demonstrated for Taylor vortices [167] and for roll solutions to the Swift-Hohenberg equation [ 178]. We also refer to [50] where the nonlinear stability of periodic patterns in the SwiftHohenberg equation is studied using invariant manifolds that discriminate between different algebraic temporal decay rates.
1042
B. Sandstede
Fronts that connect stable to unstable rest states occur in many equations. Consider, for instance, the scalar reaction-diffusion equation Ut = Uxx + F ( U ) ,
x 6 R.
The aforementioned fronts arise in the Kolmogorov-Petrovsky-Piskunov (KPP) equation, where F ( U ) - U(1 - U), and in the real Ginzburg-Landau (GL) equation, where F ( U ) = U(1 - I U I 2 ) . In fact, there is typically a continuum of fronts parametrized by their wave speeds; the wave speed of the slowest wave is denoted by c,. In the noncritical case (c :fi c,), it is possible to use exponential weights in the spatial direction to stabilize these fronts as demonstrated by Sattinger [ 162]. More refined estimates have been obtained in [93] using polynomial weights and resolvent estimates. In the critical case (c -- c,), nonlinear stability has been proved in [ 18] for the GL equation using renormalization-group techniques, in [102] for the KPP equation, and in [49], using Liapunov functionals, for general nonlinearities. In [63], optimal temporal decay estimates were obtained for general nonlinearities using renormalization-group methods [ 19]. We refer to [103,139,184] for additional references. An interesting problem in this context is which of the fronts (or, alternatively, which wave speed) is selected by a given initial condition. We refer to [48,184] for references regarding this issue. The nonlinear stability of fronts that connect spatially-periodic states has been investigated in [ 18,64] for the real Ginzburg-Landau equation using energy estimates. Energy estimates have also been used in [65,66] to establish the stability of fronts in damped hyperbolic equations. Lastly, the nonlinear stability of certain viscous shocks has been demonstrated in [92, 93] using polynomial weights and resolvent estimates. More general results can be found in [ 192] where pointwise estimates were utilized; [ 192] also contains many references to different approaches towards the stability of viscous shock profiles.
8. Equations with additional structure In this section, we give pointers to the literature for methods that are applicable to PDEs with additional structure such as Hamiltonian, monotone, and singularly-perturbed equations. Hamiltonian PDEs.
Consider an abstract evolution equation of the form
Vt -- J ~ " ( V ) ,
(8.1)
posed on a Hilbert space X, where the differentiable functional E:A" --+ IR is thought of as the energy, and J is a skew-symmetric invertible linear operator, i.e., J * - - J . Suppose that the functional C is invariant under a group ,_q(r), with r 6 R, of unitary operators, so that ~'(,_q(r) V) = ~'(V) for all V 6 ,~' and r 6 R. Such a group of symmetries generates another conserved functional that we denote by/C. The functional/C is given explicitly by
lc(v) =
s'(o) v, v)
Stability of travelling waves
1043
where ,S'(0) is the generator of the group ,S(r). Of interest are then solutions to (8.1) of the form V ( t ) -- $ ( c o t ) Q for some fixed Q E X'. Transforming Equation (8.1) using V (t) -- ,.q(cot)U (t), we obtain u, - y[E'(u)
- ~o~'(u)] - yn"
(8.2)
(u)
which is Hamiltonian with energy 7-/o)(U) -- g ( U ) - co/C(U), where co E R is a parameter. We seek stationary solutions Qo) of (8.2), i.e., critical points of 7-/o). Thus, throughout this section, we assume that Qo) is a critical point of the energy 7-/o), parametrized by co in a certain interval. Note that (8.2) is still equivariant under the group ,S(r) so that we expect group orbits {,S(r)Qo)" r E R} of stationary solutions for every fixed co. We assume that ,_q'(0)Qo) --/:0 so that the group orbit is nontrivial. As an example, consider the Korteweg-de Vries equation in Example 3. The symmetry group ,S(r) are the translations, U(.) v-~ U(. 4- r), and the parameter co is the wave speed c: solutions $(cot) Q(.) are travelling waves Q ( x - cot). In particular, they arise as families Q(. 4- r), parametrized by r E R, provided Q(.) is not a constant function. The linearization of (8.2) about Qo) is given by s - ,.77-['~ (Qo)). We expect that the essential spectrum of s resides on the imaginary axis since ,.7 is skew-symmetric. In particular, the waves Qo) will be at most marginally stable. Hence, to conclude nonlinear stability, we need to exploit the Hamiltonian nature of (8.2). If 7-t~'o(Qo)) is positive definite 12 except for the simple eigenvalue ~. -- 0, then the wave Qo) is nonlinearly stable because of conservation of energy: the Hamiltonian 7-{o) serves as a Liapunov functional. In many important applications, however, the energy will not be definite. Instead, the Hessian 7-/~(Qo)) has a unique simple eigenvalue with negative real part, so that the energy increases in all directions but one (not counting the neutral direction caused by symmetry). To compensate for this, we exploit that/C is a conserved functional and restrict (8.2) to the invariant hypersurface C - {U E A'; /C(U) -/C(Qo))}, with co fixed. If U - Qo) minimizes the energy 7-/o) locally, subject to the constraint U E C, then the family Qo) is nonlinearly stable. It suffices therefore to find conditions that guarantee that Qo) is a constrained minimizer of 7-/o) for fixed co. Alternatively, we need that the Hessian 7-/'(Qo)) restricted to the tangent space/C' (Qo))2_ of C at Qo) is positive definite except along the eigenfunction 8'(0) Qo). This issue has been analyzed in [78,79]" the criterion that guarantees that Qo) is a constrained minimizer is that
d [K~(QO))]-(7-{~(Qo))Oo)Qo) Oo)Qo))< 0
dco
(8.3)
which, in fact, is equivalent to d" (co) > 0 where
d(o2)- g~(Q~). In fact, a stronger result is true: Qo) is nonlinearly stable if, and only if, the function d(co) is convex (see [78,79]). We refer to [78, Section 3] and [ 115, Section 2] for short proofs of the sufficiency of (8.3), and to [ 108,114-116] for constrained minimization techniques and 12 This statement is, of course, also true if
~o'j(Q~o) is negative
definite.
B. Sandstede
1044
E'(Q)• ~
x+
//~' (Q)
~
I
E' (Q.~)'
/7-/" (Q)/~' (Q)•
O~Q~ ~ R(7/"(Q)) Fig. 15. Various subspaces of R ( H ~ (Qco)) are plotted in the left figure under the assumption that (8.3) is satisfied (the subscript o; is omitted in the plot). The subspaces A'_ and X+ are the unstable and stable eigenspaces of 7-/~(Qco) on which this operator is negative and positive definite, respectively. The center plot illustrates that OcoQcois mapped to KS'(Qw)= 7-t~o(Qco)OwQcounder 7-/"oj(Qco)" this is true since Qco are critical points of 7-/co (the arrows indicate how 7-/~ (QoJ) acts on vectors). As a consequence, it is easily seen that the image of the space /C'(Qoj) • is as shown in the right figure. Thus, (7-t~(Qo~)U, U) > 0 for any U ~/C~(Qco) •
their relation to stability issues. Condition (8.3) decides upon definiteness of the Hessian of 7-/~o restricted to the space/C'(Qo~) • It turns out that, if (8.3) is met, then Q~o is also a constrained minimizer of the original Hamiltonian C. The geometric meaning of the criterion (8.3) in the plane is illustrated in Figure 15. If condition (8.3) is not met, then the wave is not a constrained minimizer; in fact, the linearization s about the wave has unstable eigenvalues (this statement requires an additional analysis for which we refer to [78,79]). In fact, analogous results are true for arbitrary finite-dimensional symmetry groups [79] and for general constrained minimization problems [ 115]; the parameter co is then a finitedimensional vector. As mentioned in Section 4.2.1, the criterion in (8.3) is related to the second-order derivative D"(O)of the Evans function D(Z). This relationship has been explored first in [134,135] and has been put into an abstract framework in [21,22] by exploiting a multi-symplectic formulation of the PDE, using symplectic operators for time and space. We refer also to [20] for a condition similar to (8.3), with co ~ R e, that detects transverse instabilities in Hamiltonian PDEs with two-dimensional spatial variables. Often, the spectrum of 7-/"(Q) is relatively easy to compute. An interesting and important problem is then to infer as much as possible about the spectrum of the operator s = JT-/"(Q). We refer to [76,77] for a very general approach to this problem and to [6, 109,189,190] for applications. Note that the optimal stability result that one can hope for in the context of Hamiltonian systems is nonlinear orbital stability without asymptotic phase. In Hamiltonian PDEs on unbounded domains, it is sometimes possible to obtain asymptotic stability of travelling waves to Hamiltonian system by switching to a different norm that is not equivalent to the original one. This program has been carried out in [135] for the Korteweg-de Vries equation. We remark that the stability of e-pulses in Hamiltonian PDEs is an interesting problem. Spectral stability can again be established using the results in Section 5. Nonlinear stability, however, is often far more complicated and, in general, unsolved. The reason is that, if we switch from the primary pulse to an e-pulse, there are s rather than only one, negative eigenvalues of the operator 7-["(Qe). To compensate for this, we would need to have e independent conserved functionals/Cj with associated parameters coj, and the sta-
Stability of travelling waves
1045
bility of an g-parameter family of solutions Q,o can be concluded provided the g x g matrix with elements (7-['~(Q~o)O~oiQ,o, O~ojQ~o) is negative definite. This program has been carried out in [ 115] for the multi-solitons of the Korteweg-de Vries equation: this is possible since the Korteweg-de Vries equation is integrable and admits infinitely-many independent conserved quantities. We also refer to [ 116] for a survey on constrained minimization and stability in Hamiltonian systems. If the equation has only finitely many independent conserved functionals (or if other conserved quantities are not known), then it is not clear how to proceed to establish nonlinear stability of multi-bump pulses.
Monotone (order-preserving) PDEs. Consider a reaction-diffusion system of the form Ut= DU~ +cU~ + F(U),
C It~, U E R N.
(8.4)
In many applications, e.g., to problems arising in combustion theory, the equation is monotone, i.e., the nonlinearity satisfies
aF~
~(U)
> O,
i--~ j.
Under this assumption (in fact, under weaker conditions), the existence and stability of monotone fronts can often be proved. We refer to the monograph [182] and to [37,141] for theory and applications. In fact, [182] is mainly concerned with results for Equation (8.4) where the spatial variable ~ lives on an unbounded cylinder, i.e., where ~ ~ R x s for some bounded domain I2 C R m.
Singularly perturbed reaction-diffusion systems.
Singularly-perturbed reaction-diffusion
systems of the form 8rUt = 82Uxx -+- f
(u, v),
(8.5)
vt ~- (~21)xx + g(u, v)
often allow for the construction, and the stability analysis, of travelling waves utilizing the equations in the singular limit as e --+ 0. Travelling waves can be constructed near the singular limit using geometric perturbation theory (see [91] for a recent survey)or using asymptotic matching (see [176,112]). The stability of these travelling waves can be investigated using several different methods: One possible approach is rigorous asymptotic matching (see again [112]). A second possibility is to use the Evans function [2]. In many cases, the elephant-trunk lemma [71 ] allows to write the Evans function for the full problem (8.5) as the product of the Evans functions to the slow and fast subsystems of (8.5) in the s = 0-limit. We refer to [15,71, 142] for applications. Sometimes, the slow and fast system interact strongly, so that the Evans function is no longer computable as a product. In this situation, new and interesting phenomena occur, and we refer to [47] and [87] and the literature therein for further details. Using the approach from [47], the stability of spatially-periodic travelling waves that
1046
B. Sandstede
continue singular periodic waves has recently been investigated in [53] for the FitzHughNagumo equation. Lastly, a different approach is the SLEP method introduced in [ 128, 129]. We refer to [127] for an extensive review of the SLEP method. The SLEP method and the approach via the Evans function are related [88].
9. Modulated, rotating, and travelling waves We briefly report on extensions and generalizations of some of the theoretical results reviewed in the earlier sections. Waves in heterogeneous media. Most of the results in this survey are concerned with homogeneous media, so that the underlying PDE has no explicit dependence on the spatial variables. Waves can also occur in heterogeneous media, and we refer to the recent survey [1841. Travelling waves in cylindrical domains. We focused on travelling waves for a onedimensional spatial variable. Often, however, one would be interested in parabolic equations Ut = Uxx + A U + F ( U ) ,
(x, y) E IR x a-d,
on unbounded cylindrical domains with bounded, or unbounded, cross-section X2 C ]t~ m 9 Here, A is the Laplace operator acting on the y-variable. In a moving frame ~ = x - ct, travelling waves become solutions Q(~, y) to the elliptic problem U ~ + A U + cU~ + F ( U ) = O,
(~, y) ~ R x ~ .
The associated linearized eigenvalue-problem is given by U~$ + A U + cU~ + Ou F ( Q ( ~ , y ) ) U - ~.U.
We refer to [67,84,57,182] and the references therein for various existence results. Methods that have been used to establish existence include the Conley index and spatial discretizations, the Leray-Schauder degree, and comparison principles (i.e., the construction of upper and lower solutions). Stability results can be found in [182] and in the comprehensive list of references therein. Note that, using the reformulation
(
o - A + )~ - OuF(Q(~, .))
as a first-order system, most of the methods and techniques reviewed in this article are also applicable to PDEs on cylindrical domains. We refer to [ 113,137,157] for details. This approach is based upon using the spatial variables in the unbounded directions as evolution variables. This concept, often referred to as spatial dynamics, has been introduced
1047
Stability of travelling waves
by Kirchg~issner [ 101 ] to investigate small-amplitude solutions. We refer to [ 118,120] and references therein for many subsequent articles where spatial dynamics has been utilized. Modulated waves. Modulated waves are solutions that are time-periodic in an appropriate moving coordinate frame, i.e., solutions Q(x, t) that, for some wave speed c and a certain temporal period T, satisfy Q(x, t 4- T) = Q(x - cT, t),
t, x E N.
Such waves may arise through Hopf bifurcations (when a pair of isolated complexconjugate eigenvalues crosses the imaginary axis) or essential instabilities (when a part of the essential spectrum crosses the imaginary axis [ 152,153,158]). Another example are travelling waves in modulation equations such as the Ginzburg-Landau equation which often correspond to modulated waves of the PDE that has been reduced to the modulation equation (see, e.g., [169,179] for stability results in this context). Most of the results presented in Sections 3-6 are also applicable to modulated waves, and we refer to [ 157] for details. The key is that exponential dichotomies, the main technical tool that we exploited, can also be constructed for linearizations about modulated waves. As an example, consider again a reaction-diffusion system U t = D U ~ +cU~ + F ( U ) ,
~ ER,
in an appropriate moving frame. The linearization about a modulated wave Q(~, t) with Q(~, t 4- T) = Q(~, t) is given by Ut = D U ~ +cU~ + O u F ( Q ( ~ , t ) ) U .
Note that the coefficients of this equation are T-periodic in t. It turns out that the eigenvalues )~ of the T-map of the linearization can be characterized by exponential dichotomies to the first-order system
(
o D - ' (O, + ot - 8u F ( Q ( ~ , .)))
id _cD-1
)(U)
where ,k = e aT and (U, V) is T-periodic in t for every ~ c R. The above system is of the same form as the equations that we studied in the earlier sections except that, for each fixed ~, (U, V) take values in a certain Hilbert space of T-periodic functions that depend on t instead of in C 2N. We refer to [157] for details. Rotating waves in the plane. Ut = D A U + F ( U ) ,
Consider the reaction-diffusion system U c IRN, x E R 2,
(9.1)
on the plane. A rotating wave is a solution U (x, t) whose time-evolution is a rigid rotation with constant angular velocity c. Expressed in polar coordinates (r, q)), a rotating wave
1048
B. Sandstede
is therefore of the form U (r, ~0, t) -- Q(r, ~p - ct). In a co-rotating coordinate frame, and using polar coordinates (r, ~0), Equation (9.1) is given by Ut = DAr,~oU 4- cUp 4- F ( U ) ,
(9.2)
x E R 2.
A rotating wave with angular speed c is a stationary solution to (9.2). Examples of rotating waves are Archimedean spiral waves that are stationary solutions Q(r, ~o) to (9.2) for an appropriate value of the angular velocity c such that Q(r, ~p) --+ Q ~ ( K r 4- ~p) as r --+ oe for some 2Jr-periodic function Q ~ (Tt). The function Q ~ (Tt) is a stationary wave-train solution to Ut -- DK2U~pT~ 4- cU~ 4- F ( U ) ,
(9.3)
4/E •.
We refer to [121,176] for background on spiral waves and various other waves in two and three space dimensions. We cast (9.2) as a dynamical system in the radius r:
V~
- -
-
U~o~o +-T+
D - l(cU~o4- F ( U ) ) ] ) "
Spiral waves Q(r, ~o) can then be thought of as fronts in the radial variable r that connect the core state Q(0, ~0) at r = 0 with the r-periodic wave train Q ~ ( x r 4-cp) as r --+ oo. We refer to [ 165] where this approach has been introduced to investigate Hopf bifurcations from homogeneous rest states to spiral waves with small amplitude. To investigate the stability of spiral waves, consider the linearization of (9.2) about the spiral wave Q, written again as a first-order system in the radius r: (U,-) Vr
( - -
-- O~f2~
0 D -1 (cOco + OU F ( Q ( r , tp)) - ~.)
_
id ) ( U ) r 1
9
(9.4)
This equation can again be investigated using exponential dichotomies (see [159]). Note that the limit of (9.4) as r --+ oo is related to the linearization of (9.3) about the asymptotic wave train Q ~ . In particular, the essential spectrum of the spiral can be computed using the essential spectrum of the asymptotic wave train Q ~ . For earlier results on the stability of spiral waves, we refer to [81] and to the review [176]. Various bifurcations of spiral waves to more complicated waves have been investigated in the literature, and we refer to [56,58,74,160,161 ] and the references therein for further details.
Acknowledgment I am grateful to Bernold Fiedler, Arnd Scheel and Alice Yew for helpful comments and suggestions on the manuscript.
Stability of travelling waves
1049
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1052
B. Sandstede
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1053
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1054
B. Sandstede
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Stability o f travelling waves
1055
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Author Index Roman numbers refer to pages on which the author (or his/her work) is mentioned. Italic numbers refer to reference pages. Numbers between brackets are the reference numbers. No distinction is made between first and co-author(s).
Abarbanel, H. 374, 385 [1] Abbott, L. 13, 16, 22-24, 51, 52 [32]; 53 [65]; 123, 146 [90] Abergel, E 806, 829 [1] Ablowitz, M. 312 [18]; 631,669 [1] Abounouh, M. 909, 972, 974 [1]; 974 [2] Abraham, R. 321,322, 332, 344 [1]; 354, 371,385 [2]; 514, 590 [11 Ackers, C. 41, 51 [ 1] Adams, W. 105, 143 [1] Adelmeyer, M. 770, 832 [71 ] Adler, R. 371,385 [3] Afendikov, A. 764, 779-782, 795, 796, 811, 814, 829 [2]; 829 [31; 829 [4]; 829 [51 Ahlers, G. 338, 344 [5] Akhmediev, N. 985, 990, 1049 [ 1] Alekseev, V. 272, 311 [ 1] Alexander, J. 371, 385 [4]; 989, 1006, 1008, 1009, 1028, 1029, 1031, 1032, 1044, 1045, 1049 [2]; 1049 [3]; 1049 [41; 1049 [51; 1049 [61; 1054 [1501 Alikakos, N. 713-715, 719 [1]; 719 [21; 719 [3]; 719 [41; 719 [51; 719 [61; 851,876 [11; 876 [2] Allgower, E. 151,216 [1]; 379, 385 [5] Amann, H. 840, 842, 848, 851, 876 [31; 876 [41; 876 [5]; 876 [61 Amzica, F. 95, 139, 142, 146 [75]; 146 [76] Andelman, D. 704, 721 [801 Anderson, E. 1036, 1049 [7] Anderson, R 649, 669 [2] Andreucci, D. 735, 755 [1] Angenent, S. 756 [2]; 853,855,857, 858, 862, 876 [7]; 876 [8]; 876 [9]; 876 [101; 876 [lll; 889, 954, 958, 974 [3]; 974 [4] Ankiewicz, A. 985,990, 1049 [ 1] Anosov, D. 315, 318, 332, 344 [2] Aoki, K. 590 [2] Aranson, I. 339-341,344 [3]
Aref, H. 58, 90 [1] Argoul, F. 374, 385 [6] Aris, R. 365,388 [101]; 837, 876 [12] Arneodo, A. 374, 385 [6] Arnold, V. 298-300, 311 [2]; 376, 385 [7]; 514, 526, 590 [3]; 613,616, 661,669 [3]; 669 [4]; 669 [5] Aronson, D. 17, 18, 51 [2]; 365,385 [8] Arrieta, J. 960, 962, 963, 974 [5] Asano, K. 591 [4] Ascher, U. 157, 216 [2]; 217 [3]; 348, 350, 359, 363, 385 [9]; 385 [10]; 1035, 1036, 1049 [8] Aston, R 771,829 [6] Aubry, S. 650, 669 [6] Auerbach, D. 340, 341,344 [4] Aulbach, B. 865, 877 [13] Auslander, J. 887, 974 [6] Avez, A. 613,669 [5] Avrin, J. 591 [5] Babcock, K. 338, 344 [5] Babiano, A. 77, 91 [36] Babin, A. 509, 510, 526, 527, 577, 580, 591 [5]; 591 [6];591 [7];591 [8];591 [9];591 [10];591 [11]; 591 [12]; 591 [13]; 591 [14]; 591 [15]; 591 [16]; 591 [17]; 631,669 [7]; 763,787, 801,802, 805,806, 808, 829 [7]; 829 [8]; 861,864, 877 [14]; 877 [15]; 877 [16]; 889, 913, 915, 917, 918, 925, 939-942, 944, 951,960, 962, 963, 966, 970, 974 [7]; 974 [8]; 974 [9]; 974 [10]; 974 [11]; 974 [12]; 974 [13] Babloyantz, A. 95, 144 [18] Bacalis, N. 640, 675 [179] Bader, G. 386 [40] Baer, S. 103, 143 [2] Bahouri, H. 516, 591 [18] Bai, E 181,217 [4] Bai, Z. 1036, 1049 [7] Baik, J. 668, 669 [8]
1057
1058
A u t h o r Index
Baker, A. 441,458 [1] Bal, T. 95, 139-141,143 [3]; 144 [19]; 144 [41] Balachandar, S. 67, 90 [2] Balasuriya, S. 74, 75, 88, 90 [3]; 92 [44] Ball, J. 889, 891, 897, 899, 904, 909, 960, 961, 974 [141; 975 [151; 975 [161 Bangia, A. 1037, 1055 [ 19 l] Bar, M. 1033, 1034, 1037, 1053 [130]; 1055 [191] Baras, E 740, 741,756 [3] Bardos, C. 544, 550, 587,591 [19]; 591 [20]; 591 [21]; 591 [22]; 591 [23]; 591 [24]; 591 [25]; 925, 942, 950, 969, 970, 975 [17]; 975 [18]; 975 [19] Barrow-Green, J. 303, 311 [4] Bartucelli, M. 798, 803,829 [9]; 829 [10] Basdevant, C. 77, 91 [36] Bashir-Ali, Z. 179, 217 [5] Bates, E 627, 669 [9]; 705,709, 714, 719 [1]; 719 [7]; 719 [8]; 719 [9]; 719 [10]; 871, 877 [17]; 1041, 1049 [9] Beale, J. 517, 591 [26]; 594 [101] Beals, R. 603,611,669 [10]; 669 [11] Bebernes, J. 729, 731, 743, 756 [4]; 756 [5]; 837, 877118]
Bechouche, P. 804, 829 [ 11 ] Beigie, D. 60, 90 [4]; 90 [5] Bekki, N. 774, 795, 829 [12]; 971,980 [170] Beliakova, N. 63, 92 [39] Bellouquid, A. 550, 591 [27] Ben-Artzi, A. 565,577, 591 [29] Benabdallah-Lagha, A. 509, 591 [28] Benci, V. 420, 458 [2]; 458 [3] Benedicks, M. 240, 257, 263 [1]; 371, 385 [11]; 385 [ 12] Benettin, G. 276, 311 [3]; 372, 385 [13]; 385 [14] Benjamin, T. 614, 669 [12]; 716, 719 [ll]; 990, 1049110]
Benson, J. 105, 143 [1] Bercovici, H. 592 [30] Berens, H. 468, 497 [5] Berestycki, H. 859, 860, 877 [19]; 877 [20]; 877 [21] Bergman, K. 667, 672 [113] Bernfeld, S. 851,877 [22] Bernoff, A. 775, 776, 829 [13] Berthoz, A. 95, 138, 143 [9] Bertram, R. 95, 143 [4] Berz, M. 357, 384, 385 [15]; 389 [115] Besnard, D. 509, 592 [31] Besson, G. 876, 877 [23] B6thuel, E 763, 788, 829 [ 14] Beyn, W.-J. 151, 173, 179, 180, 182, 214, 217 [6]; 217 [7]; 217 [8]; 217 [9]; 367, 385 [16]; 385 [17]; 797, 829 [15]; 829 [16]; 1020, 1021, 1025, 1034, 1035, 1037, 1049 [11]; 1049 [12]; 1049 [13]; 1049114]
Bhatia, N. 887, 974 [6]; 975 [20] Bibbig, A. 39, 51 [31 Bikbaev, R. 630, 669 [13] Billotti, J. 888, 906, 975 [21] Birnir, B. 631,669 [14] Bischof, C. 357, 384,385 [15]; 1036, 1049 [7] Bishop, A. 618, 669 [ 15] Blahut, R. 633, 669 [16] Blank, M. 252, 263 [2] Boardman, J. 384, 385 [ 18] Bobenko, A. 630, 669 [17] Boese, F. 479, 496 [1] Bogdanov, R. 378, 385 [ 19] Bohr, T. 638, 646, 674 [172]; 762, 775, 803,830 [17] Bollerman, P. 775,780, 815, 827, 830 [18]; 830 [19] Bonami, A. 704, 719 [12] Borisyuk, R. 166, 217 [ 10] Borsuk, K. 484, 496 [2] Bose, A. 110, 113, 120-125, 128, 138-141, 143 [5]; 146 [84]; 146 [85]; 1009, 1045, 1049 [15] Bott, R. 411,458 [4] Bourgain, J. 605, 630, 640, 667, 669 [18]; 669 [19]; 669 [20]; 669 [21]; 669 [22]; 669 [23]; 669 [24]; 669 [25]; 669 [26]; 669 [27] Bowen, R. 239, 263 [3]; 315, 318, 332, 344 [6]; 354, 371,385 [20]; 385 [21] Bower, A. 62, 63, 91 [6]; 91 [7]; 91 [8] Bower, J. 13, 23, 40,52 [12]; 52 [13] Bowman, C. 718, 721 [66] Bowman, K. 77, 78, 91 [9]; 91 [10]; 91 [11]; 91 [12]; 91 [13] Bowtell, G. 8, 54 [73] Brachet, M. 203,217 [26] Brand, H. 700, 721 [77]; 721 [78] Brandst~iter, A. 374, 385 [22] Brazovskii, S. 701,719 [13] Bressan, A. 725, 731,732, 756 [6]; 756 [7] Bressloff, P. 15, 22, 23, 51 [4]; 51 [5]; 51 [6] Brevdo, L. 1005, 1006, 1049 [ 16] Brezis, H. 573, 592 [32]; 763,788, 829 [14] Bricher, S. 729, 731,756 [4] Bricmont, J. 733, 756 [9]; 786, 792, 830 [20]; 1041, 1042, 1049 [17]; 1049 [18]; 1049 [19] Bridges, T. 789, 826, 830 [21]; 1001, 1005, 1006, 1009, 1011, 1044, 1049 [16]; 1049 [20]; 1049 [21]; 1049 [22]; 1049 [23] Briggs, R. 1000, 1002, 1005, 1006, 1049 [24] Brodzik, M. 379, 385 [23] Broer, H. 268, 311 [5]; 368, 385 [24] Bronsard, L. 714, 719 [2]; 719 [14] Bronski, J. 631, 646, 650, 661, 663, 667, 669 [28]; 670 [29]; 670 [30]; 670 [31]; 670 [32]; 672 [113] Bronstein, I. 871,877 [24] Browder, E 463,484, 487, 494, 496 [3]; 497 [4]
Author Index
Brown, M. 64, 86, 91 [14] Brown, R. 374, 385 [ 1] Brunovsk3), R 749, 755, 756 [10]; 853, 858, 865, 866, 877 [25]; 877 [26]; 877 [27]; 877 [28]; 877 [29]; 896, 937, 951, 955-957, 965, 975 [22]; 975 [23]; 975 [24]; 975 [25]; 975 [26]; 975 [27] Budd, C. 726, 756 [8] Buffoni, B. 1032, 1049 [25] Btiger, M. 791,830 [22] Buhl, E. 5, 34, 53 [62] Bullough, R. 640, 670 [33] Bunimovich, L. 631,669 [7] Burkardt, J. 166, 219 [63] Bush, E 123, 143 [6] Butera, R. 100, 116, 143 [7]; 143 [8] Butte, M. 95, 143 [4] Butzer, P. 468, 497 [5] Buzsfiki, G. 24, 54 [68]; 95, 138, 143 [9] Buzyna, G. 374, 387 [76] Caffarelli, L. 521,592 [33] Caflisch, R. 549, 555, 592 [34]; 592 [35]; 592 [36]; 592 [37] Caginalp, G. 715, 719 [15]; 719 [16] Caglioti, E. 580, 587, 589, 592 [38]; 592 [39] Cahn, J. 708, 709, 719 [17]; 719 [18]; 719 [19] Cai, D. 631,646, 648, 670 [34]; 670 [35] Calabrese, R. 5, 53 [41 ] Calderon, C. 523,592 [40] Callaway, J. 49, 54 [74] Calvin, W. 58, 91 [15] Camassa, R. 60, 91 [16] Campbell, S. 45, 51 [7] Cannell, D. 338, 344 [5] Cannone, M. 523,592 [41] Cao, Y. 495,497 [6] Caponeri, M. 638, 670 [39] Carbinatto, M. 447, 458 [5]; 458 [6] Carleson, L. 371,385 [11] Carr, J. 713, 719 [20]; 1034, 1049 [26] Carslaw, H. 837, 877 [30] Carvalho, A. 960, 962, 963,974 [5] Casartelli, M. 372, 385 [13] Casdagli, M. 385 [25]; 926, 981 [195] Casten, R. 850, 877 [31] Cawley, E. 371,385 [26] Cazenave, T. 605,606, 670 [36]; 972, 975 [28] Cerami, G. 863,877 [32] Cercignani, C. 529, 530, 536, 592 [42]; 592 [43] Ceron, S. 908, 911,968, 975 [29] Chae, D. 569, 592 [44] Chafee, N. 955, 975 [30] Champneys, A. 157, 181, 182, 187-189, 191, 193, 217 [4]; 217 [11]; 217 [12]; 217 [20]; 379, 385 [27];
1059
1020, 1032, 1034-1036, 1049 [14]; 1049 [25]; 1050 [27]; 1050 [28]; 1050 [29]; 1050 [44] Chandler, S. 103, 145 [52] Chang, H.-C. 1006, 1050 [30]; 1050 [31] Chang, Y. 351,386 [32] Chapman, S. 542, 592 [45]; 763,776, 830 [23] Charach, C. 715,716, 719 [21] Chat6, H. 638, 674 [170] Chay, T. 95, 97, 100, 101,143 [10]; 143 [11] Chemin, J. 516, 591 [18] Chen, M. 749, 756 [11]; 857, 858, 877 [33]; 975 [31] Chen, S. 638, 672 [112] Chen, T.-E 858, 877 [34] Chen, X. 705,714, 719 [22]; 719 [23] Chen, X.-Y. 729, 746, 749, 756 [11]; 756 [12]; 756 [13]; 853, 855, 857, 858, 862-865, 877 [33]; 877 [35]; 877 [36]; 877 [37]; 877 [38]; 877 [39]; 877 [40]; 939, 954, 975 [31]; 975 [32]; 975 [33] Chen, Y. 640, 670 [33] Cheng, S. 876, 878 [41] Chepyzhov, V. 889, 915, 975 [34]; 975 [35] Chernoff, P. 891,901,975 [36]; 975 [37] Chiang, C. 38, 54 [67] Chmaj, A. 705, 719 [24] Choe, W. 362, 387 [77] Cholewa, J. 894, 902, 903,911,912, 975 [38] Chorin, A. 509, 556, 592 [47]; 640, 670 [37] Chory, M. 365, 385 [8] Chossat, P. 249, 263 [4]; 558, 592 [46]; 763,778, 779, 830 [24] Chow, C. 20-24, 26-30, 51 [8]; 51 [9]; 51 [ 10]; 54 [71]; 123, 143 [12]; 146 [97]; 638, 646, 670 [38] Chow, S. 355,386 [28]; 386 [29] Chow, S.-N. 315, 318, 320, 343, 344 [7]; 344 [8]; 344 [9]; 344 [10]; 344 [11]; 344 [12]; 344 [13]; 463,484, 495,497 [7]; 497 [8]; 697, 719 [25]; 856, 857, 872, 878 [42]; 878 [43]; 878 [44]; 934, 951, 975 [23]; 975 [39]; 987, 1002, 1020, 1032, 1034, 1050 [32]; 1050 [33] Christen, Y. 95, 138, 143 [9] Christiansen, J. 157, 217 [3]; 219 [67] Christiansen, P. 640, 674 [154]; 674 [155] Chueshov, I. 931,975 [40] Ciliberto, S. 638, 670 [39] Cliffe, K. 212, 219 [74] Cockburn, B. 931,970, 975 [41 ] Coddington, E. 887, 975 [42]; 1025, 1026, 1050 [34] Cohen, A. 95, 143 [13] Cohen, L. 740, 741,756 [3] Coifman, R. 611,669 [10] Coleman, B. 701,702, 719 [26] Collet, P. 700, 702, 719 [27]; 763, 786, 791, 792, 801, 804-806, 815, 830 [25]; 830 [26]; 830 [27];
1060
Author Index
830 [28]; 830 [29]; 830 [301; 970, 976 [431; 1041, 1050 [35] Collins, G. 726, 756 [8] Colonna, J. 579, 593 [69] Conley, C. 224, 232, 263 [5]; 399, 406, 407, 420, 445, 447, 457, 458 [7]; 458 [8]; 458 [9]; 458 [10]; 458 [111; 458 [12]; 458 [131; 458 [14]; 458 [151; 458 [16]; 865,878 [45]; 922, 976 [46] Constantin, P. 510, 565, 570-574, 578, 592 [30]; 592 [48]; 592 [49]; 592 [50]; 592 [51]; 592 [52]; 592 [53]; 592 [54]; 592 [55]; 592 [56]; 593 [57]; 593 [58]; 593 [59]; 798, 803,829 [9]; 925,964, 970, 976 [44]; 976 [45] Contreras, D. 95, 139, 142, 143 [14]; 144 [20]; 146 [75]; 146 [76] Cook, L. 100, 143 [15] Coombes, B. 354, 355, 386 [30]; 386 [31] Coombes, S. 15, 22, 23, 51 [4]; 51 [5]; 51 [6] Cooperman, G. 894, 902, 976 [47] Coppel, W. 993, 1003, 1050 [36] Corliss, G. 351, 355, 357, 384, 385 [15]; 386 [32]; 386 [33]; 386 [34] Coullet, P. 203,217 [13]; 217 [26] Cowan, J. 22, 23, 25, 52 [20]; 123, 144 [25] Cowling, T. 542, 592 [45] Craig, W. 630, 670 [40]; 774, 830 [31] Crandall, M. 696, 697, 720 [28]; 720 [29] Crauel, H. 889, 976 [48] Crawford, C. 700, 720 [30] Crisanti, A. 638, 675 [176] Crook, S. 13, 23, 40, 51 [11]; 52 [12]; 52 [13] Crooks, E. 989, 1045, 1050 [37]
Cross, C. 585,596 [138] Cross, M. 631, 632, 638, 670 [41]; 679, 700, 717, 720 [31]; 720 [321 Crutchfield, J. 387 [63] Cruz Pacheco, G. 625, 670 [42] Cuong, P. 580, 594 [97] Cushman, R. 29 l, 311 [6] Dafermos, C. 512, 593 [60]; 889, 915, 968, 976 [49]; 976 [50] Dahlquist, G. 350, 386 [35] Damodaran, K. 640, 672 [106] Dancer, E. 850, 859-861, 863, 869, 871, 873-875, 878 [46]; 878 [47]; 878 [48]; 878 [49]; 878 [50]; 878151]
Daners, D. 840, 843, 865, 878 [52]; 878 [53] Date, E. 613,670 [43] Davey, A. 762, 775, 776, 779, 780, 830 [32] Dawson, S. 315, 316, 321, 323-325, 332, 335, 338, 344 [14] Dayawansa, W. 316, 329, 332, 333,344 [19] de Boor, C. 157, 217 [14]
de Melo, W. 371, 386 [36]; 870, 881 [151]; 937, 980 [ 179] de Vries, G. 18, 19, 53 [66] Debnath, L. 985, 990, 1050 [38] Debussche, A. 889, 976 [51 ] Decker, W. 775, 776, 830 [33] Degasperis, A. 603,670 [44] Degiovanni, M. 420, 458 [3] Deift, P. 603, 611,612, 630, 651-653, 659, 660, 665, 668, 669 [8]; 669 [11]; 670 [45]; 670 [46]; 670 [47]; 670 [48]; 670 [49]; 670 [50]; 670 [51]; 670 [52]; 670 [53]; 670 [54]; 830 [34] Deimling, K. 848, 878 [54]; 910, 976 [52] Deissler, R. 338, 339, 341,344 [15]; 344 [23] Del Negro, C. 103, 145 [52] del-Castillo-Negrete, D. 63, 91 [17] Dellnitz, M. 226, 228, 232, 234, 243, 244, 249-254, 262, 263 [6]; 263 [7]; 263 [8]; 263 [9]; 263 [10]; 263 [11]; 263 [12]; 263 [13]; 263 [14]; 263 [22]; 368, 386 [37] DeLorey, T. 139, 141,144 [36] DeMasi, A. 593 [61 ] Demay, Y. 764, 771,779, 813, 815, 832 [74] Demekhin, E. 1006, 1050 [30]; 1050 [31] Demmel, J. 1036, 1049 [7] Deng, B. 178, 217 [15]; 1026, 1032, 1050 [32]; 1050 [39]; 1050 [40]; 1050 [41] Deng, K. 737, 756 [14] Denjoy, A. 365,386 [38] Derks, G. 1009, 1011, 1044, 1049 [21 ]; 1049 [22] Desai, R. 711,720 [34] Desch~nes, M. 120, 146 [77] Destexhe, A. 95, 96, 103, 138-143, 143 [14]; 144 [17]; 144 [18]; 144 [19]; 144 [20]; 144 [21]; 144 [22] Deuflhard, P. 155, 217 [16]; 223, 249, 263 [14]; 263 [ 15] Devaney, R. 1032, 1050 [42] Devillard, P. 650, 670 [55] deVries, G. 101,143 [ 16] Dias, E 985, 1050 [43] Dieci, L. 356, 366, 373, 386 [39]; 386 [40]; 386 [41]; 386 [42]; 386 [43]; 386 [44]; 386 [45]; 386 [46] Diekmann, O. 463, 464, 468, 469, 495, 497 [9] Diener, E 356, 361,386 [47] Diener, M. 356, 361,386 [47]; 386 [48] Ding, J. 238, 239, 244, 263 [16]; 263 [17] DiPerna, R. 550, 593 [62]; 593 [63] DiPrima, R. 762, 776, 830 [35]; 834 [140] Dockery, J. 851,878 [55] Doedel, E. 151, 155, 157, 162, 173, 179, 181, 182, 217 [17]; 217 [18]; 217 [19]; 217 [20]; 217 [21]; 217 [22]; 217 [23]; 217 [24]; 217 [25]; 359, 363, 369, 379, 381, 386 [49]; 386 [50]; 386 [51];
Author Index 386 [52]; 386 [53]; 1020, 1034-1036, 1049 [14]; 1050 [44]; 1051 [61]
Doelman, A. 773, 774, 782, 789, 795, 798, 799, 826, 830 [36]; 830 [37]; 830 [38]; 830 [39]; 830 [40]; 830 [41]; 830 [42]; 989, 1009, 1017, 1045, 1050 [45]; 1050 [46]; 1050 [47] Doering, C. 565, 592 [50]; 786, 798, 803, 804, 824, 829 [9]; 830 [43]; 830 [44]; 832 [91] Doerner, R. 79, 91 [ 18] Dold, J. 731,756 [15] Domich, L. 120, 146 [77] Dongarra, J. 1036, 1049 [7] Dormand, J. 237, 263 [ 18] Dorroh, J. 891,976 [53] Douady, A. 571,593 [64]; 925, 976 [54] Drazin, P. 603, 671 [56] Du, Q. 238, 244, 263 [ 16]; 763,776, 831 [45] Du Croz, J. 1036, 1049 [7] Duan, J. 60, 91 [19]; 774, 831 [46] Duff, G. 357, 386 [54] Dugundji, J. 484, 488,497 [10] Dumortier, F. 379, 386 [55]; 386 [56] Easton, R. 420, 458 [9] Eberly, D. 743, 756 [5]; 837, 877 [18] Ebert, U. 1042, 1050 [48] Eckhaus, W. 356, 386 [57]; 716, 720 [33]; 762764, 774, 776, 785, 786, 795, 818, 820, 830 [35]; 830 [38]; 831 [47]; 831 [48]; 831 [49] Eckmann, J.-P. 369, 372, 373,387 [58]; 387 [59]; 609, 631, 638, 671 [59]; 671 [80]; 700, 702, 719 [27]; 763, 765, 786, 787, 790-792, 801, 804, 806, 815, 830 [26]; 830 [27]; 830 [28]; 830 [29]; 830 [30]; 831 [50]; 831 [51]; 831 [52]; 970, 976 [43]; 1041, 1042, 1050 [35]; 1050 [49]; 1050 [50] Eden, A. 509, 565, 570-572, 574-578, 591 [29]; 593 [65]; 593 [66]; 593 [67]; 593 [68]; 925, 926, 970, 976 [55]; 976 [56] Efendiev, M. 763, 801,806, 831 [53]; 831 [54] Egolf, D. 636, 638, 671 [57]; 671 [58]; 674 [159] Ehmanuilov, O. 567, 594 [86] Ei, S.-I. 1033, 1034, 1050 [51] Eidenschink, M. 231,263 [19] Eirola, T. 356, 386 [41 ] Eiswirth, M. 1037, 1055 [ 191 ] Elder, K. 711,720 [34] Elphick, C. 203,217 [26]; 1033, 1050 [52] Engel, A. 38, 53 [50] Engelborghs, K. 497 [ 11]; 498 [37] Enright, W. 382, 388 [95] Epstein, H. 609, 671 [59]; 786, 830 [30]; 970, 976 [43]; 1041, 1050 [35] Epstein, S. 49, 52 [39]
1061
Ercolani, N. 613, 614, 616, 617, 666, 667, 671 [60]; 671 [61]; 671 [62]; 671 [63]; 718, 720 [35]; 721 [66] Ermentrout, B. 5, 6, 8, 9, 13, 16-18, 20, 22-24, 26, 3237, 40, 41, 51 [2]; 51 [11]; 52 [12]; 52 [13]; 52 [14]; 52 [15]; 52 [16]; 52 [17]; 52 [18]; 52 [33]; 52 [34]; 52 [35]; 53 [49]; 53 [65]; 95, 101, 103, 105, 109, 110, 113, 116, 119, 123, 138, 144 [42]; 144 [43]; 145 [59]; 145 [61]; 146 [88]; 146 [90] Erneux, T. 103, 143 [2] Esposito, R. 593 [61] Eszter, E. 989, 1046, 1050 [53] Eubank, S. 385 [25] Evans, J. 1008, 1041, 1050 [54] Eyre, D. 709, 720 [36]; 720 [37] Faddeev, L. 603, 61 l, 671 [64] Fairgrieve, T. 157, 181,217 [20]; 217127]; 1035, 1036, 1050 [44] Falkovich, G. 641,675 [ 197] Fan, H. 976 [57] Farge, M. 579, 593 [69]; 593 [70]; 593 [71]; 594 [102] Faria, T. 868, 878 [56] Farmer, J. 374, 385 [22]; 387 [63] FagangovL E. 864, 865,878 [57] Faulkner, H. 5, 34, 39, 51 [3]; 53 [62] Fauve, S. 679, 720 [38] Fefferman, C. 510, 571,592 [51]; 592 [52] Feigenbaum, M. 365, 379, 387 [60]; 387 [72] Feir, J. 614, 669 [12]; 716, 719 [11] Feireisl, E. 763, 805, 831 [55]; 853, 858, 864-866, 878 [58]; 878 [59]; 878 [60]; 878 [61]; 878 [62]; 878 [63]; 913, 964, 968, 970, 973, 976 [58]; 976 [59]; 976 [60]; 976 [61] Fellen, B. 382, 388 [95] Feller, W. 335, 344 [16] Fenichel, N. 107, 144 [23]; 628, 671 [65]; 671 [66]; 671 [67]; 671 [68]; 871,878 [64] Fenske, C. 463, 484, 497 [12]; 497 [13] Fermanian Kammerer, C. 733, 756 [16] Ferrari, A. 933, 951,970, 976 [62] Fibich, G. 804, 831 [56] Fiedler, B. 155, 187, 217 [16]; 218 [28]; 270, 276, 312 [9]; 431, 458 [17]; 463, 484, 497 [14]; 743, 749, 755, 756 [10]; 756 [17]; 853-855, 857, 858, 868, 875, 876 [11]; 877 [26]; 878 [65]; 878 [66]; 878 [67]; 878 [68]; 878 [69]; 879 [70]; 879 [71]; 956-958, 974 [4]; 975 [24]; 976 [63]; 976 [64]; 976 [65]; 976 [66]; 976 [67]; 976 [68]; 1019, 1021, 1025, 1032, 1041, 1046, 1048, 1050 [32]; 1050 [55]; 1051 [56]; 1051 [57]; 1051 [58]; 1055 [181] Field, M. 381,387 [61 ]; 387 [62] Fife, P. 680, 682, 693, 703, 705, 709, 714-716, 718, 719 [7]; 719 [8]; 719 [16]; 719 [21]; 720 [39];
1062
A u t h o r Index
720 [40]; 720 [411; 720 [421; 720 [431; 720 [44]; 720 [451; 837, 879 [72]; 985, 1051 [59] Fila, M. 743, 745, 756 [18]; 756 [19]; 858, 879 [73] Filippas, S. 725, 727-731, 733, 756 [20]; 756 [21]; 756[22] Firle, S. 1037, 1055 [191]
Fischer, A. 223,263 [ 15] Fischer, E 338, 344 [35] Fitzgibbon, W. 968, 976 [69] Flandoli, E 889, 976 [48] Flaschka, H. 613, 665, 666, 671 [69]; 671 [70]; 671 [71] Flierl, G. 64, 65, 91 [20] Floer, A. 399, 442, 458 [18]; 458 [19]; 458 [20]; 458 [21]; 458 [22]; 458 [23] Focant, S. 989, 1051 [60] Foias, C. 509, 553, 560, 564, 565, 567-578, 591 [29]; 592 [30]; 592 [53]; 592 [54]; 592 [55]; 592 [56]; 593 [57]; 593 [58]; 593 [59]; 593 [65]; 593 [66]; 593 [67]; 593 [68]; 593 [72]; 593 [73]; 593 [74]; 593 [75]; 593 [76]; 593 [77]; 593 [78]; 593 [79]; 593 [80]; 593 [811; 593 [821; 594 [831; 594 [84]; 594 [85]; 924-927, 931, 933, 934, 951, 964, 970, 976 [44]; 976 [45]; 976 [55]; 976 [56]; 976 [70]; 976 [71]; 977 [72]; 977 [73]; 977 [74]; 977 [75] Fokas, A. 603, 671 [72] Fontich, E. 289, 293,311 [7]; 312 [8] Forest, M. 614, 616, 618,640, 661,664-666, 669 [15]; 671 [60]; 671 [61]; 671 [70]; 671 [73]; 671 [74]; 671 [75] Forster, D. 646, 671 [76] Franks, J. 414, 420, 458 [24] Franzosa, R. 426, 428, 447, 458 [25]; 459 [26]; 459 [27]; 459 [28]; 459 [29] Freadrich, K. 78, 91 [26] Freire, E. 179, 214, 219 [64] Freitas, P. 855, 879 [74] Friedman, A. 725, 756 [23]; 840, 843, 879 [75]; 879 [76] Friedman, M. 181, 182, 217 [21]; 217 [22]; 217 [23]; 218 [29]; 1035, 1051 [61] Friesen, W. 119, 144 [24] Frisch, U. 561,594 [88]; 594 [89] Froehling, H. 387 [63] Froyland, G. 239, 250, 251,263 [6]; 263 [20] Fujii, H. 1009, 1046, 1053 [128]; 1053 [129] Fujita, H. 725, 737, 748, 756 [24]; 756 [25] Fuller, A. 160, 218 [30] Fuller, F. 463,484, 497 [15] Furioli, G. 523,594 [87] Fursikov, A. 553,567, 594 [86]; 596 [163]; 596 [164]; 597 [ 165]; 597 [ 166]
Fusco, G. 713-715, 719 [1]; 719 [2]; 719 [3]; 719 [4]; 719 [5]; 719 [6]; 720 [46]; 858, 859, 879 [77]; 879 [78]; 879 [79]; 956, 977 [76]; 1034, 1051 [62] Gahinet, E 356, 388 [111] Galaktionov, V. 725, 726, 731,733, 736, 737, 739, 740, 756 [8]; 756 [26]; 756 [27]; 756 [28]; 756 [29]; 756 [30]; 757 [31]; 757 [32]; 757 [33]; 758 [73]; 858, 879 [80] Galgani, L. 372, 385 [13]; 385 [14] Gallay, T. 765, 787, 789-792, 831 [50]; 831 [51]; 831 [57]; 831 [58]; 831 [59]; 898, 940, 952, 963, 964, 973, 977 [77]; 989, 1042, 1051 [60]; 1051 [63]; 1051 [64]; 1051 [65]; 1051 [66] Gallouet, T. 573, 592 [32] Gardner, C. 609, 611,671 [77]; 671 [78] Gardner, R. 447, 458 [10]; 459 [30]; 789, 826, 830 [39]; 989, 1002, 1006, 1008, 1009, 1014, 1016, 1017, 1021, 1042, 1045, 1046, 1049 [2]; 1050 [45]; 1050 [46]; 1050 [47]; 1051 [67]; 1051 [68]; 1051 [69]; 1051 [70]; 1051 [71]; 1051 [72]; 1052 [92] Garfinkel, A. 103, 145 [52] Garratt, T. 383, 387 [64] Gaspard, P. 216,218 [31] Gatermann, K. 381,387 [65] Gear, C. 348, 387 [66] Gedeon, T. 431,445, 459 [31]; 459 [32]; 459 [33] Gelfand, I. 744, 757 [34] Gelfreich, V. 273, 294, 295, 297, 298, 302, 312 [10]; 312 [11]; 312 [12]; 312 [13]; 312 [14]; 312 [15]; 312 [16] Georg, K. 151,216 [1]; 379, 385 [5] G6rard, P. 512, 554, 594 [90] Gershenfeld, N. 374, 390 [ 149] Gerstein, V. 887, 906, 977 [78] Gerstner, W. 21-25, 52 [19]; 52 [20]; 52 [30]; 123, 144 [25] Ghidaglia, J.-M. 804, 831 [60]; 909, 925, 942, 960, 970, 972, 974, 977 [79]; 977 [80]; 977 [81]; 977 [82]; 977 [83]; 977 [84]; 977 [85] Gibbon, J. 786, 798, 803, 824, 829 [9]; 829 [10]; 830 [43]; 830 [44] Gidas, B. 743, 757 [35]; 859, 860, 879 [81]; 879 [82] Gierer, A. 703, 720 [47] Giga, Y. 725, 726, 730, 757 [36]; 757 [37]; 757 [38] Gill, G. 715, 720 [41]; 720 [42] Ginibre, J. 798, 800, 801,805, 831 [61] Ginzburg, V. 763,776, 831 [62] Giorgilli, A. 276, 311 [3]; 372, 385 [ 13] Gisself~ilt, M. 798, 803,829 [9] Glassey, R. 606, 671 [79] Glendinning, P. 1032, 1051 [73] Glimm, J. 594 [93]
Author Index
Gobbino, M. 895, 899, 900, 977 [86] Goedde, C. 640, 671 [73] Goirand, E. 579, 593 [70] Goldstein, J. 468, 497 [ 16] Goldstein, R. 704, 720 [48] Gollub, J. 338, 344 [27] Golomb, D. 95, 96, 123, 138-142, 144 [26]; 144 [27]; 144 [28]; 144 [29]; 146 [93]; 339-341,344 [3] Golse, E 537, 544, 550, 591 [19]; 591 [20]; 591 [21]; 591 [22]; 594 [91]; 594 [92] Golub, G. 327, 344 [ 17] Golubitsky, M. 249, 263 [4]; 374, 376, 387 [67]; 996, 1048, 1051 [74]; 1051 [75] Goren, G. 631,638, 671 [80] G6rniewicz, L. 451,459 [34]; 459 [35] Goubet, O. 934, 973,974, 977 [87]; 977 [88]; 977 [89]; 977 [90]; 977 [91] Govaerts, W. 151, 163, 169, 193, 218 [32]; 218 [33]; 218 [34]; 374, 382, 384, 387 [68]; 387 [69]; 387 [70]; 1020, 1034, 1035, 1049 [14] Grace, T. 103, 145 [48] Grad, H. 594 [94] Granas, A. 451,459 [34]; 463,484, 487, 497 [17] Grant, C. 709, 714, 720 [49]; 720 [50] Grasseau, G. 374, 385 [6] Gray, C. 5, 52 [21]; 95, 144 [30]; 144 [31] Grebogi, C. 315-318, 320, 321, 323-325, 327, 329, 332, 333, 335, 338, 339, 343, 344 [14]; 344 [17]; 344 [18]; 344 [19]; 344 [21]; 344 [25]; 344 [26]; 344 [33]; 354, 355,371,387 [71]; 388 [88] Greenbaum, A. 1036, 1049 [7] Greene, J. 365,387 [72]; 609, 611,671 [77]; 671 [78] Greenside, H. 636, 638, 671 [57]; 671 [58]; 674 [159] Grenier, E. 667, 671 [81] Griewank, A. 173, 212, 218 [35]; 218 [36]; 351,357, 363, 384, 385 [15]; 387 [73] Grigoriev, R. 638, 671 [82] Grillakis, M. 608, 609, 671 [83]; 671 [84]; 989, 1009, 1043, 1044, 1049 [6]; 1051 [76]; 1051 [77]; 1051 [78]; 1051 [79] Grillner, S. 95, 143 [13] Grischkowsky, D. 661,674 [163] Grosch, C. 90, 92 [46] Grossman, S. 79, 91 [ 18] Guckenheimer, J. 59, 80, 91 [21]; 160, 163, 182, 217 [21]; 218 [34]; 218 [37]; 235, 263 [21]; 357, 358, 362, 367, 368, 370, 372-374, 383, 384, 387 [69]; 387 [74]; 387 [75]; 387 [76]; 387 [77]; 387 [78]; 387 [79]; 387 [80]; 387 [81]; 387 [82]; 387 [83]; 387 [84]; 388 [85]; 388 [103]; 603, 621, 629, 672 [85]; 870, 879 [83]; 987, 1002, 1020, 1034, 1051 [80] Guder, R. 256, 262, 263 [22]; 264 [23] Guillemin, V. 374, 376, 387 [67]
1063
Gunzburger, M. 763,776, 831 [45] Guo, Y. 587, 591 [23] Gutkin, B. 41, 52 [ 18] Gwinn, E. 327, 344 [20] Hagan, R 763, 788, 831 [63]; 1048, 1051 [81] Hairer, E. 312 [17]; 348-350, 388 [86]; 388 [87] Hajek, O. 887, 975 [20] Hale, J. 214, 218 [38]; 430, 459 [36]; 459 [37]; 463, 464, 467-470, 479, 480, 483, 495, 497 [18]; 497 [19]; 497 [20]; 697, 713, 719 [25]; 720 [46]; 749, 756 [11]; 787, 802, 805, 807, 808, 822, 831 [64]; 831 [65]; 839, 842, 853, 857, 858, 863, 865, 868, 873, 877 [33]; 879 [84]; 879 [85]; 879 [86]; 879 [87]; 879 [88]; 879 [89]; 879 [90]; 879 [91]; 887-890, 893-898, 901-904, 906, 908, 911-916, 918-922, 924, 927, 928, 930-934, 936, 937, 939, 940, 942, 944-948, 951-954, 960, 962970, 973, 974 [5]; 975 [31]; 975 [33]; 975 [38]; 976 [57]; 977 [92]; 977 [93]; 977 [94]; 977 [95]; 977 [96]; 977 [97]; 977 [98]; 977 [99]; 977 [100]; 978 [101]; 978 [102]; 978 [103]; 978 [104]; 978 [105]; 978 [106]; 978 [107]; 978 [108]; 978 [109]; 978 [110]; 978 [111]; 978 [112]; 978 [113]; 978 [114]; 978 [115]; 978 [116]; 987, 1002, 1020, 1034, 1050 [33]; 1051 [62] Hall, G. 318, 319, 325,344 [28]; 365,385 [8] Hailer, G. 74, 75, 79, 87, 91 [22]; 91 [23]; 91 [24]; 91 [25]; 92 [38]; 630, 672 [86]; 672 [87]; 672 [88] Hamel, E 809, 831 [66] Hammarling, S. 1036, 1049 [7] Hammel, S. 315, 318, 320, 321, 325, 332, 333, 335, 339, 344 [18]; 344 [21]; 354, 355,388 [88] Han, S. 19, 52 [22] Hanner, O. 484, 497 [21]; 497 [22] Hansel, D. 41, 52 [23]; 638, 672 [89] Haragus-Courcelle, M. 829, 831 [67] Haraux, A. 862, 865-867, 879 [92]; 879 [93]; 889, 915, 952, 960, 961,965, 972, 975 [28]; 978 [117]; 978 [118]; 978 [119]; 978 [120] Harlow, E 509, 592 [31 ] Harterich, J. 1032, 1051 [82] Hartman, P. 319, 344 [22]; 1001, 1009, 1025, 1026, 1051 [831 Harvey, G. 608, 674 [ 152] Hasegawa, A. 608, 672 [90] Hasimoto, H. 774, 831 [68] Hassard, B. 179, 184, 218 [39]; 218 [40] Hasselblatt, B. 372, 388 [99] Hayes, N. 479, 497 [23] Hayot, F. 646, 672 [93] Heinze, S. 1046, 1052 [84] H61ein, E 763, 788, 829 [ 14]
1064
Author Index
Helfrich, K. 87, 88, 91 [33] Henon, M. 354, 371,388 [89] Henrici, P. 348, 388 [91] Henry, D. 708, 720 [51]; 733, 743, 749, 757 [39]; 840, 841, 843, 858, 872, 879 [94]; 879 [95]; 889, 917, 921,927, 942, 950, 951,954, 955, 978 [121]; 978 [122]; 978 [123]; 978 [124]; 986, 989, 1004, 1040, 1041, 1052 [85] Herbst, B. 312 [18]; 631,669 [1] Herman, M. 365,388 [90] H6ron, B. 804, 831 [60]; 970, 977 [83] Herrero, M. 730, 732-735, 755 [ 1]; 756 [26]; 757 [40]; 757 [41]; 757 [42]; 757 [43]; 757 [44]; 757 [45]; 757 [46]; 757 [47] Herz, A. 495,497 [24] Hess, P. 841-843,847, 850-852, 861,876 [ 1]; 876 [2]; 878 [49]; 878 [50]; 879 [96]; 879 [97]; 879 [98]; 879[99]
Hildebrand, M. 1037, 1055 [191] Hilhorst, D. 704, 719 [12] Hilliard, J. 708, 719 [19] Hindmarsh, J. 101,144 [32] Hirsch, M. 365,372, 376, 379, 380, 388 [92]; 388 [93]; 843, 845, 851, 852, 871, 880 [100]; 880 [101]; 880 [1021; 880 [1031 Hlava~ek, V. 175, 219 [66]; 384, 389 [128] Hockett, K. 327, 344 [20] Hocking, L. 733, 757 [48]; 762, 775, 776, 779, 780, 830 [32]; 831 [69] Hodgkin, A. 96, 139, 144 [33] Hofer, H. 399, 459 [38] Hoffmann-Ostenhof, M. 876, 880 [104] Hoffmann-Ostenhof, T. 876, 880 [104] Hohenberg, E 631, 632, 638, 670 [41]; 672 [91]; 674 [170]; 679, 700, 701, 720 [31]; 721 [81]; 774, 795,834 [144]; 990, 1009, 1055 [177] Hohmann, A. 226, 228, 234, 263 [7]; 263 [8]; 368, 386 [37] Holen, M. 590, 595 [132]; 638, 674 [170] Holland, C. 850, 877 [31] Holland, W. 88, 91 [27] Holm, D. 786, 798, 824, 830 [44] Holmes, E 59, 80, 91 [21]; 235, 263 [21]; 327, 344 [20]; 367, 370, 387 [78]; 389 [135]; 603, 621, 629, 672 [85]; 763, 774, 795, 831 [46]; 831 [70]; 833 [103]; 870, 879 [83]; 987, 1002, 1020, 1034, 1051 [801 Holodniok, M. 173,218 [41] Holschneider, M. 579, 593 [69] Homburg, A. 1032, 1052 [86] Hopkins, W. 100, 143 [15] Hoppensteadt, E 6, 52 [24]; 103, 105, 144 [34] Howard, L. 628, 672 [92]
Howard, P. 1041, 1042, 1055 [192] Howison, S. 763,776, 830 [23] Hsiao, C.-F. 103, 145 [52] Hsu, H. 223, 259, 264 [24] Hu, Y. 851,877 [22] Huang, W. 479, 480, 483, 497 [ 18] Hubbard, J. 358, 388 [94] Htibinger, B. 79, 91 [ 18] Huguenard, J. 139, 141,144 [35]; 144 [36] Huisinga, W. 223,263 [15] Hull, T. 382, 388 [95] Humphries, A. 352, 390 [143]; 915, 981 [201] Hunt, E 262, 264 [25] Huntsman, M. 139, 141,144 [36] Hurewicz, W. 924, 978 [125] Hurley, M. 407, 459 [39]; 459 [40]; 865,880 [105] Hutson, V. 447, 459 [53]; 851,878 [55] Huxley, A. 96, 139, 144 [33] Hwa, T. 638, 646, 670 [38] Ikeda, H. 1009, 1045, 1046, 1052 [87]; 1052 [88]; 1053 [129] Ikeda, T. 1045, 1052 [87] Illner, R. 529, 530, 536, 592 [43]; 594 [95] Ilyashenko, Y. 367, 388 [96] Imai, K. 534, 596 [141] Indik, R. 718, 720 [35]; 721 [66] Infante, E. 955, 975 [30] Ingersoll, A. 580, 594 [96]; 594 [97] Iooss, G. 203, 217 [26]; 558, 592 [46]; 763, 764, 770, 771, 774, 778, 779, 811, 813-815, 830 [24]; 831 [49]; 832 [71]; 832 [72]; 832 [73]; 832 [74]; 872, 883 [2051 Its, A. 651,653,660, 670 [45] Iwasaki, N. 968, 978 [ 126] Izhikevich, E. 6, 7, 41, 45, 52 [24]; 52 [25]; 52 [26]; 52 [27]; 95, 96, 103, 105,144 [34]; 144 [37] Jacklet, J. 95, 144 [38] Jaeger, J. 837, 877 [30] Jayaprakash, C. 45, 51 [7]; 638, 646, 672 [93]; 674 [ 172] Jeffreys, J. 5, 23, 27, 32, 34, 53 [62]; 53 [63]; 53 [64]; 54 [72]; 95, 120, 144 [39]; 146 [98] Jendoubi, M. 866, 867, 879 [92]; 880 [106]; 965, 978 [ 120] Jensen, M. 638, 646, 674 [172]; 762, 775,803,830 [17] Jepson, A. 157, 212, 217 [27]; 219 [74] ....... : i--~ ~"": ~7.;] Jessell, T. 95, 144 [40] Jiang, J.-E 851,880 [107]; 880 [108] Jin, S. 665-667, 671 [62]; 672 [94]; 672 [95] Johansson, K. 668, 669 [8]
Author Index
John, E 664, 672 [96] Johnson, M. 368, 388 [97] Johnson, R. 603,671 [56]; 985,990, 1052 [89] Johnson, S. 372, 387 [79] Jolly, M. 368, 388 [97]; 858, 880 [109]; 955,978 [127] Jones, C. 64, 67, 68, 70-72, 74, 75, 87-89, 90 [3]; 91 [32]; 91 [33]; 92 [41]; 92 [44]; 628, 672 [97]; 789, 826, 830 [39]; 989, 1006, 1008, 1009, 1028, 1029, 1031, 1032, 1041, 1042, 1044, 1045, 1049 [2]; 1049 [3]; 1049 [4]; 1049 [6]; 1049 [9]; 1049 [15]; 1051 [70]; 1051 [71]; 1052 [90]; 1052 [91]; 1052 [92]; 1054 [150]; 1055 [190] Jones, D. 565, 594 [98]; 931,970, 975 [41]; 978 [128] Jones, E. 95, 120, 138, 146 [78]; 146 [79] Jones, G. 489, 497 [25] Jones, S. 38, 39, 52 [28]; 53 [48] Jorba, A. 388 [98] Joseph, D. 744, 757 [49] Joyce, G. 584, 594 [99] Juedes, D. 351,363, 387 [73] Junge, O. 232, 234, 237, 243, 244, 249, 252-256, 262, 263 [9]; 263 [10]; 263 [11]; 263 [12]; 263 [14]; 264 [26]; 264 [27] JUngel, A. 804, 829 [11] Kaashock, M. 469, 497 [26] Kaczyfiski, T. 455,459 [41 ]; 459 [42] Kalyakin, L. 774, 832 [77] Kamvissis, S. 663,667, 674 [150] Kan, I. 371,385 [4] Kandel, E. 95, 144 [40] Kaneko, K. 338, 341,344 [23]; 344 [36] Kaniel, S. 522, 594 [100] Kaper, T. 38, 39, 52 [28]; 60, 91 [28]; 91 [29]; 989, 1009, 1017, 1045, 1050 [45]; 1050 [46]; 1050 [47] Kapitanski, L. 964, 978 [129] Kapitula, T. 763, 774, 786, 795-797, 832 [78]; 832 [79]; 832 [80]; 832 [81]; 832 [82]; 990, 1009, 1010, 1014, 1016, 1017, 1032, 1041, 1042, 1052 [92]; 1052 [93]; 1052 [94]; 1052 [95]; 1052 [96]; 1052 [97]; 1052 [98]; 1054 [151] Kaplan, J. 495,496, 497 [27]; 497 [28]; 497 [29] Kaplan, S. 725, 757 [50] Kappeler, T. 613, 672 [98]; 672 [99] Kapral, R. 338, 344 [36]; 985, 990, 1052 [99] Karbowski, J. 34, 52 [29] Kardar, M. 646, 672 [100] Kast, A. 640, 670 [37] Kath, W. 989, 101 O, 1052 [ 106] Kato, T. 469, 497 [30]; 517, 522, 591 [26]; 594 [101]; 594 [104]; 865,880 [110]; 1007, 1016, 1052 [100] Katok, A. 372, 388 [99] Kawasaki, K. 704, 721 [72] Kawohl, B. 860, 880 [ 111 ]
1065
Kazarinoff, N. 184, 218 [40] Keener, J. 608, 672 [101]; 989, 990, 1009, 1045, 1048, 10551176]
Keizer, J. 95, 100, 143 [10] Keller, G. 238, 252, 263 [2]; 264 [28] Keller, H. 151, 154, 155, 157, 162-165, 173, 217 [24]; 217 [25]; 218 [42]; 218 [43]; 218 [44]; 223, 264 [29]; 363,386 [52]; 386 [53] Keller, J. 639, 672 [102]; 681, 682, 689, 721 [69]; 721 [70] Kenyon, R. 371,388 [ 100] Kern6vez, J.-E 151, 155, 157, 162, 173, 179, 182, 217 [ 19]; 217 [24]; 217 [25]; 363,386 [52]; 386 [53] Kevlahan, N. 579, 593 [71]; 594 [102] Kevrekidis, I. 365, 368, 388 [97]; 388 [101]; 1033, 1034, 1037, 1053 [130]; 1055 [191] Kharif, C. 985, 1050 [43] Khesin, B. 514, 526, 590 [3] Khibnik, A. 163, 169, 187, 193, 218 [34]; 218 [45]; 218 [46]; 360, 384, 387 [69]; 388 [102] Kiemel, T. 95, 143 [4] Kiessling, M. 587, 589, 594 [105] Kifer, Y. 239, 241,244, 264 [30] Kim, S.-H. 370, 388 [103] Kim, U. 95, 144 [41] Kinoshita, S. 488, 497 [31] Kirane, M. 952, 978 [ 119] Kirchgfissner, K. 763, 764, 778, 811, 832 [83]; 832 [84]; 832 [85]; 832 [86]; 833 [104]; 1042, 1047, 1052 [101]; 1052 [102]; 1052 [103]; 1053 [139] Kirrmann, P. 764, 774, 815,832 [87] Kirwan, A.D., Jr. 90, 92 [46] Kisaka, M. 1032, 1052 [104] Kistler, W. 24, 52 [30] Klainerman, S. 509, 594 [106] Klein, R. 640, 672 [103]; 672 [104]; 672 [105]; 672 [106]; 672 [107] Kleinkauf, J. 385 [16] Klef~, W. 367, 385 [ 17]; 797, 829 [ 15] Knapp, B. 650, 672 [ 108] Kodak, H. 354, 355,386 [30]; 386 [31] Koch, H. 858, 880 [ 112] Koch Medina, E 840, 843,878 [52] Kodama, Y. 608, 661,672 [90]; 672 [109] Kohn, R. 521,592 [33]; 714, 719 [14]; 725-730, 733, 756 [20]; 757 [36]; 757 [37]; 757 [38] Kokubu, H. 431,436, 445,459 [32]; 459 [43]; 459 [44]; 1032, 1052 [86]; 1052 [104] Kolmogorov, A. 594 [ 107] K6nig, E 38, 53 [50]; 54 [67] Kopanski~, A. 871,877 [24] Kopelevich, D. 1006, 1050 [30]; 1050 [31] Kopell, N. 6, 8, 9, 17, 18, 20, 23, 24, 26-30, 3239, 41, 45-49, 51, 51 [1]; 51 [2]; 51 [9]; 51 [10];
1066
A u t h o r Index
[14]; 52 [15]; 52 [16]; 52 [17]; 52 [28]; 52 [29]; [31]; 52 [32]; 52 [33]; 52 [34]; 52 [35]; 52 [36]; [38]; 52 [39]; 53 [42]; 53 [44]; 53 [47]; 53 [48]; [56]; 53 [58]; 53 [59]; 54 [71]; 95, 105, 109, ll0, 113, 116, 119-125, 128, 138-141,143 [5]; 144 [42]; 144 [43]; 144 [44]; 145 [45]; 145 [49]; 145 [68]; 146 [71]; 146 [72]; 146 [73]; 146 [84]; 146 [85]; 146 [97]; 628, 672 [92]; 672 [110] Kostelich, E. 316, 329, 332, 333, 344 [19]; 368, 389 [ 114] Kostin, I. 575, 594 [108]; 915, 940, 944, 966, 967, 979 [130]; 979 [131] Kova6i6, G. 60, 91 [29]; 628, 672 [111 ] Kowalczyk, M. 680, 682, 693, 713, 718, 719 [6]; 52 52 52 53
218 [48]; 218 [49]; 218 [50]; 218 [51]; 218 [52]; 360, 379, 385 [27]; 388 [102]; 388 [106]; 987, 1002, 1020, 1032, 1034-1036, 1049 [14]; 1050 [28]; 1050 [29]; 1050 [44]; 1052 [107] Kwapisz, J. 447, 458 [5]
Lacey, A. 740, 741, 745, 747, 748, 755, 757 [51]; 757 [52]; 757 [53]; 757 [54] Ladyzhenskaya, O. 570, 575, 595 [111]; 595 [112]; 595 [113]; 595 [114]; 595 [115]; 748, 757 [55]; 887-889, 897, 904, 912, 924, 925, 931, 963, 970, 979 [133]; 979 [134]; 979 [135]; 979 [136]; 979 [137]; 979 [138]; 979 [139]; 979 [140] Lai, Y. 327, 343, 344 [17]; 344 [25]; 344 [26] 7201431 Laing, C. 771,829 [6] Kozlov, V. 298-300, 311 [2] Lamb, G. 603,673 [ 120] Kozono, H. 523,594 [103] Kraichnan, R. 561, 562, 594 [109]; 595 [110]; 638, Landau, L. 511,538, 595 [116]; 763,776, 831 [62] Landauer, B. 522, 527, 551,595 [117] 6721112] Lanford, O. 357, 366, 388 [108]; 388 [109]; 388 [110]; Krasnoselskii, M. 848, 880 [113]; 887, 906, 977 [78] 529, 595 [ 118] Krauskopf, B. 368, 388 [104] Larsson, S. 315, 318, 320, 344 [32] Krein, M. 880 [114] Kreuzer, E. 223, 256, 262, 263 [22]; 264 [23]; 264 [31] LaSalle, J. 887, 888, 891,896, 903,904, 906, 975 [21]; 977 [100]; 979 [141] Kriecherbauer, T. 668, 670 [46]; 670 [47] Lasota, A. 240, 242, 245,264 [32]; 264 [33] Krisztin, T. 497 [32]; 497 [33] Laub, A. 356, 388 [111 ] Krug, J. 638, 646, 674 [172] Laurenqot, Ph. 972, 973, 979 [142] Krupa, M. 989, 1032, 1052 [86]; 1052 [105] Kruskal, M. 609, 611, 631, 671 [77]; 671 [78]; Lauterbach, R. 381,387 [65] Lavrentiev, M.M. 851,865,883 [213] 674 [ 166]; 674 [ 167] Lax, P. 609, 610, 664, 665,667, 673 [121]; 673 [122]; Krylov, D. 667, 672 [113] 673 [123]; 673 [124]; 673 [125]; 673 [126] Kuang, Y. 494, 497 [34] Lazutkin, V. 273, 293, 295, 297, 302, 312 [14]; Kubi6ek, M. 151,173,218 [41]" 218 [47] 312 [15]; 312 [19]; 312 [20] Kubo, R. 646, 675 [ 182] Kuksin, S. 630, 667, 669 [13]; 669 [17]; 672 [114]; Le Dung 576-578, 595 [119] 672 [115]; 673 [116]; 673 [117]; 673 [118]; Le Roux, G. 312 [18] Lebeau, G. 969, 975 [ 17] 6731119] LeBlanc, V. 1048, 1051 [74] Kunin, B. 182, 217 [22] Lebowitz, J. 593 [61]; 640, 673 [127]; 673 [128] Kunkel, P. 155, 217 [16] Lecar, H. 17, 42, 53 [46]; 122, 145 [51] Kupferman, R. 640, 670 [37] Kupiainen, A. 733, 756 [9]; 786, 792, 830 [20]; 1041, Lee, E. 95, 100, 106, 107, 113, 117, 124, 145 [46]; 146 [86] 1042, 1049 [17]; 1049 [18]; 1049 [19] Lee, J. 664-666, 671 [74] Kupka, I. 388 [105] Lee, Y. 105, 145 [60] Kuramoto, Y. 19, 52 [22]; 52 [37]; 338, 344 [24] Lega, J. 700, 720 [52]; 720 [53]; 721 [63]; 762, Kuratowski, K. 917, 979 [132] 833 [ 110] Kurdyumov, S. 736, 756 [27]; 758 [73] Lehoucq, R. 262, 264 [34] Kurrer, C. 19, 52 [22] Leibovich, S. 527, 595 [131] Kurths, J. 343, 344 [26] Kutz, J. 667,672 [113]; 763,795,833 [103]; 989, 1010, Leizarowitz, A. 701,702, 720 [54] Lemarri6, G. 523, 594 [87] 1052 [106]; 1054 [151] LeMasson, G. 95, 144 [44] Kuznetsov, E. 774, 834 [148] L6meray, E. 476, 497 [35] Kuznetsov, V. 380, 388 [ 107] Kuznetsov, Y. 151, 157, 162, 165, 166, 169, 179, Leng, L. 667, 672 [113] 181, 182, 184, 187-189, 191, 193, 203, 204, 209, Leonard, A. 60, 90 [4]; 90 [5]; 92 [42] 215, 216, 217 [11]; 217 [12]; 217 [20]; 218 [46]; Leray, J. 595 [120]; 595 [121]; 595 [122]
Author Index
Levandosky, S. 1043, 1052 [108] Levermore, D. 544, 550, 591 [19]; 591 [20]; 591 [21]; 591 [22]; 591 [24]; 625, 630, 665-667, 670 [42]; 670 [48]; 671 [62]; 672 [94]; 672 [95]; 673 [123]; 673 [ 124]; 673 [ 125]; 673 [ 126]; 798-800, 830 [34]; 830 [43]; 832 [88]; 832 [89] Levin, J. 946, 979 [143] Levine, H. 737, 756 [14]; 757 [56]; 858, 877 [34]; 880 [ 115] Levinger, B. 469, 498 [36] Levinson, N. 887, 975 [42]; 979 [144]; 1025, 1026, 1050134]
Levitin, V. 157, 162, 166, 169, 187, 193, 218 [46]; 218 [52]; 360, 380, 388 [102]; 388 [107] Levy, D. 804, 831 [56] Li, C. 859, 880 [ 116] Li, L. 668, 670 [49] Li, T.-Y. 238, 244, 263 [16]; 264 [35] Li, Y. 612-614, 617, 618, 621, 627, 629, 630, 673 [129]; 673 [130]; 673 [131]; 673 [132]; 6731133]
Li, Y.A. 1017, 1044, 1052 [109]; 1052 [110] Lichtenstein, L. 514, 595 [123] Lichtman, E. 608, 674 [ 152] Lieb, E. 573, 595 [124] Lifshitz, E. 511,538, 595 [116] Lin, E 521,595 [125] Lin, F.-H. 788, 789, 832 [90] Lin, G. 1042, 1049 [19] Lin, X. 355,386 [28] Lin, X.-B. 343, 344 [7]; 915, 918, 978 [101]; 1009, 1021-1023, 1025, 1028, 1032, 1034, 1045, 1052 [lll]; 1053 [1121 Lions, J.-L. 595 [126] Lions, E-L. 550, 580, 587, 589, 592 [38]; 592 [39]; 593 [63]; 595 [127]; 640, 673 [134]; 848, 880 [117] Lipphardt, B. 90, 92 [46] Liu, J. 338,344 [27] Liu, L. 179, 218 [53]; 218 [54] Liu, V. 570, 595 [129] Liu, W. 729, 73 l, 756 [21]; 757 [57] Liu, Y. 179, 218 [54] Llinfis, R. 95, 103, 120, 138, 143 [9]; 145 [47]; 145 [48]; 146 [79] LoFaro, T. 45, 48, 52 [38]; 116, 145 [49] Logak, E. 704, 719 [12]; 720 [55] L6hner, R. 389 [ 112] Lojasiewicz, S. 938, 979 [145] Lopes, O. 860, 880 [118]; 880 [119]; 906, 908, 91 l, 946, 964, 968, 975 [29]; 978 [102]; 979 [146] Lord, G. 1046, 1053 [113] Lorenz, E. 355, 358, 366, 389 [113]; 558, 595 [128] Lorenz, J. 366, 386 [42]; 386 [44]; 386 [45]; 797, 829 [16]; 1037, 1049 [13]
1067
Lou, Z. 368, 389 [ 114] Lozier, M. 62, 63, 91 [7]; 92 [39] Lu, K. 627, 669 [9]; 856-858, 871, 872, 877 [17]; 878 [42]; 878 [43]; 878 [44]; 880 [120]; 880 [121]; 934, 975 [39] Lubich, C. 312 [ 17] Luce, B. 625,670 [42]; 803, 804, 832 [91] Lucheroni, C. 640, 673 [138] Lunardi, A. 840-843, 880 [122] Lundgren, T. 744, 757 [49] Lunkeit, E 78, 91 [26] Lust, K. 498 [37] Luzyanina, T. 498 [37] Lvov, V. 641,675 [197] Lynden-Bell, D. 585,595 [130] MacEvoy, W. 667, 671 [62] MacKay, R. 60, 91 [30]; 365, 370, 387 [72]; 388 [103] Mackey, M. 242, 245,264 [32] Maddocks, J. 1043-1045, 1053 [114]; 1053 [115]; 1053 [ 116] Magalhfies, L. 430, 459 [37]; 868, 878 [56]; 887, 897, 920-922, 924, 978 [ 103] Maginu, K. 1009, 1032, 1055 [ 187] Magnus, W. 612, 673 [135] Mahaffy, J. 479, 480, 498 [38] Mahalov, A. 510, 526, 527, 577, 580, 591 [5]; 591 [6]; 591 [7]; 591 [8]; 591 [9]; 591 [10]; 591 [11]; 591 [12]; 591 [13]; 595 [131] Maier-Paape, S. 710, 711,720 [56]; 720 [57]; 795,796, 832 [811 Majda, A. 509, 517,590, 591 [26]; 592 [52]; 594 [101]; 594 [106]; 595 [132]; 640, 641,643,645, 672 [103]; 672 [104]; 672 [105]; 672 [106]; 672 [107]; 673 [134]; 673 [136] Makarov, M. 613,672 [98] Makino, K. 357, 389 [ 115] Malanotte-Rizzoli, P. 64, 65, 91 [20] M~ilek, J. 574, 595 [133] Mallet-Paret, J. 463, 467, 484, 489, 494-496, 497 [7]; 497 [14]; 498 [39]; 498 [40]; 498 [41]; 498 [42]; 498 [43]; 498 [44]; 498 [45]; 498 [46]; 498 [47]; 498 [48]; 498 [49]; 854, 856-858, 878 [43]; 878 [44]; 878 [67]; 880 [ 123]; 880 [ 124]; 880 [ 125]; 924, 929, 934, 951,958, 970, 976 [65]; 979 [147]; 979 [148] Malo, S. 357, 387 [80] Manakov, S. 603,674 [ 158] Marl& B. 568, 595 [134] Marl& R. 871,880 [126]; 924-926, 979 [149] Mangus, N. 77, 91 [ 11 ] Manley, O. 567, 568, 592 [30]; 592 [55]; 592 [56]; 593 [75]; 593 [76]; 593 [77]; 593 [78]; 596 [135]; 931,976 [70]
1068
A u t h o r Index
Manneville, E 638, 673 [137]; 762, 775, 776, 803, 832 [92] Manor, Y. 48, 49, 51, 52 [39]; 52 [40]; 53 [47] Marchesoni, E 640, 673 [138] Marchioro, C. 509, 580, 587-590, 592 [38]; 592 [39]; 596 [136] Marcus, M. 701, 702, 719 [26]; 721 [58]; 721 [59]; 721 [601; 721 [611 Marcus, P. 579, 596 [137] Marder, E. 5, 46-49, 52 [39]; 53 [41]; 53 [42]; 53 [47]; 53 [53]; 53 [56]; 123, 145 [68] Marek, M. 151,218 [47] Marsden, J. 184, 218 [55]; 370, 389 [ 135]; 514, 590 [ 1]; 891,901,975 [37] Martel, Y. 740, 741,757 [58] Martienssen, W. 79, 91 [18] Martin, R. 852, 880 [ 127] Massatt, E 855, 865, 879 [88]; 881 [128]; 899, 903, 911, 958, 968, 979 [150]; 979 [151]; 979 [152]; 97911531
Masuda, K. 741,742, 758 [59] Matano, H. 729, 743, 756 [12]; 756 [18]; 843, 850-853, 855, 858, 876 [2]; 877 [37]; 877 [38]; 881 [129]; 881 [130]; 881 [131]; 881 [132]; 881 [133]; 881 [134]; 881 [135]; 881 [136]; 881 [137]; 954, 958, 979 [154]; 979 [155] Mather, J. 377, 389 [116]; 389 [117]; 389 [118]; 389 [119]; 389 [120]; 389 [121] Matkovsky, B. 773,832 [93] Mato, G. 41, 52 [23] Matos, J. 735, 758 [60]; 758 [61] Mattheij, R. 359, 363,385 [9]; 1035, 1036, 1049 [8] Matveev, V. 616, 673 [139] Maxey, M. 67, 90 [2] McCauley, J. 316, 344 [29] McCord, C. 426, 431, 434, 447, 459 [45]; 459 [46]; 459 [47]; 459 [48]; 463,484, 498 [50] McCormick, D. 95, 103, 138-141, 143 [3]; 144 [19]; 144 [21]; 144 [31]; 144 [41]; 146 [78]; 146 [80] McCracken, M. 184, 218 [55] McDuff, D. 399, 460 [66] McGehee, R. 365, 385 [8] Mclntyre, M. 80, 91 [31 ] McKean, H. 613, 631, 640, 669 [14]; 673 [140]; 673 [141]; 673 [142]; 673 [143] McKenney, A. 1036, 1049 [7] McLaughlin, D. 608, 612-614, 616-619, 621, 627631, 638, 639, 641, 643, 645, 646, 648, 650, 661, 663, 665-667, 669 [15]; 669 [28]; 670 [31]; 670 [34]; 670 [35]; 671 [60]; 671 [61]; 671 [63]; 671 [70]; 671 [71]; 672 [94]; 672 [95]; 672 [101]; 672 [102]; 673 [130]; 673 [131]; 673 [136]; 673 [144]; 674 [145]; 674 [146]; 674 [147]; 674 [148]; 674 [149]
McLaughlin, K. 661,668, 670 [46]; 670 [47]; 671 [75] McLaughlin, R. 640, 672 [ 107] McLeod, J. 725, 756 [23] Medvedev, G. 49, 53 [43]; 53 [44] Meerbergen, K. 383,389 [ 122] Mehlhorn, K. 231,264 [36] Mei, Z. 165,219 [56] Meinhardt, H. 703, 720 [47]; 721 [62] Meiss, J. 60, 91 [30] Melbourne, I. 764, 771, 813, 827, 832 [94]; 1048, 1051 [74] Melnikov, V. 214, 219 [57]; 370, 389 [123] Meloon, B. 362, 387 [81] Mendelson, N. 700, 721 [63] Mendelson, E 887, 979 [156] Merino, S. 865,878 [53] Merle, E 725, 730, 731,733, 736, 756 [16]; 756 [22]; 758 [62]; 758 [63]; 758 [64] Meron, E. 679, 721 [64]; 1002, 1033, 1034, 1050 [52]; 1053 [ 1171 Meunier, C. 41, 52 [23] Meyer, K. 318, 319, 325, 344 [28] Meyer, Y. 579, 593 [70] Mielke, A. 763-765, 771,774, 776, 779-783,786, 792, 795, 796, 798-803, 805, 806, 811, 813-815, 820822, 824-826, 829 [2]; 829 [3]; 829 [4]; 829 [5]; 831 [58]; 832 [72]; 832 [73]; 832 [74]; 832 [87]; 832 [95]; 832 [96]; 833 [97]; 833 [98]; 833 [99]; 833 [100]; 833 [101]; 833 [102]; 833 [103]; 833 [104]; 833 [105]; 833 [106]; 833 [107]; 833 [108]; 865, 881 [138]; 913, 919, 980 [157]; 980 [158]; 980 [159]; 1001, 1042, 1047, 1049 [23]; 1051 [64]; 1053 [118]; 1053 [119]; 1053 [120] Mierczyfiski, J. 848, 849, 851, 857, 881 [139]; 881 [140]; 881 [141]; 881 [142]; 881 [143]; 881 [144]; 881 [145] Mikhailov, A. 736, 756 [27]; 758 [73] Miles, R. 20, 53 [61]; 95, 146 [89] Miller, J. 585,596 [138] Miller, P. 64, 67, 68, 70-72, 74, 75, 87-89, 91 [32]; 91 [33]; 92 [41]; 92 [44]; 663, 667, 674 [150] Miller, R. 889, 915, 980 [160]; 980 [161] Milnor, J. 411,459 [49]; 887, 980 [162] Mimura, M. 1009, 1046, 1053 [ 129] Miranville, A. 831 [53] Mirollo, R. 40, 53 [45] Mischaikow, K. 426, 430, 431,434, 436, 445,447, 455, 458, 458 [5]; 458 [61; 458 [171; 459 [281; 459 [291; 459 [32]; 459 [33]; 459 [43]; 459 [44]; 459 [46]; 459 [47]; 459 [48]; 459 [50]; 459 [51]; 459 [52]; 459 [53]; 459 [54]; 460 [55]; 460 [56]; 460 [57]; 460 [58]; 460 [59]; 460 [60]; 463, 484, 498 [50]; 851,878 [55]; 922, 966, 967, 980 [163]; 980 [164] Mischenko, E. 127, 145 [50]
A u t h o r Index
Miura, R. 101,143 [16]; 609, 611,671 [77]; 671 [78]; 67411511
Mizel, V. 685, 693, 701, 702, 719 [26]; 720 [54]; 721 [65] Mizoguchi, N. 737, 738, 758 [65]; 758 [66] Mohammadi, B. 596 [139] Moise, I. 910, 974, 977 [91]; 980 [166] Mollenauer, L. 608, 674 [152] Moloney, J. 608, 674 [157]; 700, 720 [52]; 720 [53] Monteiro, A. 182, 217 [23] Montgomery, D. 584, 594 [99] Montgomery, R. 666, 671 [61] Moore, G. 165, 172, 179, 218 [53]; 219 [58]; 219 [59]; 219 [60]; 383,387 [64] Mora, X. 858, 865,877 [27]; 936, 966, 967, 980 [167] Morita, Y. 431,459 [54] Morris, C. 17, 42,53 [46]; 122, 145 [51] Morrison, P. 63, 91 [ 17] Moser, J. 603,629, 674 [153] Movahedi-Lankarani, H. 926, 980 [ 165] Mrozek, M. 420, 445, 447, 455, 458, 459 [41]; 459 [42]; 459 [48]; 460 [55]; 460 [56]; 460 [57]; 460 [58]; 460 [59]; 460 [61]; 460 [62]; 460 [63]; 460 [64]; 460 [65]; 922, 980 [163] Muraki, D. 704, 720 [48] Murray, J. 837, 881 [146]; 985,990, 1048, 1053 [121] Murray, R. 257, 264 [37] Musher, S. 614, 641,675 [ 198] Muto, V. 640, 674 [154]; 674 [155] Myers, M. 160, 218 [37]; 383,384, 387 [82]; 387 [83] Nadim, E 48, 49, 52 [39]; 53 [47] Nadirashvili, N. 809, 831 [66]; 848, 881 [147] Nagasaki, K. 744, 758 [67] Namah, G. 864, 881 [148] Natiello, M. 1037, 1055 [191] Neishtadt, A. 103, 145 [53]; 276, 279, 285, 298-300, 311 [2]; 312 [21] Nelkin, M. 561,594 [89] Nelson, D. 646, 671 [76] Neubelt, M. 608, 6 74 [ 152] Newell, A. 603, 608, 674 [156]; 674 [157]; 700, 716718, 720[32]; 720[35]; 720[52]; 720[53]; 721 [66]; 721 [67]; 721 [68]; 721 [73]; 762, 775, 776, 785, 786, 833 [109]; 833 [110]; 833 [111]; 833 [112] Newton, P. 681,682, 689, 721 [69]; 721 [70] Ni, W.-M. 739, 743, 747, 748, 757 [35]; 758 [68]; 758 [69]; 758 [70]; 859, 860, 879 [81]; 879 [82] Nickel, K. 852, 881 [149]; 954, 980 [168] Nicolaenko, B. 509, 510, 526, 527, 565,574-578, 580, 591 [5]; 591 [6]; 591 [7]; 591 [8]; 591 [9]; 591 [10]; 591 [11]; 591 [12]; 591 [13]; 591 [29]; 593 [57]; 593 [65]; 593 [66]; 593 [67]; 593 [79]; 595 [119];
1069
786, 798, 824, 830 [44]; 925, 926, 970, 975 [18]; 976 [56]; 980 [ 169] Nii, S. 989, 1009, 1031, 1032, 1053 [122]; 1053 [123]; 1053 [124]; 1053 [125]; 1053 [126] Nikolaev, E. 169, 187, 193,218 [46]; 360, 388 [102] Nirenberg, L. 521,592 [33]; 743, 757 [35]; 859, 860, 877 [20]; 877 [21]; 879 [81]; 879 [82] Nishida, T. 534, 549, 596 [140]; 596 [141] Nishiura, Y. 431,459 [43]; 704, 721 [71]; 1009, 1046, 1052 [88]; 1053 [127]; 1053 [128]; 1053 [129] Nitecki, Z. 288, 312 [22] Nohel, J. 946, 979 [143] NCrsett, S. 348, 349, 388 [86] Novikov, S. 603,674 [ 158] Nozaki, K. 971,980 [ 170] Nozakki, B. 774, 795,829 [12] Nussbaum, R. 463, 467, 479, 480, 483, 484, 487, 489, 490, 494-496, 498 [40]; 498 [41]; 498 [42]; 498 [43]; 498 [44]; 498 [45]; 498 [46]; 498 [47]; 498 [51]; 498 [52]; 498 [53]; 498 [54]; 498 [55]; 498 [56]; 498 [57]; 498 [58]; 498 [59]; 498 [60]; 498 [61]; 498 [62]; 499 [63]; 499 [64]; 499 [65]; 889, 911,945,946,9801171]; 9801172]; 9801173]; 9801174]
Nusse, H. 70, 91 [34] Nyman, B. 608, 674 [ 152] Oakson, G. 120, 146 [77] Ochs, G. 223,264 [29] Ockendon, J. 763,776, 830 [23] O'Donovan, M. 100, 146 [81] Oesterl6, J. 571,593 [64]; 925, 976 [54] Ogiwara, T. 851,881 [136]; 881 [137] O'Hern, C. 636, 674 [159] Ohkitani, K. 570, 596 [142] Ohnishi, I. 704, 721 [71 ] Ohta, T. 704, 721 [72] Oka, H. 431,436, 445, 459 [32]; 459 [43]; 459 [44]; 1032, 1052 [104] Okubo, A. 837, 881 [150] Oliva, W. 430, 459 [37]; 858, 879 [77]; 887, 897, 920922, 924, 978 [103]; 980 [175] Oliver, M. 798-800, 803,829 [10]; 832 [88]; 832 [89]; 933,973,980 [ 176] Olson, E. 565, 578, 593 [80]; 976 [71] Ono, H. 774, 831 [68] Onsager, L. 587, 596 [143] Or-Guil, M. 1033, 1034, 1053 [130] Orszag, S. 509, 596 [144]; 646, 675 [194] Osborn, J. 244, 264 [38] Oseledec, V. 372, 389 [ 124] Osinga, H. 367, 368, 385 [24]; 388 [104]; 389 [125]; 389 [ 126] Osipenko, G. 231,264 [39]
1070
A u t h o r Index
Ostrouchov, S. 1036, 1049 [7] Ott, E. 316, 329, 332, 333, 344 [19]; 371,387 [71] Ottino, J. 58, 78, 91 [35] Ovchinnikov, Y. 788, 789, 833 [113] Overman, E. 613, 614, 618, 619, 628, 629, 638, 669 [15]; 674 [145]; 674 [149] Packard, N. 387 [63] Paladin, G. 762, 775, 803, 830 [ 17] Palis, J. 870, 881 [151]; 920, 922, 937, 980 [177]; 980 [ 178]; 980 [ 179] Palmer, K. 315, 318, 320, 343, 344 [7]; 344 [8]; 344 [9]; 344 [10]; 354, 355, 386 [28]; 386 [30]; 386 [31]; 870, 881 [152]; 997, 998, 1053 [131]; 1053 [ 132] Palmore, J. 316, 344 [29] Pandit, R. 646, 672 [93] Papanicolaou, G. 650, 672 [108] Paparella, F. 77, 91 [36] Paraskevopoulos, P. 467, 489, 494, 498 [47]; 499 [66] Parisi, G. 646, 672 [ 100] Pascal, E 579, 593 [70] Pascal, M. 41, 52 [ 18] Passot, T. 718, 720 [35]; 721 [66]; 721 [67]; 721 [73]; 762, 833 [110] Patera, A. 338, 344 [35] Payne, L. 858, 880 [ 115] Pazy, A. 468,499 [67]; 840, 881 [153]; 980 [180]; 986, 1041, 1053 [133] Pego, R. 713, 719 [20]; 990, 1005, 1006, 1008, 1009, 1011, 1014, 1015, 1034, 1044, 1049 [26]; 1053 [134]; 1053 [135]; 1053 [136] Peletier, L. 685, 693,702, 721 [65] Pelikan, S. 365, 388 [ 101] Penrose, O. 715, 720 [44] Percival, I. 60, 91 [30] Perko, L. 324, 344 [30] Pernarowski, M. 101,145 [54]; 851,878 [55] Pesch, W. 775,776, 830 [33] Peterhof, D. 822, 827, 828, 833 [114]; 994, 995, 1046, 1053 [113]; 1053 [137] Peterson, J. 763, 776, 831 [45] Petrich, D. 704, 720 [48] Petzeltov& H. 864, 865, 878 [60] Petzold, L. 348, 350, 385 [10] Pierrehumbert, R. 79, 92 [37] Pillet, C.-A. 609, 674 [160] Pilyugin, S. 917, 974 [7] Pinsky, P. 116, 125, 145 [55] Pinto, D. 38, 39, 52 [28]; 53 [48] Pironneau, O. 596 [139] Pitaevskii, L. 603, 674 [ 158] Plant, R. 105, 145 [56] Pliss, V. 914, 980 [181]
Poincar6, H. 303, 312 [23] Poje, A. 74, 75, 87, 90, 91 [23]; 91 [24]; 92 [38]; 92 [46] Pol~i~ik, P. 743, 745-747, 749, 756 [13]; 756 [19]; 758 [71]; 758 [72]; 847-851, 853, 855, 858, 861866, 868, 869, 871, 873-875, 877 [27]; 877 [28]; 877 [29]; 877 [39]; 877 [40]; 878 [51]; 878 [61]; 878 [68]; 879 [73]; 879 [93]; 879 [98]; 879 [99]; 881 [145]; 881 [154]; 881 [155]; 881 [156]; 882 [157]; 882 [158]; 882 [159]; 882 [160]; 882 [161]; 882 [162]; 882 [163]; 882 [164]; 882 [165]; 882 [166]; 896, 937, 951,952, 958, 965, 975 [25]; 975 [26]; 976 [66]; 980 [182]; 980 [183]; 980 [ 1841 Ponce, E. 179, 214, 219 [64] Ponce, G. 522, 594 [104] Pontryagin, L. 214, 219 [61]; 470, 480, 482, 499 [68] Porcello, D. 139, 141,144 [36] Posashkov, S. 725, 726, 731,733, 756 [28]; 756 [29] Poschel, J. 603,612, 613, 672 [99]; 674 [161] Po2niak, M. 442, 460 [67] Pratt, L. 63, 64, 67, 68, 70-72, 74, 87-89, 91 [32]; 91 [33]; 92 [39]; 92 [41] Pra25_k, D. 574, 595 [133] Prince, D. 141,144 [35] Prince, P. 237, 263 [ 18] Prizzi, M. 869, 874, 875, 882 [167]; 882 [168]; 882 [ 169] Procaccia, I. 631,638, 671 [80] Prodi, G. 553,593 [81]; 931,970, 977 [72] Promislow, K. 927, 933, 951,980 [185]; 1017, 1044, 1052 [109]; 1052 [110] Protter, M. 843, 882 [170] Provenzale, A. 77, 78, 91 [26]; 91 [36]; 92 [40] Prtiss, J. 1041, 1053 [138] Pryce, J. 169, 218 [33]; 382, 387 [70] Pugh, C. 354, 365, 371, 388 [93]; 389 [127]; 871, 88O[lO3]
Pulvirenti, M. 509, 529, 530, 536, 580, 587-590, 592 [38]; 592 [39]; 592 [43]; 594 [95]; 596 [136] Pumir, A. 638, 674 [ 170] Qin, w.-x. 913,980 [ 186] Rabinowitz, E 696, 697, 720 [28]; 720 [29]; 721 [74] Rauch, J. 512, 596 [145]; 969, 975 [17] Rauenzahn, R. 509, 592 [31] Raugel, G. 527, 596 [146]; 787, 805, 807, 822, 831 [65]; 833 [115]; 853, 858, 863, 865, 879 [89]; 879 [90]; 879 [91]; 882 [171]; 896, 898, 912, 913, 915, 918-920, 922, 927, 930-934, 936, 937, 940, 942, 944, 948, 951-954, 962-970, 973, 975 [27]; 977 [77]; 978 [101]; 978 [104]; 978 [105];
Author Index 978 [106]; 978 [107]; 978 [108]; 978 [109]; 978 [110]; 978 [111]; 978 [112]; 978 [113]; 980 [164]; 981 [187]; 981 [188]; 981 [189]; 1042, 1051 [65]; 1051 [66]; 1052 [103]; 1053 [139] Reddien, G. 173, 212, 218 [35]; 218 [36] Reddy, S. 1005, 1054 [ 140] Reineck, J. 431, 432, 445, 447, 459 [32]; 460 [58]; 460 [60]; 460 [68] Ren, X. 705, 719 [8]; 719 [24] Reyna, L. 713, 721 [75] Reynolds, O. 338, 344 [31] Reynolds, W. 568, 596 [ 154] Rheinboldt, W. 151,155, 166, 219 [62]; 219 [63]; 379, 385 [231 Rhines, P. 88, 91 [27] Richeson, D. 414, 420, 426, 447,458 [24]; 460 [69] Riecke, H. 700, 720 [30]; 721 [76] Rinzel, J. 5, 17, 24, 41, 42, 47, 51, 52 [40]; 53 [49]; 53 [54]; 53 [57]; 54 [69]; 54 [70]; 95-98, 100,
101, 103, 105, 113, 116, 119, 122, 123, 125, 138142, 143 [2]; 143 [7]; 143 [8]; 143 [11]; 144 [26]; 144 [27]; 144 [28]; 144 [29]; 145 [55]; 145 [57]; 145 [58]; 145 [59]; 145 [60]; 145 [61]; 145 [69]; 145 [70]; 146 [81]; 146 [93]; 146 [94]; 146 [95]; 146 [96] Ritt, J. 23, 24, 26, 27, 29, 30, 49, 51 [10]; 52 [39]; 54 [71]; 123, 146 [97] Robbin, J. 420, 426, 447, 453,460 [70]; 460 [71] Robert, R. 585,596 [147]; 596 [148] Robinson, C. 407, 460 [72]; 460 [73]; 870, 871, 882 [172]; 926, 927, 981 [190] Rocha, C. 743, 749, 756 [17]; 858, 878 [69]; 879 [70]; 879 [78]; 882 [173]; 955-957, 976 [67]; 976 [68]; 977 [76]; 981 [191] Rodrfguez-Luis, A. 179, 214, 219 [64] Roelfsema, P. 38, 53 [50] Rogers, T. 711,720 [34] Rogerson, A. 64, 67, 68, 70-72, 74, 89, 91 [32]; 92 [41] Rom-Kedar, V. 60, 92 [42]; 92 [43] Romeiras, R. 316, 329, 332, 333,344 [19] Roose, D. 175, 212, 219 [65]; 219 [66]; 383, 384, 389 [ 122]; 389 [ 128]; 497 [ 11]; 498 [37] Roquejoffre, J.-M. 864, 881 [148]; 882 [174]; 882 [175]; 989, 1045, 1054 [141] Rosa, R. 567, 593 [75]; 910, 974, 980 [166] Rose, H. 640, 673 [127]; 673 [128] Rose, R. 101,144 [32] Rossby, H. 62, 91 [8] Rossignol, S. 95, 143 [13] Rothenberg, J. 661,674 [162]; 674 [163] Rottschafer, V. 773, 830 [40] Rougemont, J. 787, 831 [52] Roussarie, R. 379, 386 [55]; 386 [56] Rowlands, G. 789, 826, 830 [21 ]
1071
Rozov, N. 127, 145 [50] Rubenchik, A. 614, 641,675 [198] Rubin, J. 7, 41-46, 50, 53 [51]; 53 [52]; 95, 109, 113, 123, 124, 129, 131, 132, 134, 136-142, 145 [62]; 145 [63]; 145 [64]; 145 [65]; 1009, 1017, 1045, 1052 [96]; 1054 [142] Ruelle, D. 239, 263 [3]; 264 [40]; 369, 372-374, 387 [58]; 387 [59]; 389 [129]; 389 [130]; 849, 882 [ 176] Ruiz, A. 969, 981 [192] Rumpf, M. 232, 263,263 [12]; 264 [41] Russell, R. 157, 179, 217 [3]; 218 [53]; 219 [67]; 359, 363, 366, 373, 385 [9]; 386 [45]; 386 [46]; 1035, 1036, 1049 [8] Rutman, M. 880 [114] Rybakowski, K. 395,420, 460 [74]; 463,484, 499 [69]; 866, 868, 869, 872-875, 882 [161]; 882 [162]; 882 [168]; 882 [169]; 882 [177]; 882 [178]; 882 [ 179]; 882 [ 180]; 897, 937, 947, 948, 978 [ 114]; 980 [184]; 981 [193]; 981 [194] Rychlik, M. 367, 389 [ 131] Sachs, R. 1009, 1043-1045, 1049 [5]; 1053 [115]; 1053 [ 116] Sacks, P. 739, 747, 748, 758 [68]; 758 [69]; 758 [70]; 858, 877 [34]; 880 [115] Saint Raymond, L. 550, 594 [92] Saito, N. 646, 675 [ 182] Sakaguchi, H. 700, 721 [77]; 721 [78] Salamon, D. 399, 407, 420, 426, 440, 447, 453, 460 [66]; 460 [70]; 460 [71]; 460 [75] Salle, M. 616, 673 [139] Samarskii, A. 736, 756 [27]; 758 [73] Samelson, R. 63, 64, 86, 91 [14]; 92 [45] Sammartino, M. 555,592 [37] Sander, E. 711,721 [79] Sanders, J. 200, 219 [68]; 367, 370, 381,389 [132]; 3891133]
Sandstede, B. 74, 75, 88, 90 [3]; 92 [44]; 157, 180182, 190, 191, 193, 217 [12]; 217 [20]; 219 [69]; 219 [70]; 379, 385 [27]; 774, 797, 798, 832 [82]; 833 [116]; 853, 855, 875, 877 [29]; 879 [71]; 882 [181]; 989, 990, 994, 995,997, 999, 1005, 1006, 1009, 1010, 1014, 1016, 1017, 1020-1022, 10251029, 1031-1039, 1041, 1044, 1046-1048, 1049 [6]; 1049 [14]; 1050 [29]; 1050 [44]; 1051 [56]; 1052 [97]; 1052 [98]; 1052 [105]; 1053 [113]; 1053 [126]; 1053 [137]; 1054 [143]; 1054 [144]; 1054 [145]; 1054 [146]; 1054 [147]; 1054 [148]; 1054 [149]; 1054 [150]; 1054 [151]; 1054 [152]; 1054 [153]; 1054 [154]; 1054 [155]; 1054 [156]; 1054 [157]; 1054 [158]; 1054 [159]; 1054 [160]; 1054 [161]; 1055 [190]
1072
Author Index
Sanz-Serna, J. 315, 318, 320, 344 [32]; 352, 389 [134] Sardella, M. 895, 899, 900, 977 [86] Satin, L. 100, 143 [15] Sattinger, D. 1005, 1006, 1041, 1042, 1054 [162] Sauer, T. 315-318, 320, 321,323-325, 332, 333, 335, 338, 344 [14]; 344 [18]; 344 [33]; 344 [34]; 926, 981 [ 195] Sauzin, D. 302, 312 [ 16] Scarpellini, B. 765, 783, 786, 833 [117]; 1001, 1054 [163]; 1054 [164] Schachmanski, I. 297, 302, 312 [20] Schaeffer, D. 996, 1051 [75] Schatz, M. 338, 344 [35] Schecter, S. 180, 181,192, 219 [71]; 219 [72] Scheel, A. 763, 798, 833 [116]; 833 [120]; 834 [121]; 989, 994, 995, 997, 999, 1005, 1006, 1021, 1022, 1025-1027, 1032, 1036-1039, 1041, 1046-1048, 1051 [56]; 1051 [57]; 1053 [113]; 1053 [137]; 1054 [1521; 1054 [153]; 1054 [154]; 1054 [155]; 1054 [156]; 1054 [157]; 1054 [158]; 1054 [159]; 1054 [160]; 1054 [161]; 1054 [165] Scheffer, V. 581,596 [149] Scheurer, B. 970, 980 [ 169] Scheurle, J. 270, 276, 312 [9]; 370, 389 [135]; 927, 928, 932, 978 [115] Schmidt, L. 365,388 [ 101 ] Schmitt, E. 463, 499 [70] Schneider, G. 763-765, 773-775, 777, 779, 801,802, 805, 810, 815, 816, 818-822, 824-829, 830 [19] 831 [67]; 832 [87]; 833 [105]; 833 [106]; 833 [107] 833 [108]; 833 [114]; 834 [122]; 834 [123] 834 [124]; 834 [125]; 834 [126]; 834 [127] 834 [128]; 834 [129]; 834 [130]; 834 [131] 834 [132]; 834 [133]; 834 [134]; 834 [135] 834 [136]; 913, 980 [159]; 1041, 1047, 1054 [166] 1054 [167]; 1055 [168]; 1055 [169] Schneider, K. 579, 593 [71 ] Schober, C. 631,669 [ 1] Schuster, H. 638, 671 [82] Schfitte, C. 223, 249, 263 [14]; 263 [15]; 264 [42] Schfitze, O. 228, 263 [13] Schwartz, J. 95,144 [40] Scott, A. 608, 631, 640, 674 [146]; 674 [154]; 6741155]
Sedgwick, A. 382, 388 [95] Segel, L. 762, 776, 830 [35]; 834 [137] Segev, I. 51, 52 [40] Segur, H. 631, 661,674 [ 164]; 674 [ 165]; 674 [ 166]; 674 [ 167] Seibert, N. 887, 974 [6] Sejnowski, T. 95, 96, 103, 123, 138-142, 143 [6]; 144 [18]; 144 [19]; 144 [20]; 144 [21]; 144 [22]; 146 [80]
Sell, G. 495, 498 [48]; 498 [49]; 527, 573, 574, 593 [79]; 593 [82]; 596 [146]; 858, 861, 864, 877 [16]; 880 [123]; 880 [124]; 889, 897, 915, 934, 951, 977 [73]; 979 [148]; 980 [161]; 981 [196]; 981 [197]; 981 [198] Senn, W. 100, 146 [81 ] Serrin, J. 522, 596 [150]; 860, 882 [182] Sertl, S. 228, 250, 251,263 [6]; 263 [13] Seul, M. 704, 721 [80] Seydel, R. 151,219 [73] Shabat, A. 609, 610, 675 [199] Sharp, A. 48, 53 [53] Shatah, J. 608, 609, 621,627-631,646, 648, 670 [34]; 670 [35]; 671 [83]; 671 [84]; 673 [131]; 674 [147]; 674 [148]; 674 [168]; 1043, 1044, 1051 [78]; 1051 [79] Shaw, R. 387 [63] She, Z. 575,593 [66] Shelley, M. 650, 669 [28] Shen, W. 849, 857, 858, 882 [183] Sheng, P. 649, 674 [ 169] Shepeleva, A. 774, 833 [118]; 833 [119] Sherman, A. 17-19, 51, 53 [54]; 53 [57]; 53 [66]; 95, 96, 100, 143 [4]; 145 [66]; 145 [67]; 145 [69] Shilnikov, L. 1026, 1032, 1055 [ 170] Shinbrot, M. 522, 594 [100] Shnirelman, A. 581,596 [ 151 ] Showalter, K. 985,990, 1052 [99] Shraiman, B. 638, 672 [91]; 674 [170] Shub, M. 227, 264 [43]; 354, 356, 365, 371,388 [93]; 389 [127]; 389 [136]; 871,880 [103] Sideris, T. 512, 596 [152] Sidorov, N. 371,389 [137] Sidorowich, J. 374, 385 [ 1] Sigal, I. 788, 789, 833 [113] Sigvardt, K. 46, 53 [42] Sim6, C. 268, 279, 289, 293,311 [5]; 311 [7]; 312 [8]; 312 [24]; 370, 389 [138]; 389 [139] Simon, L. 866, 867, 883 [184]; 951,981 [199] Simondon, E 858, 878 [62]; 878 [63] Sinai, Y. 239, 264 [44]; 274, 312 [25] Singer, W. 5, 38, 53 [50]; 53 [55]; 95, 138, 143 [9] Sinha, A. 640, 671 [73] Skinner, F. 47, 48, 53 [53]; 53 [56]; 123, 145 [68] Slemrod, M. 887, 888, 896, 903,904, 977 [ 100] Slijep~evi6, S. 787, 831 [59]; 834 [138] Smagorinsky, J. 596 [153] Smale, S. 321, 322, 332, 344 [1]; 353, 354, 358, 369-371,385 [2]; 389 [140]; 390 [141]; 618, 629, 674 [171]; 920, 922, 980 [178] Smith, H. 494, 497 [34]; 499 [71]; 843, 845, 848, 851, 852, 858, 880 [125]; 880 [127]; 883 [185]; 883 [ 186]; 883 [ 187]; 883 [ 188] Smith, J. 100, 116, 143 [7]; 143 [8]
Author Index
Smith, L. 568, 596 [154] Smolen, P. 51, 53 [57]; 100, 101,145 [69]; 145 [70] Smoller, J. 438, 447, 458 [11]; 458 [12]; 458 [13]; 458 [14]; 460 [76]; 843, 858, 863, 883 [189]; 883 [190]; 951,981 [200]; 985, 990, 1055 [171] Sneppen, K. 638, 646, 674 [172] Sobolev, S. 526, 596 [155] Softer, A. 609, 631,674 [173]; 675 [174]; 675 [175] Sohr, H. 521,596 [156] Solh-Morales, J. 858, 865, 877 [27]; 936, 966, 967, 980 [ 167] Somers, D. 45, 46, 52 [36]; 53 [58]; 53 [59]; 113, 116, 145 [45]; 146 [71]; 146 [72] Sommeria, J. 585,596 [148] Sompolinsky, H. 339-341, 344 [3]; 638, 672 [89]; 6751176]
Sone, Y. 590 [2] Soner, H. 788, 789, 832 [76] Sorensen, D. 262, 264 [34]; 383, 390 [142]; 1036, 1049 [7] Sorger, R 763, 832 [86] Sogovi6ka, V. 855, 868, 882 [163] Soto, C. 123, 146 [97] Soto-Trevino, C. 23, 24, 26, 29, 30, 51, 52 [32]; 54 [71]; 105, 146 [73] Sotomayor, J. 379, 386 [55]; 386 [56] Souillard, B. 650, 670 [55] Souli, M. 718, 721 [67] Spalding, D. 522, 527, 551,595 [ 117] Spanier, E. 411,460 [77] Speer, E. 640, 673 [127]; 673 [128] Spence, A. 172, 212, 219 [60]; 219 [74]; 383,387 [64]; 389 [ 122] Spencer, T. 649, 675 [177] Spiegel, E. 203,217 [13]; 1033, 1050 [52] Spiteri, R. 157, 216 [2] Stanford, I. 32, 53 [63] Stasheff, J. 411,459 [49] Stein, E. 596 [157] Steinberg, V. 338, 344 [5] St6phanos, C. 160, 219 [75] Stephen, M. 646, 671 [76] Steriade, M. 95, 120, 138, 139, 141, 142, 143 [14]; 144 [20]; 146 [74]; 146 [75]; 146 [76]; 146 [77]; 146 [78]; 146 [79]; 146 [80] Stewartson, K. 733, 757 [48]; 762, 775,776, 779, 780, 795,830 [32]; 831 [69]; 834 [139] Straughan, B. 858, 880 [115] Strauss, W. 587, 591 [23]; 603, 605, 606, 608, 609, 671 [83]; 671 [84]; 675 [178]; 1043, 1044, 1051 [78]; 1051 [79] Strelcyn, J.-M. 372, 385 [13] Strogatz, S. 40, 53 [45] Strzodka, R. 232, 263 [12]
1073
Stuart, A. 352,390 [143]; 915,981 [201] Stuart, J. 733, 757 [48]; 762, 795,834 [139]; 834 [140] Stubbe, J. 970, 976 [43] Sturmfels, B. 383,387 [83] Sulem, C. 774, 830 [31 ] Sulem, P.-L. 561,594 [89]; 774, 830 [31] Suris, Y. 291,312 [26] Suzuki, T. 744, 758 [67] Suzuki, Y. 1009, 1046, 1052 [88] Svanidze, N. 297, 312 [14] Swartz, B. 157, 217 [14] Swift, J. 374, 385 [22]; 700, 701,721 [81] Swinney, H. 338, 344 [35]; 374, 385 [22]; 637, 675 [1831 Szmolyan, R 989, 1052 [105] Szymczak, A. 420, 444, 455, 458, 460 [59]; 460 [78]; 460 [791; 460 [801 Tabak, E. 641,643,645,673 [136] Tabak, J. 100, 146 [81] Tabanov, M. 273, 295, 297, 302, 312 [15]; 312 [20] Tagg, R. 338, 344 [35] Takfi6, R 848, 850, 851, 883 [191]" 883 [192]" 883 [193]; 883 [194]; 883 [195]; 883 [196] Takaishi, T. 431,459 [43] Takens, F. 372, 374, 378, 389 [130]; 390 [144]; 390 [145]; 565,596 [158] Takhtajan, L. 603, 611,671 [64] Tan, B. 939, 975 [33] Tanaka, S. 613,670 [43] Tang, T. 179, 218 [54] Tanga, P. 77, 91 [36] Taniuchi, Y. 523,594 [ 103] Tartar, L. 554, 596 [159]; 942, 950, 975 [19] Tavantzis, J. 739, 758 [70] Temam, R. 430, 460 [81 ]; 509, 565,567-578, 592 [56]; 593 [57]; 593 [58]; 593 [59]; 593 [67]; 593 [68]; 593 [75]; 593 [76]; 593 [77]; 593 [78]; 593 [79]; 593 [82]; 594 [83]; 594 [84]; 594 [85]; 596 [160]; 596 [161]; 787, 802, 805,808, 824, 834 [141]; 889, 904, 908, 923-927, 931, 933, 934, 951, 960, 964, 967, 970, 976 [45]; 976 [55]; 976 [56]; 976 [70]; 977 [73]; 977 [74]; 977 [75]; 977 [84]; 977 [85]; 980 [169]; 981 [202] Tereg6~ik, I. 847, 849-851, 855-858, 882 [164]; 882 [165]; 883 [197]; 883 [198] Terman, D. 7, 41-46, 50, 53 [51]; 53 [52]; 53 [60]; 95, 97, 100, 101, 106, 107, 109, 110, 113, 116, 117, 120-125, 128, 129, 131,132, 134, 136-142, 143 [5]; 145 [46]; 145 [61]; 145 [62]; 145 [63]; 145 [64]; 145 [65]; 145 [70]; 146 [82]; 146 [83]; 146 [84]; 146 [85]; 146 [86]; 146 [87]; 146 [88]; 146 [91]; 989, 1045, 1054 [141]
1074
Author Index
Terraneo, E. 523,594 [87] Theodorakopoulos, N. 640, 675 [ 179] Thieme, H. 845, 848, 851,883 [ 187]; 883 [ 188] Thieullen, P. 925, 981 [203] Thomae, S. 79, 91 [ 18] Thtimmler, V. 797, 829 [ 15] Tian, E 667, 675 [ 180] Timonen, J. 640, 670 [33] Tirapegui, E. 203, 217 [26] Titi, E. 527, 565, 592 [50]; 594 [98]; 595 [131]; 774, 798, 799, 830 [41]; 831 [46]; 931, 933, 951, 970, 973,975 [41]; 976 [62]; 978 [128]; 980 [176] Toda, M. 603, 646, 675 [181]; 675 [182] Toland, J. 1032, 1049 [25] Tomei, C. 603, 611,668, 669 [ 11]; 670 [49] Toner, M. 90, 92 [46] Traub, R. 5, 20, 23, 27, 32, 34-37, 39, 51 [3]; 52 [35]; 53 [61]; 53 [62]; 53 [63]; 53 [64]; 54 [72]; 95, 120, 144 [39]; 146 [89]; 146 [98] Trefethen, L. 1005, 1054 [140]; 1055 [175] Treve, Y. 931,976 [70] Treves, Y. 593 [78] Troy, W. 457, 460 [82]; 685, 693, 702, 721 [65]; 859, 883 [ 199] Trubowitz, E. 603, 611-613, 670 [50]; 674 [161] Tsameret, A. 338, 344 [5] Tsimring, L. 374, 385 [ 1] Tu, L. 411,458 [4] Tucker, W. 240, 264 [45]; 358, 390 [146] Turaev, D. 1048, 1051 [58] Turing, A. 679, 721 [82] Tyson, J. 989, 990, 1009, 1045, 1048, 1055 [ 176] Tzanetis, D. 740, 741, 747, 748, 755, 757 [52]; 757 [53]; 757 [54]; 758 [74] Uecker, H. 765, 833 [107]; 1041, 1047, 1055 [178]; 1055 [ 179] Ukai, S. 534, 550, 591 [25]; 596 [162] Ulam, S. 238, 264 [46] Ural'ceva, N. 748, 757 [55] Ushiki, S. 289, 312 [27] Utke, J. 351,363,387 [73] Vakulenko, S. 868, 883 [200]; 883 [201]; 883 [202] van Gils, S. 184, 219 [76]; 463, 464, 468, 469, 495, 497[9]
van Harten, A. 763, 775, 815, 830 [19]; 830 [42]; 834 [142]; 834 [143] van Hemmen, J. 123, 144 [25] van Hemmen, L. 22-25, 52 [20]; 52 [30] Van Loan, C. 327, 344 [ 17] van Neerven, J. 468, 499 [72] van Saarloos, W. 638, 674 [170]; 774, 795, 834 [144]; 990, 1009, 1042, 1050 [48]; 1055 [177]
van Strien, S. 371,386 [36] van Veldhuizen, M. 365, 390 [ 147] Van Vleck, E. 343, 344 [11]; 344 [12]; 344 [13]; 355, 366, 373,386 [29]; 386 [43]; 386 [46] van Vreeswijk, C. 13, 16, 22-24, 53 [65] Vanderbauwhede, A. 727, 758 [75]; 863, 872, 883 [203]; 883 [204]; 883 [205]; 996, 1019, 1021, 1025, 1055 [180]; 1055 [181] Vanier, M. 13, 51 [ 11] Vaninsky, K. 640, 673 [143 ] vanVreeswijk, C. 123, 146 [90] Varadhan, S. 860, 877 [21] Vastano, J. 637, 675 [183] V~izquez, J. 736, 737, 739, 740, 756 [30]; 757 [31]; 757 [32]; 757 [33]; 858, 879 [80] Vegter, G. 368, 385 [24] Vel~izquez, J. 729, 730, 732-735, 755 [1]; 756 [26]; 757 [40]; 757 [41]; 757 [42]; 757 [43]; 757 [44]; 757 [45]; 757 [46]; 757 [47]; 758 [76]; 758 [77]; 758 [78]; 758 [79] Velo, G. 798, 800, 801,805, 831 [61] Venakides, S. 665, 667, 668, 670 [46]; 670 [47]; 670 [51]; 673 [126]; 675 [184] Verduyn-Lunel, S. 463, 464, 467-470, 479, 495, 497 [9]; 497 [19]; 497 [26]; 859, 879 [79]; 889, 890, 903, 945,978 [116] V6ron, L. 858, 877 [38] Vershik, A. 371,388 [ 100]; 389 [ 137] Villanueva, J. 388 [98] Vishik, M. 509, 527, 553, 567, 591 [14]; 591 [15]; 591 [16]; 591 [17]; 596 [163]; 596 [164]; 597 [165]; 597 [166]; 763, 787, 801, 802, 805, 806, 808, 829 [7]; 829 [8]; 889, 913, 915, 918, 925,939-942, 944, 951,960, 962, 963, 966, 970, 974 [8]; 974 [9]; 974 [10]; 974 [11]; 974 [12]; 974 [13]; 975 [34]; 975 [35]; 981 [204]; 1046, 1051 [57] Vishnevskii, M. 848, 851,865, 883 [206]; 883 [207]; 883 [213] Vivaldi, E 365, 387 [72] Volpert, A. 985,986, 989, 990, 1045, 1046, 1055 [182] Volpert, V. 773, 832 [93]; 985, 986, 989, 990, 1045, 1046, 1054 [141]; 1055 [182] Volterra, V. 945,981 [205]; 981 [206] von Hardenberg, J. 78, 91 [26] von Stein, A. 38, 54 [67] von Wahl, W. 521,596 [156] Vuillermot, E 851,877 [22]; 883 [208] Vulpiani, A. 762, 775, 803,830 [17] Waller, I. 338,344 [36] Wallman, H. 924, 978 [ 125] Walther, H.-O. 463, 464, 468, 469, 495, 497 [8]; 497 [9]; 497 [32]; 497 [33]; 499 [73]
Author Index
Wan, Y.-H. 184, 218 [40] Wang, D. 45, 51 [7]; 53 [60]; 113, 116, 129, 146 [87]; 146 [91] Wang, X. 705, 719 [8]; 720 [45]; 910, 974, 980 [166]; 1035, 1036, 1050 [44] Wang, X.-J. 24, 41, 42, 47, 54 [68]; 54 [69]; 54 [70]; 95, 96, 103, 119, 122, 123, 138-142, 144 [28]; 144 [29]; 145 [61]; 146 [92]; 146 [93]; 146 [94]; 146 [95]; 146 [96]; 157, 181,217 [20] Wanner, G. 348-350, 388 [86]; 388 [87] Wanner, T. 710, 711,720 [56]; 720 [57]; 721 [79] Ward, M. 713, 715, 721 [75]; 721 [83]; 721 [84]; 721 [85]; 722 [86] Warner, E 348, 390 [148] Wasserman, A. 863,883 [190]; 951,981 [200] Watson, D. 105, 146 [73] Wayne, C. 609, 630, 670 [40]; 670 [48]; 671 [59]; 674 [160]; 675 [185]; 765, 790, 791, 830 [34]; 831 [51]; 1041, 1042, 1050 [49]; 1050 [50] Webb, G. 908, 960, 965, 968, 981 [207]; 981 [208]; 981 [209] Weichman, P. 585,596 [138] Weigend, A. 374, 390 [ 149] Weinberger, H. 843,882 [170] Weinstein, A. 631,669 [14] Weinstein, M. 608, 609, 631, 674 [173]; 675 [174]; 675 [175]; 675 [186]; 675 [187]; 675 [188]; 788, 834 [145]; 990, 1005, 1006, 1008, 1009, 1011, 1014, 1015, 1044, 1053 [134]; 1053 [135]; 1053 [136]; 1055 [183] Weissler, E 523,597 [167]; 725, 758 [80] Wells, J. 939, 944, 981 [210] Werner, B. 175,219 [77] West, B. 358, 388 [94] Westervelt, R. 327, 344 [20] White, B. 650, 672 [ 108] White, J. 23, 24, 26, 27, 29, 30, 41, 51 [1]; 51 [10]; 54 [71]; 123, 146 [97] Whitehead, J. 716, 721 [68]; 762, 775,776, 833 [111]; 833 [1121 Whitham, G. 603, 607, 664, 666, 675 [1891; 675 [190] Whittington, M. 5, 23, 27, 32, 34-37, 39, 51 [3]; 52 [35]; 53 [62]; 53 [63]; 53 [64]; 54 [72]; 95, 120, 144 [391; 146 [981 Wickerhauser, M. 579, 593 [70] Wierse, A. 263, 264 [41] Wiggins, S. 60, 90 [4]; 90 [5]; 91 [16]; 91 [19]; 91 [28]; 92 [42]; 92 [43]; 92 [47]; 603, 618, 621, 627-630, 672 [881; 673 [1311; 673 [1321; 673 [1331; 674 [1451; 675 [191]; 870, 871,883 [2091; 883 [2101 Wilkinson, J. 383,390 [ 150] Williams, R. 358, 387 [84] Williams, T. 8, 54 [73] Wilson, C. 49, 54 [74]
1075
Wilson, R. 804, 834 [146]; 834 [147] Winfree, A. 31, 54 [75] Winkler, W. 612, 673 [135] Wisdom, J. 352, 390 [151] Witt, I. 966, 981 [211] Wittenberg, R. 636, 675 [ 192] Wittwer, R 1041, 1050 [50] Wolf, A. 356, 374, 385 [22]; 390 [152] Wolfrum, M. 981 [212] Wolibner, W. 515,597 [168] Worfolk, R 368, 381,388 [85]; 390 [153] Wright, E. 476, 499 [74] Wu, J. 497 [33]; 499 [75]; 852, 883 [211] Wulff, C. 798, 833 [116]; 1041, 1048, 1051 [56]; 1054 [160]; 1054 [161] Xiang, Y. 384, 390 [154] Xie, X. 495,499 [76]; 499 [771; 499 [781; 499 [79] Xin, J. 788, 789, 832 [90]; 834 [145]; 986, 1042, 1046, 1055 [1841 Xiong, C. 628, 629, 674 [1451 Xun, J. 714, 719 [91; 719 [10] Yakhot, V. 509, 596 [144]; 646, 675 [193]; 675 [194] Yakovenko, S. 367, 388 [96] Yanagida, E. 737, 738, 758 [65]; 758 [66]; 850, 882 [166]; 989, 1009, 1032, 1055 [185]; 1055 [186]; 1055 [ 187] Yang, C. 262, 264 [34] Yang, H. 79, 92 [37] Yarom, Y. 51, 52 [40]; 103, 145 [48] Yew, A. 95, 146 [88]; 1032, 1044, 1055 [188]; 1055 [189]; 1055 [190] Yi, Y. 849, 857, 858,882 [183] Yoccoz, J.-C. 365,390 [155] Yorke, J. 70, 91 [34]; 240, 264 [33]; 315-318, 320, 321, 323-325, 327, 329, 332, 333, 335, 338, 339, 344 [14]; 344 [17]; 344 [18]; 344 [19]; 344 [21]; 344 [33]; 344 [34]; 354, 355, 368, 371, 385 [4]; 387 [71]; 388 [88]; 389 [114]; 495, 496, 497 [27]; 497 [28]; 497 [29]; 926, 981 [195] Yosid, K. 468, 469, 499 [80] Yosida, K. 243,264 [47] You, Y. 889, 897, 915, 981 [198] You, Z. 371,385 [4] Young, L.-S. 240, 257, 263 [1]; 369, 371-373, 385 [12]; 390 [156]; 390 [157]; 390 [158] Yu, S.-X. 851,880 [108] Yuan, G. 79, 91 [25] Yudovich, V. 514, 515,597 [169] Zaag, H. 733,736, 756 [16]; 758 [63]; 758 [64] Zabusky, N. 64, 65, 91 [20]
1076
Author Index
Zakharov, V. 603, 609, 610, 614, 641,645, 671 [72]; 674 [158]; 675 [195]; 675 [196]; 675 [197]; 675 [198]; 675 [199]; 774, 834 [148] Zaleski, S. 638, 646, 647, 675 [200] Zaslavski, A. 701,702, 721 [60]; 721 [61]; 722 [87] Zauderer, E. 959, 981 [213] Zehnder, E. 399, 458 [15]; 458 [16]; 459 [38] Zelawski, M. 458, 460 [65] Zelenyak, T. 851,853, 865, 883 [212]; 883 [213]; 951, 953,981 [214] Zelik, S. 763, 801,804, 806, 831 [54]; 834 [149] Zemach, C. 509, 592 [31 ] Zeng, C. 627, 630, 669 [9]; 674 [168]; 871,877 [17]
Zhang, Y. 646, 672 [100] Zhou, A. 239, 263 [17] Zhou, X. 611, 630, 651-653, 659, 660, 665, 668, 670 [45]; 670 [46]; 670 [47]; 670 [51]; 670 [52]; 670 [53]; 670 [54] Zhu, H.-R. 18, 19, 53 [66] Ziegra, A. 824-826, 833 [108] Zimmermann, M. 1037, 1055 [191 ] Zl~mal, M. 966, 981 [215]; 982 [216] Zuazua, E. 968, 970, 976 [61]; 982 [217] Zumbrun, K. 1009, 1014, 1016, 1017, 1041, 1042, 1051 [72]; 1055 [192]
Subject Index
a-condensing, 910 a-contracting, 910 - of order k, 911 or-limit set, 353, 893 ot.:-limit set, 893, 898 g-pseudo-trajectory, 317 e-k model, 507 e-shadowing trajectory, 317 co-limit set, 353, 583,841,855, 864, 871,893
- dimension, 804 -exponential, 506-509, 565,573-578 -generalized limit, 807, 823 - global, 224, 505-507, 558, 565, 566, 575, 576, 580, 802, 805,887, 898, 904, 906, 940, 941, 943,946, 950, 961,973 - - minimal, 912 - hyperbolic, 244 - local, 898 - Lorenz, 235 -relative global, 225 - topological, 576 attractor-repeller pair decomposition, 399 attracts points locally, 900 augmented system, 176 - fold, 171 - Hopf, 173 -minimally, 168, 169, 185, 187, 193, 198, 212 -standard, 171,172, 185, 199 AUTO, 359 automatic differentiation, 351 averaging. 279 Avogadro number, 505, 529
absolutely continuous, 240, 241 absorbing set, 902 activator-inhibitor model, 680, 703 adaptive subdivision algorithm, 254-257 adjoint equation, 998 algebraic independence, 872, 874, 876 almost cyclic behavior, 250 almost invariant, 249-253 almost synchronous solutions, 125 alpha rhythm, 38 amalgamation, 444 amplitude equations, 716 analytic semigroup, 840 Anosov flows, 870, 871 Anosov system, 239 anticyclone of the Acores, 580 antiphase solution, 19, 109, 116, 118, 123 approximation property, 814 Arnoldi method, 262 asymptotic behavior, 838, 850-852, 867 asymptotically compact, 897 asymptotically smooth, 887, 896, 897, 904, 907910, 961,962 asymptotically smooth gradient system, 940 asymptotically smooth semigroup, 907 asymptotically stable, 901 attract, 894 attracting set, 224 attractivity property, 818 attractor, 399, 563,565, 887 -comparison of, 806
backward uniqueness, 892, 942, 950 ball Bx(e, r), 939 basic sets, 353 basin of attraction, 224 BBGKY hierarchy, 507, 529, 533, 536 B6nard convection, 700 bi-focus, 188 bifurcating solutions, 695-697 bifurcation, 374, 679, 745,748, 1026 - Bautin, 182, 186, 187, 193, 206, 216 - Bogdanov-Takens, 182, 185-187, 193, 200, 205 - codimension-2 equilibrium, 182 - codimension-3, 193 - cusp, 185, 187, 197, 200, 205 - detected, 158. 169 - double-Hopf, 171, 186, 187, 196, 200, 208, 216 -flip, 161-164, 167, 168, 176, 177 1077
1078
Subject Index
-fold, 158, 161-163, 176-178, 182, 183, 189, 193, 205 - continuation of, 172 numerical continuation of, 172 - simple, 171, 172 - f o l d + double-Hopf, 196, 197 - fold-Hopf, 185-187, 195, 200, 201,207 - Gavrilov-Guckenheimer, 185 generalized Hopf, 186, 198, 201 homoclinic - codimension-2, 187 - - saddle, 201 flip, 189 - Hopf, 6, 7, 16, 98, 103, 104, 158, 159, 163, 167, 175, 182, 184, 186, 189, 201,383 - direction of, 184 numerical continuation of, 173 - simple, 160, 167, 173, 174 194, 196, 197 - Hopf-Hopf, 187 inclination-flip, 191 - Neimark-Sacker, 161, 176, 177, 202, 203 homoclinic and heteroclinic, 1032 orbit-flip, 190, 191 point, 157 resonant double-Hopf, 197 - Shil'nikov homoclinic, 202, 203 Shilnikov-Hopf, 189 torus, 162 - triple zero eigenvalue, 194, 196 bifurcation, global, 463,494, 495 bifurcation curve, 169, 171,173 Birkhoff Ergodic Theorem, 239 Bloch-Floquet ansatz, 782 blocked connections, 957 Bogdanov-Takens bifurcation, 378, see also bifurcation Boltzmann equation, 505, 507, 510, 511,527-529, 533-536, 538, 541,544, 545, 549-551,553, 589 Boltzmann hierarchy, 533,534 Boltzmann-Grad limit, 505, 528, 529 bordered LU-decomposition, 155 bordered system, 162, 170, 171,204, 205,211,212 bordering function, 163, 177 bordering technique, 162, 169, 176, 193 Borel measure, 569 boundary condition non-transverse, 1038 -periodic, 1035, 1036 -projection, 180, 1035 -separated, 952, 1037 boundary map, 422 boundary operator, 427 -
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h
o
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n
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H
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o
p
f
-
B
o
g
d
a
n
o
v
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T
a
k
e
n
s
,
boundary value method, global, 363 boundary-value problem, 359 - for PDE-spectra, 1001, 1035 - for PFDE-spectra, 1021 - for travelling waves, 1035 -truncated, 180-182, 188, 189, 191-193 bounded dissipative, 888, 902 Boussinesq equation, 558, 700 branch of homoclinic orbits, 188 branch point, simple, 165, 187, 214 branch switching, 168 at a Bautin bifurcation, 214 at a Bogdanov-Takens point, 212 at a cusp point, 210 - at codimension-2 bifurcation, 209 at simple bifurcation point, 165 at simple binors points, 164 branching equation, algebraic, 164 brittleness 13, 323, 324 bursting neurons, 41 oscillation, 95 elliptic, 96, 102 - - parabolic, 96, 104 - - square-wave, 96, 98, 106 oscillators, 7 -
-
-
-
-
-
-
-
-
-
C0-semigroup, 468, 470, 1041 C-slow entrance point, 446 C-slow exit point, 445 Cahn-Hilliard equation, 680, 708-715,774 canonical measure, 506 cascading, 956 cell-mapping techniques, 259 center manifold, 8, 31, 151, 725-729, 731, 732, 748, 872 chain recurrent set, 224, 230, 406 chaotic advection, 58, 64 chaotic dynamics, 97, 100 chaotic invariant sets, 369 Chapman-Enskog expansion, 538 characteristic equation, 469-471,492, 496 characteristic length, 709, 710, 712 Chua's circuit, 230, 252, 253 circular restricted three body problem, 237 classical solution, 949, 959 clustering, 95, 109 coarsening, 709, 712 coherent structures, 506, 507, 509, 579, 580 compact dissipative, 902 compact semiflow, 841 comparison principle, 843, 1046 compartments, 49
1079
Subject Index
complete blow-up, 740, 741, 751 computer-generated trajectory, 315 concave hull, 690 cone condition, 934 conjugacy, 920, 922 Conley index, 922 homological, 409 - homology, 415 - homotopy, 409, 415 map, 414 pair, 408 - triple, 421 connecting homomorphism, 422 connecting orbit, 400, 954, 955,958 connection matrix, 427 connections between equilibria, 725, 736 conservation laws, 1009, 1012, 1017, 1041, 1042 conservative systems, 1006 conserved functional, 1043 consistency equations, 22 constrained minimization, 1043 continuation, 396, 397 equilibrium of codimension-2, 193 - fold bifurcation, 168 homoclinic, 181, 182 Hopf bifurcation, 168 - travelling waves, 1034 unique, 969 continuation method, 151,233,245,379 continuous -diffusion process, 335 system, 890, 891 -gradient system, 936, 947 -group, 890, 892 semigroup, 890 shadowability, 317 spiking, 100 control parameter, 679, 680 convergence, 847, 848, 853, 865, 866 quasiconvergent, 844, 845 convergent orbit, 937, 953 convolution, 691,704, 705 Coriolis force, 526 correlation dimension, 373 Couette-Taylor problem, 777, 827 critical exponents, 737 Cross-Newell equation, 718 curl, 515, 525 cyclic behavior, 242, 246 -
-
-
-
-
-
degree - 1 map, 427 delays, 12 delta rhythm, 138 dendrite, 49 dense output, 351 depressing synapse, 48 determining modes, 927, 928, 931,964, 973 differential-delay equations, 463-496 diffusion, 6, 16, 709 diffusive mixing, 792 dimension per unit volume, 805 Hausdorff, see Hausdorff dimension dimension topological, 923 dimensionless parameters, 28 Diophantine condition, 526 Dirac measure, 240 discrete dynamical system, 223, 890 discrete semigroup, 890 dispersion relation, 1000, 1002, 1025, 1034 dispersion relative, 78-80, 82 dispersive wave turbulence, 640-645, 668 dissipative wave numbers, 507 distance metrics, 43 domain of attraction, 739, 746 double bordering, 171 double rotor system, 329 doublets, 32 -
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d
y
n
a
m
i
c
a
l
-
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damped wave equation, 933,958 defect measure, 507, 553,554 defects, 718 defining equations, 375
Eckhaus criterion, 786 eddies, 561,564 eddy diffusivity, 64, 86 effective invariant manifold, 70, 71, 89 effective stochastic dynamics, 646-649 eigenmeasure, 242 eigenvalue -critical, 1021, 1025, 1027, 1031 -isolated, 997, 1008, 1037 - leading, 1025 - double real, 188 -multiplicity, 996-997, 1002, 1008 991, 1014-1017, 1038 -temporal, 987, 996 eigenvalue problem, 987 electrical coupling, 27, 49 electrical synapses, 9, 16 electrically coupled neurons, 6 elliptic equations, 859 energies, 699 energy, 691,702, 707 energy functional, 681,682, 691, 701, 718 energy spectra of turbulence, 553, 559 enstrophy, 521,525,560-562 entropy, 506, 507, 512, 528, 529, 540, 541, 715, 716 -
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s
p
a
t
i
a
l
,
1080
Subject Index
epileptiform oscillations, 138, 142 equation adjoint variational, 1019, 1025 - homological, 203, 210, 212, 215 - variational, 1019, 1025 equilibria - hyperbolic, 193 - non-hyperbolic, 188, 189 equilibrium -hyperbolic, 157, 177, 180 ergodic, 558 - hypothesis, 528, 552 - measure, 239 -theorem, 506, 567 ergodicity, 558, 559, 568 Euler equation, 505, 506, 509-515, 517-519, 526, 528, 529, 538-540, 549, 554, 555, 581,582, 587-589 - 2d, 583 incompressible, 513 Euler limit, 538, 539, 541 Eulerian stagnation point, 74 Evans function, 997, 1002, 1006, 1046 - definition, 1007 essential spectrum, 1012 -multi-bump pulse, 1027, 1031 - on bounded intervals, 1038 evolution laws, 679 evolution operator, 992 excitable media, 109, 113, 118, 121, 123,679 excitation, 23 excitatory synapse, 15 exhaustion technique, 262 exit set, 408 explicit method, 350 exponential asymptotics, 715 exponential attractor, 506, 507, 509, 565, 573-578 exponential decay, 568 exponential dichotomy, 992, 997-999, 1007, 1014, 1016, 1047, 1048 adjoint equation, 999 exponential separation, 849 extended gradient flow, 787 extended system, 155, 161,165 extremum point, 159, 162
Floquet theory, 1001 flow box theorem, 285 fluid dynamics limits, 537 fluid exchange, 58, 60, 63, 68 fold simple, 153 - definition of, 159 fold curve, 211, 213 forced damped pendulum, 327 FourLegs map, 250 fractal dimension, 565, 574, 924, 926, 951, 964, 974 Fredholm index, 996, 998, 1003, 1004 frequency, 10 frequency regulation, 26 functional differential equation, 463-496, 928, 931 fundamental neighborhood, 224
-
-
-
-
fast threshold modulation (FTM), 7, 42, 116, 138 finite-dimensional combined trajectory, 941 finite-time transport, 70, 72, 74 FitzHugh-Nagumo equation, 703,989, 1027, 1032 fixed point, 463-466, 484, 485, 491,492, 494 fixed point index, 463, 484-488, 492, 494 Floquet bundles, 856 Floquet exponent, 1001, 1019, 1021
-
-
GAIO, 258 Galerkin method, 241,243, 244 gamma rhythm, 31 gap condition, 934, 935, 951 gap junctions, 9 gap lemma, 1016 generic, 917 Gevrey regularity, 933, 973 Gibbs measure, 587, 589 Gierer-Meinhardt system, 715 Ginzburg-Landau -energy, 710, 712 -equation 712, 786, 970, 990, 1009, 1041 --derivation of, 767 - - complex, 795, 798 - - cubic-quintic, 774 -formalism, 762, 765, 809 manifold, 820 operator, 716 - vortices, 788 glitch, 321 gradient flow, 680-682, 684, 701,706, 716, 718 gradient structure, 865, 866 gradient system, 889, 936, 942, 953 gradient-like, 864, 874 granular materials, 700 GRAPE, 263 graph transform, 368 Grashoff number, 558, 568-570, 576 Green's function, 704, 705 group velocity, 766, 1000, 1002 -
-
"half-center" oscillators, 46 Hamilton-Jacobi equation, 736, 737 Hamiltonian, 588, 589
Subject Index Hamiltonian PDEs, 1009, 1011, 1042 Hamiltonian system, 213,214, 237, 505,507, 527, 528, 588, 589 hard sphere model, 529 Hausdorff dimension, 565,572, 923,941, 951 Hausdorff distance, 889 Hausdorff measure, 521 heat equation, 839, 948 Hdnon attractor, 229 Hdnon map, 228, 248, 252, 253,256, 257 Hermite polynomial, 727, 738 heteroclinic, 940, 1031 heterogeneity, 24 hierarchy, 505,507, 510, 527-529, 537, 553 hippocampal slice, 31 Hocking-Stewartson equation, 776 Hodgkin-Huxley model, 6, 96, 139 "hold-and-fire" systems, 31 homoclinic bifurcation, 103 intersection, 318 -orbit, 16, 97, 98, 104, 107, 613-618, 621-631 -points, 274, 288 - solution, 790, 795 tangency, 318 homoclinic orbit, 179, 190, 191, 201, 213, 216, 1020, 1023 -transverse, 870, 871 homotopic, 408 homotopy class, 408 homotopy continuation, 323 Hopf bifurcation, 6, 7, 16, 98, 103, 104, 158, 159, 163, 167, 175, 182, 184, 186, 189, 201,383 Hopf curve, 213 Hopf equation, 567 horseshoe, 413, 419, 441 hydrodynamic limit, 538 hyperbolic, 318 attractor, 244 point, 938, 951 point, 938 - matrix, 992 -normally, 365, 865 - set, 227 - structure, 353 -
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q
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u
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b
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m
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ignition variables, 731 implicit methods, 350 inertial dynamical system, 565 inertial form, 934 inertial manifold, 507, 509, 573-575, 934, 935, 955,967 inertial range, 507, 560, 563 infinite delay, 928
1081
infinite horizon variational problems, 701 infinitesimal phase response curve, 9 infinitesimal PRC, 41 inheritable, 450 inhibition, 23 inhibitory coupling, 16 instability absolute vs. convective, 1005 lntegrable systems, 1008, 1012 integrate-and-fire model, 6, 20 integrator, 267 lntegrodifferential equation, 705 interaction kernel, 2 l interaction of pulses, 1018, 1033 interfaces, 712, 713,715 lntermittency, 507 internal energy, 715 internal layers, 712 intersection test, 259, 260 interval arithmetic, 448 lnvariance -principle of LaSalle, 937 translation, 988-989, l018, 1028 invariant, 89 l, 892, 895 invariant cylinder, 50 invariant manifold, 58, 59, 71, 72, 233, 574, 868, 870, 874, 875 -center, 8, 31, 151, 183-185, 200-206, 210, 725729, 73 l, 732, 748, 872 center-stable, 178 center-unstable, 178 - inertial, 507, 509, 573-575, 824, 934, 935, 955, 967 spatial center, 811 - stable, 227, 318, 366, 921,936 -stable, theorem, 227 strong unstable, 190 -unstable, 227, 245, 318, 366, 679, 713,714, 921, 939 invariant, positively, 408, 891,934 invariant measure, 238-240, 244, 246, 247, 507 -absolutely continuous, 245,256 - attractors, 557 -natural, 239, 257 invariant set, 224, 245, 353, 395 - maximal compact, 903 lnvariant tori, 365 lnvariantly connected, 891,895, 899 inverse scattering theory, 1008 ionic currents, 38 lrreversibility, 505 isolated invariant set, 396 isolating block, 408 -
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-
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1082
Subject Index
isolating neighborhood, 396 singular, 445 isotropy hypothesis, 559
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J-stable, 922 Joyce-Montgomery equation, 584, 586 Jupiter red spot, 580 k-e model, 507, 509, 550-552 KAM theory, 365 kinetics of diffusion, 709 Kirchg/issner reduction, 810 knotted flow, 232 Knudsen and Reynold number, 538 Knudsen number, 510, 511, 528, 529, 537, 545, 549, 550, 553 Kolmogorov dissipation wave number, 563 exponent, 528 - flow, 775 inertial range, 563 - Kraichnan wave, 556, 562, 563,568 - scaling law, 563,570 - spectrum, 563 - wave number, 563,564 Kolmogorov-Petrovsky-Piskunov (KPP) equation, 1042 Korteweg-de Vries equation, 989, 1008, 1013 Kraichnan inertial range, 563 - scenario, 561 - spectrum, 563 Krein-Rutman theorem, 486, 487 Kullback entropy, 585 Kuramoto-Sivashinsky equation, 774, 970 (Kuratowski)-measure of noncompactness, 910 -
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L l-connection, 743,750, 751,753,754 L 1-continuation, 741,755 L 1-solution, 739-741,747, 751,754, 755 Lagrange multiplier, 513 Lagrangian coordinates, 513, 516 Lagrangian transport, 58, 64, 70 Landau equation, 770 lap number, 954 Leray, 566 Leray reduction, 415 Leray-Schauder degree, 485, 487, 488 limit capacity, 924 linear stability, 689 Lin's method, 1022, 1034 lobe dynamics, 60, 68 localized patterns, 700 locally compact dissipative, 902
LOCBIF, 36O logistic map, 245,256 long-range synchronization, 31, 35 Lorenz system, 235, 245, 246, 558 Lyapunov coefficient 184, 186, 199 - second, 201,206 Lyapunov dimension, 576 Lyapunov exponents, 79, 331,571,576, 849 Lyapunov functional, 50, 679, 682, 716, 861,864, 889, 936, 941, Lyapunov number, 571,576 Lyapunov-Schmidt reduction, 714 - f i r s t ,
Mach number, 509, 511,538, 545, 553 Marl6 projection, 577, 578 Mafir's theorem, 577 manifold, see also invariant manifold Ginzburg-Landau, 820 manifold approach, 714 marginals, 589 Markov partitions, 371 matched asymptotic expansions, 726, 733 materials science, 701, 712 Maxwellian, 534, 535, 538, 539-541, 545, 548, 550, 553 mean field equation, 584 mean free path, 528 measure - ergodic, 239 - invariant, 239 - natural, 372 - Sinai-Ruelle-Bowen (SRB), 238, 239, 244, 257, 372 Melnikov integral, 1020 - method, 370 - theory, 1023 metastable patterns, 680, 708 metric, Euclidean, 115 metric, time, 116 mild solution, 890, 959 Miller-Robert equation, 586 minimizer, 681,684, 691,695, 698, 699, 702 680, 684-688, 690, 691, 694, 698, 699, 706-708, 716 minimizing sequence, 706 modeling, 342 models of turbulence, 505 modes of growth, 709 modulation, 717 ansatz, 768 monodromy matrix, 157, 162 monotone, strongly, 844
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- g l o b a l ,
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Subject Index monotone mapping, 844 monotone (order-preserving) PDEs, 1045 monotone systems, 843, 989, 1045, 1046 Monte-Carlo simulations, 711 Morris-Lecar equations, 42, 122 Morse decomposition, 403, 855,940 Morse index, 416, 749, 993,998, 1003, 1007, 1032, 1037 Morse sets, 403 Morse-Smale map, 920, 922 Morse-Smale system, 951,954, 965,967 moving hyperplanes, 860 multi-pulse solutions, 795 multi-step method, 350 multiple shooting algorithm, 360 multiplier Floquet, 157, 161 simple, 167 trivial, 157 multivalued map, 448
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1083
numerical integration, 348 - one-step method, 350 operator adjoint, 998 - Fredholm, 996 - null space, 990 - range, 990 -sectorial, 840, 1040 -self-adjoint, 727, 731,735 orbit, 937 heteroclinic, 193,216, 1031 heteroclinic orbit, 189 - homoclinic, 179, 190, 191, 201, 213, 216, 1020, 1023 - central, 192 - codimension-1,214 - - complete, 892 - non-central, 181, 192 - non-central saddle-node, 192 - regular, 177, 179-181 - - saddle-node, 188 Shilnikov, 216 - - to a saddle-focus, 1030 - - to a saddle-node, 178, 181 - to non-central saddle-node, 178 n-homoclinic, 187 negative, 892 - periodic, 358, 1022 - existence, 440 point, 921 solution, 489, 496 positive, 891 -pseudo-orbit, 354, 821 - pseudoperiodic, 224 order, admissible, 403 order, flow defined, 404 order preserving, 843 oscillations, synchronized, 5 oscillatory, 5, 108, 113, 117, 121, 123
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Nagumo equations, 438 Navier-Stokes and Euler equations, 509, 513, 527 Navier-Stokes equation, 505-507, 509-512, 518, 519, 521,523, 526, 527, 538, 541,543,545, 547, 549-555, 557, 558, 563, 565, 566, 574, 970 - compressible, 544 - incompressible, 545 Navier-Stokes limit, 541 negative temperature, 506 Nemitskii operator, 840 neurons, "type 1", 24 neurons, "type 2", 24 Newell criterion, 786 Newell-Whitehead equation, 775 Newton-Keller criterion, 681 noise-sustained structures, 338 nonconvergent trajectories, 875 nonhyperbolic systems, 320 nonwandering set, 353, 921 normal form, 7, 18, 151, 182, 184, 200, 205, 209, 214, 216, 367 - coefficient, 204, 206, 215 of codimension-2, 200 - simple fold, 183 truncated, 211 normalization technique, 203, 205 numerical computation - of PDE-spectra, 1035 numerical continuation - of travelling waves, 1034 numerical evaluations of brittleness, 327 -
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q
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i
l
i
b
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u
m
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p
e
r
i
o
d
i
c
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p
e
r
i
o
d
i
c
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pancreatic beta cells, 96, 100 parabolic equation, 837, 852, 859, 868, 948, 952 paroxysmal discharges, 138 pattern, 686, 693, 707, 716, 717 - basic periodic, 761 761,820 patterning, 712 periodic process, 914 Perron-Frobenius operator, 238, 241, 242, 245, 248 -discretized, 243,247, 249, 253,255,262 phase condition, 156, 167, 180, 191, 214
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m
o
d
u
l
a
t
e
d
,
1084
Subject Index
evolution, 716 - l a g , 10 parameter, 715 - separation, 708 slip, 791 phase-difference equations, 7 phase-field equations, 716 phase-field models, 715 phase-locked oscillations, 5 phase-response curve (PRC), 31 "phasic" regime, 25 Poincar6 continuation, 156 Poincar6-Bendixson theorem, 854 Poincar6 map, 6, 163, 164, 175, 275 Poincar6 section, 438 point dissipative, 887, 902, 904 pointed space, 407 pointwise - dimension, 373 - estimates, 1042 - shadowing distance, 333 Poiseuille flow, 779 Poiseuille problem, 827 Poisson bracket, 513 population dynamics, 837 post-inhibitory rebound, 47, 119 potential vorticity, 64, 80, 85, 87 power spectrum, 569 Prandtl number, 511 product - bi, 384 bialternate, 160, 162, 169, 186, 198 - definition of, 152 propagation of chaos, 534 -
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random averages, 552 random fluctuations, 556 random perturbation, small, 240, 241,244 Rayleigh number, 700 Rayleigh-Brnard problem, 776, 827 reaction-diffusion systems, 868, 948, 988, 1006, 1041 -singularly perturbed, 1009, 1017, 1045 realization of vector fields, 868, 873-875 recurrent set, 224 reduction techniques, 6 refractory period, relative, 33 regularity, 509, 927 repeller, 399 representable numbers, 448 representable set, 449 representation of, 449 residual, 917 resolvent, 998
resonance condition, 367 resonant periodic orbits, 298 resonant saddle, 188 retarded functional differential equation, 890, 945 return map, 360 reversible systems, 1019, 1026, 1029, 1032 Reynold stress tensor, 507 Reynolds number, 505, 509, 511, 518-520, 538, 545, 550, 553, 558, 564, 570 Reynolds tensor, 507, 551,553, 554, 556 Riemann-Hilbert problem, 651-660, 668 rolls, 717 rotating waves, 958 Rouchr's theorem, 470, 471,474, 476, 483 roughness theorem, 993 Runge-Kutta method, 348 saddle, neutral, 175, 186, 188 saddle-focus, 188 - equilibrium, 1030 saddle-node, 181 - bifurcation, 383 SBR measure, see measure scaling parameters, 509 Schrrdinger equation, nonlinear, 774, 789, 990, 1009, 1012, 1017, 1032 Schur factorization, 181 semicontinuity -lower, 916, 942, 943 -upper, 806, 822, 916 semi-classical limit, 660--668 semiflow, 505, 841 semilinear heat equation, 726 sensitive dependence on initial conditions, 315 separatrix, 269, 287 set - almost invariant, 249, 250 -invariant, 224, 245, 353,395 stable, 178 unstable, 178 set-wise image, 260 shadowing, 317, 354, 820 - distance, 324 lemma, 318 -time, 332, 336 shift dynamics, 443 shift equivalent, 414, 415 shooting method, simple, 360 shooting permutation, 956, 957 sideband instability, 782 - vector, 783 similarity variables, 726, 730 -
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Subject Index
singular index pair, 446 skew-product flow, 915 SLEP method, 1046 slow motion, 680, 713,714 slowly oscillating periodic solution (SOP), 463, 490, 491,494-496 small denominator, 526 Sobolev spaces uniformly local, 784 solenoid, 358 solution, see also orbit - branch, 152, 153 heteroclinic, 177 homoclinic, 177 - periodic - - hyperbolic, 157, 201 152, 155, 165 SOP solution, 463,490, 491,494-496 spatial - average, 552 - dynamics, 1046-1048 eigenvalue, 1014 -pattern, 679, 703 spatially unstable systems, 338 spatially-periodic wave trains, 1021 spatiotemporal chaos, 631-638 spectral - assumption, 766 -gap, 995, 1014, 1038 stability, 1044 spectrum, 996 -absolute, 1032, 1038, 1039 -essential, 996, 998, 1012, 1018, 1021, 1037 -point, 996-998, 1007, 1018, 1037, 1039 spike, 804 spike adding, 100, 107 spike response method, 6, 20 spike shape, 27 spike-frequency adaptation, 10 spike-time maps, 31 spindle rhythm, 138 spinodal decomposition, 708, 709, 715 spiral waves, 797 squeezing property, 575 SRB measure, see measure stability, 509, 512, 679 - fronts, 1002, 1005 rest state, 1000 analysis, 681,709 -linear criterion, 680, 686, 689, 690 -multi-bump pulses, 1017, 1027, 1031, 1032, 1044 - nonlinear, 1039, 1043 - of waves, 985, 1039 - pulses, 1004 - spectral, 987, 1040 -
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- r e g u l a r ,
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- h o m o g e n e o u s
- l i n e a r
1085
transition to instability, 1026 wave train, 100 l, 1021, 1048 stable, 900, 901 stable uniformly asymptotically, 901 standard map, 269, 295 star-shaped, 451 state dependent delays, 467, 494 stationary statistical solutions, 566 statistically sharp, 590 stiff integrators, 350 stochastic stability - of the spectrum, 252 stochastic transition function, 240 straddle algorithm, 368 straddling, 70, 77, 89 streamfunction, 59, 64 Strichartz inequalities, 963,972 strongly connected component, 231 structurally stable, 954 Sturm-Liouville properties, 954 subdivision algorithm, 225 subdivision algorithm, adaptive, 254 subharmonic solutions, 850 subshift, 369 succession map, 275 summation property, 412 support of solutions, 987 suppressed solution, 109, 124 Swift-Hohenberg equation, 680, 700, 702, 824, 1041 symmetry, 859-863 symplectic integrators, 352 synapse, 9, 111, 117, 120, 131 synaptic coupling, 109 - escape, 47 - kernel, 21 release, 47 reversal potential, 111, 126 synchronized oscillations, 5 synchronous solution, 109
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Taylor series method, 351 temporal chaos, 618-631 test brittleness, 325 test function, 158, 160, 162-165, 169, 182, 185188, 192 codimension- 1 bifurcations of maps, 161 - Hopf bifurcation, 160 inclination-flip, 191 homoclinic orbit, 192 orbit-flip, 190, 191 -well-defined, 161, 162, 187, 188, 191-193, 196, 197 -
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- n o n - c e n t r a l
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1086 thalamic reticular (RE), 139 thalamocortical relay (TC), 139 thalamus, 96, 129 thermal diffusivity, 511,545 Thom-Boardman stratification, 377 threshold results, 685 time average, 372 time delay, 372 time metric, 7, 44 time regularity, 927 time scales, 6 time step, 348 time t map, 348 "tonic" regime, 24 topological equivalence, 915, 920, 922, 957 trajectory, 348, see also orbit, solution trajectory, true, 317 transition density function, 241 transition matrix, singular, 433 transition matrix, topological, 433 transitive, 871 translation eigenvalue, 1028 transverse intersection, 59, 68 travelling - fronts, 985 -multi-bump pulses, 1028, 1032 - on cylindrical domains, 1046 - pulses, 985 wave trains, 985, 1017 - waves, 705, 985 turbulence, 507, 509, 550, 553, 563,580 - modelling, 505,528, 550 statistical, 551,555 turbulent, 505, 506, 554, 558, 580 diffusion, 551 - energy, 551 - energy spectra, 580 - intermittencies, 574 - spectra, 506, 507, 509, 556, 568 Turing patterns, 1001 two-cycle, 248, 253 -
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-
Ulam's method, 238 uniformly ultimately bounded, 888 upper triangular, 427
Subject Index variable time step, 350 variational equation adjoint, 190 "virtual" delay, 46 viscosity, 71, 85, 88, 511,513, 520, 525, 545, 554, 558 voltage-gated conductance equations, 5 vortex, 72, 73, 77, 80 vorticity, 509, 510, 513, 515, 523, 524, 550, 580, 581,583
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wandering point, 353 wave
- modulated, 1047 plane, 784 - rotating, 1047 spiral, 1006, 1048 - travelling, 705, 985 wave train, 1048 Wa2ewski Property, 397 Wa2ewski Theorem, 232 weak interaction of pulses, 1033 weak solution, 718 weak turbulence (see dispersive wave turbulence), 640 weakly coupled oscillators, 6 weakly damped KdV equation, 974 weakly damped Schrrdinger equation, 933, 971 weakly nonlinear analysis, 682, 700, 716 weakly nonlinear stability theory, 679 wedge product, 186 weight - exponential, 1005, 1006, 1015, 1041 polynomial, 1041 weighted energy estimate, 801 - norm, 784 Wigner transform, 507, 509, 553 -
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-
-
Young measure, 585, 586 Zakharov equations, 974 zero, regular, 160, 163, 165, 185-187 definition of, 158 zero number, 745, 852, 853, 858, 862, 954, 955, 958
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