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
517
(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 s