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increases with each oscillation period by an amount 2n ■ 2", where n = 1,2,3,..., and 2 n is the periodicity of the attractor. Fig. 13.5 is a plot
Coexistence of regular patterns and chaos
213
Fig. 13.5 Coherent patterns in the chaotic medium described by Eqs. (13.3). Taken from (Goryachev and Kapral, 1996).
of the local phase angle <j>(r) in a two-dimensional medium described by Eqs. (13.3). One sees a complex pattern of spiral defects whose number varies with time. Close to the defects the local behavior is periodic with period T, and at different distances from the defect the local behavior is reminiscent of different stages of the bifurcation sequence shown in Fig. 13.4. Far from the defect the local periodic dynamics changes from period T to period 2T, etc., and becomes chaotic through the period-doubling sequence at suffi ciently large radial distance from the defect. Thus in a homogeneous chaotic medium coherent patterns like spirals can be stable and such patterns may coexist with turbulent states in other parts of the two-dimensional medium. Now it is easy to take the next step to spatially inhomogeneous chaotic media. Owing to the periodicity of oscillations near the defect, the disper sion of their parameters does not change the situation significantly. Neigh bors characterized by different periods of oscillation will be synchronous, but naturally the synchronization domain will be smaller; see Appendix A.4. Such patterns in lattices of weakly-coupled chaotic oscillators may called phase synchronization patterns because, in the synchronous state, the os cillation phases <j> are locked, while their amplitudes remains chaotic. The study of cluster formation in a chain of coupled Rossler oscillators with dif ferent distributions of natural frequencies was reported by (Osipov et al., 1997). The authors observed several clusters of phase synchronization with
214
Patterns in Chaotic Media
different principal frequencies for randomly distributed natural frequencies. Such clusters can arise at arbitrary places in the chain and can coexist with oscillators that belong to no cluster.
13.4
Coarse grain spatio-temporal patterns
Patterns observed in a lattice of chaotic neurons (c/. Fig 13.2) have an ab solutely different origin in comparison with the patterns discussed above, which naturally could be called chaos suppressed patterns. Chaos is not sup pressed in Fig. 13.2; rather, the structures just modulate the chaos as may be seen in the time series of a lattice node as in middle rows of Figs. 13.1a,b. The nature of these patterns is connected with the presence of two distinct time scales in the neural oscillations: fast chaotic spikes rising out of the background of slow yet chaotic pulsations. The coherent patterns observed in numerical experiments, like those displayed in Fig 13.2, vary periodi cally in time. Therefore, one of the principal questions that needs to be answered in order to clarify the nature of these patterns is: Are fast chaotic oscillations able to change the dynamics of large-scale collective motion by making it regular? By large-scale patterns we understand structures whose characteristic size is much larger than a cell of the lattice. These together with the pattern topology, which appears like waves of switching between states in regular excitable media, leads to a possible mechanism for the formation of patterns at moderate values of neural coupling. This may be analyzed by introducing the concept of neuron clusters, as discussed below. The cluster with average time periodic behavior will be called a coarse grain (eg). We suppose that the regular spatio-temporal patterns observed in the computer simulations are strongly related to the existence of the coarse grain for a moderate value of the coupling. The cooperative be havior of diffusively coupled coarse grains (periodic oscillators in our case) produce many different regular spatio-temporal patterns similar to those obtained with the discrete analog of the complex Ginzburg-Landau or the FitzHugh-Nagumo models. To analyze this behavior we need an equation that describes the average dynamics of the coarse grain. The coarse grain dynamics are described using the cluster variables 1
M
Coarse grain spatio-temporal
patterns
215
where M is the number of elements in the cluster. An approximate system of equations for X, Y, Z is obtained by substituting Xi = X{t) + Si(t),
yt(t)=Y(t)
+ Vi(t),
Zi(t) = Z(t) + Q(t)
into Eqs. (13.2). Ignoring terms of order higher than ff gives, after aver aging over M elements, the governing equations
~dt dT dt u dt
Y + aX2 + ar(t)-X3 -3Xr(t)-Z -cX2
- Y - cr(t) - b
-Z + s{X + d)
+e
(13.4) (13.5) (13.6)
where e = (ei)cg. We have taken into account from the definition of X, Y, and Z that (£i(t))cg = {r)i(t))cg = (Q(t))cg = 0, and consequently the only function left to be determined is r(t) = {£2)CgIn order to describe the slow dynamics, we need to make a reason able assumption about the nonautonomous terms on the right-hand side of Eqs. (13.4) and (13.5). Since r(t) varies much more rapidly than the slow coarse-grain oscillation, we suppose that the dynamics of an individ ual coarse grain will depend on the time-averaged value of r(t) defined as 1 ft+T R(t) = - / r(t')dt' T
(13.7)
h
with tr "C T < T, where tr is the characteristic time scale of the fast oscillation r(t) and T is the characteristic time scale of X(t). In Eqs. (13.4) and (13.5) we now replace r(t) by the slow function of time R(t) given by Eq. (13.7) which also depends on the strength of the diffusive coupling g between elements and on the size M of a coarse grain. If our hypothesis is correct, R is very nearly a nonzero constant for small values of g, and R is almost zero for large values of the coupling; for moderate values of the coupling, prediction of the behavior of R is not intuitively clear. Computer simulations shown in Fig. 13.6a at g = 0.1, however, indicate that for moderate values of g the behavior of R becomes periodic. This in dependent behavior of R infers that the averaged dynamics X(t) will vary as the coupling parameter is varied, as may be seen in Fig. 13.6b. Note the appearance of periodic average activity X(t) at g = 0.1.
Patterns in Chaotic Media
216
H(t) ■0.2
-
l
"IN.N-KN.N.N.IS.^ 1000
2000
3000
« ^/WVWWl 0
1000
2000
3000
It Fig. 13.6 (a) Evolution of R(t) as denned by Eq. (13.7) for three different coupling strengths g in a network of 10 x 10 HR elements; (b) Average activity X(t) in this network for the same values of g. The parameter values are a — 3, 6 = 1, c = 5, d = 1.6, e,- = 3.281 ± 0.05, /i = 0.0021, and 3 = 4; units are dimensionless in this model. Taken from (Rabinovich et at, 1999a).
The appearance of the periodic average behavior is illustrated by the phase portraits and corresponding time series of the full dynamical system shown in Fig. 13.7. For sufficiently small values of g, R is nearly con stant taking on values in the range 0.4-0.5, and only a single stable fixed point appears corresponding to steady-state behavior of the cluster. For R < Rc (g > gc), this fixed point becomes unstable and the limit cycle in the three-dimensional phase space of the average coarse-grain system undergoes a supercritical and sharp Andronov-Hopf bifurcation to a stable fixed point. Strictly speaking, at the moment of this bifurcation R becomes a periodic function of time; see Fig. 13.6. However, as the numerical results confirm, slightly above the threshold for bifurcation, the influence of this periodicity on the existence of the limit cycle is not important. The dynamical mechanism giving rise to the ordering behavior of the coarse grain relies on the synchronization and regularization of the activity of the M elements inside the grain. The degree of synchronization of a single neuron with the average activity of the whole grain depends on the strength of the coupling, as one can see in Figs. 13.8a,b. For g = 0.1 in Fig. 13.8a, the activity of a single neuron is highly synchronized with the periodic mean field. For g = 0.05 in Fig. 13.8b, however, the synchronization between mean field and individual behavior is absent and spatio-temporal disorder is observed. Thus, for intermediate values g = 0(1O - 1 ), the coarse grain behaves as a single element with periodic slow dynamics.
Coarse grain spatio-temporal
1000
2000
patterns
3000 4000 time
5000
217
6000
Fig. 13.7 Phase space portraits (top rows) and corresponding X(t) time series (bottom rows) of a single coarse-grain element described by Eqs. (13.4)—(13.6) computed for the parameter values given in the caption of Fig. 13.6; (a) R = 0.47, a stable fixed point; (b) R(t) is the periodic function shown in Fig. 13.7a for g — 0.1, a limit cycle; and (c) R = 0.0007, a strange attractor. Taken from (Rabinovich et al., 1999a).
Patterns in Chaotic Media
218 g-o.1
8 =0
0
0
-0.5
■0.5 *1
•1
-1 •1.5
Jf, •1.5
-1.5
•1
-0.5
X
0
-1.5
05
.
if
40
m
30
w, -1
-0.5
X
K,
/
20 10
0
wJ ■
08
^
0.7
-
0.6
0.5
^
04
0.3
0.2
0.1
0
g
Fig. 13.8 Activity of a single HR unit Xi versus the average activity X (defined in the text) for (a) g = 0.1 and (b) g — 0.05; (c) Kolmogorov-Sinai entropy as a function of the coupling strength g in a network of 7 x 7 HR elements. Taken from (Rabinovich et a/., 1999a)
Using the above observations, we are now in a position to explain the existence of large (TV 3> 1) regular spatio-temporal patterns in a discrete diffusive medium. First, the existence of regular structures is impossible in weakly diffusive media because local oscillations of neighboring elements are not correlated for small couplings g, and the mean field of the coarse grains becomes homogeneous and stable. Direct computation of the KolmogorovSinai entropy presented in Fig. 13.8c confirms that the level of spatially homogeneous chaos increases as g —> go <£. 1. For moderate coupling, the coarse-grain assembly should exhibit regu lar spatio-temporal patterns. As confirmation of this conjecture we have checked the behavior of a lattice medium consisting of coarse-grain elements with slow periodic behavior. The description of this medium is analogous to that given by the network of HR elements wherein (n ,yt,Zi) are replaced by (Xi, Yi,Zi). We are looking for patterns in the coarse-grain system that have the same space scale, relative to the size of the lattice, as the pattern in the original HR lattice. Thus, the pattern in the coarse-grain lattice should be of identical structure but with a smaller absolute size. Since the patterns on the two lattices have the same time scale, the speed of front propagation in the HR lattice must be larger than in the coarse-grain lat tice. The propagation speed of the front increases with increasing values of the diffusion. We surmise, based on this scaling argument, that a coarsegrain pattern with the same relative size as the original may be found only in the case when the coarse-grain lattice coupling G is smaller than the dif fusive coupling g of the original HR network. Verification of this conjecture
Coarse grain spatio-temporal patterns
219
Fig. 13.9 (a) Evolution of a periodic spatio-temporal pattern observed in a network of 100 x 100 HR elements; parameter values are those specified in Fig. 13.6 with g=1.5; (b) periodic spatio-temporal patterns observed in a network of 30 x 30 coarse-grain elements computed for R = 0.23 and G = 0.5 (the rest of the parameters have the same values used in the HR lattice). The value of R is close to the bifurcation point and the individual coarse grain dynamics is periodic. Taken from (Rabinovich et al., 1999a).
is given by the sequence of patterns obtained for the 30 x 30 network of coarse-grain units shown in Fig. 13.9. These patterns, plotted for G = 0.5 in Fig. 13.9b, are clearly similar to those produced by the original heterogeneous lattice of chaotic HR elements for g = 1.5 in Fig. 13.9a. Thus, identical periodic boundary conditions applied to both the HR and the square coarse-grain networks give the same topology of the patterns observed. Furthermore, additional computations have revealed the same correspondence of pattern topology when hexagonal lattices were used. In the latter case, the strength of the coupling was reduced to take into account the larger number (six) of nearest neighbors. We conclude that the formation of large-scale coherent structures in nonequilibrium media consisting of discrete and chaotic HR elements with fast and slow oscillations exhibits two key features. The first is the regularization phenomena in small assemblies of chaotic elements, i.e. coarse grains. This regularization of behavior is the result of the action of the av-
220
Patterns in Chaotic Media
4 2 x,(t)
x«(t>
i r r n n i n - i n -
-] >
0 -2 -4
4000
\J 1J U U U.U \J \! ,1 J \j I. 2000 Time
4000
Fig. 13.10 Checkerboard patterns (top row) in a network of 100 x 100 chaotic HR elements with negative electrical coupling between nearest neighbor units and the regular slow oscillations of a single neuron's activity (bottom row) for (a) g = —0.95 and for (b) a stronger absolute value of the coupling, g = -1.12. Taken from (Rabinovich et al., 1999c).
eraged activity of fast oscillations in the slow coarse grain dynamics. The second feature is the instability of homogeneously oscillating modes in a me dia considered to be a coarse-grain lattice. It is important to keep in mind that the coarse grains are a temporal assembly of neurons whose relaxation time is smaller than the relaxation time of the coherent structures. 13.5
Coherent patterns on a chaotic checkerboard
Another example of regular spatio-temporal behavior in a nonregular lattice described by Eqs. (13.4)-(13.6) is shown in Fig. 13.10. The origin of this regular behavior is absolutely different from that of the regular behavior of the coarse-grain patterns discussed in the previous section. The behavior we shall discuss now is typical for negative electrical couplings that model
Coherent patterns on a chaotic checkerboard
221
2.0 1.0
* 1 - 2 o.o -1.0 -2.0
t Fig. 13.11 Antiphase regularization in a system of two negatively coupled HR neurons. Taken from (Rabinovich et a/., 199b).
inhibitory connections in neural networks (Abarbanel et al., 1996). This regularization phenomena is intimately related to the behavior of two neg atively coupled chaotic HR neurons. When the two oscillators are coupled with negative conductance (g ~ —1), the antiphase regularization shown in Fig. 13.11 is observed. The two neurons regulate their slow oscillations in the sense that the lengths and shapes of the bursts are kept uniform. This happens because the origin of the chaoticity of the model is related to the interaction of the fast subsystem (x, y) with the slow variable z: the homoclinic nature of the fast oscillations are regulated by the slow oscillations. In the absence of inhibitory action from other neurons to limit the rise of this slow variable, the system will be driven to a near homoclinic orbit which is unstable. Negative electrical coupling, however, will not permit individual neurons to reach a fast oscillation which is unstable. When we have a lattice of such chaotic generators with negative cou pling, they will form antiphase behavior with their nearest neighbors to form stable, regular checkerboard patterns as in in Fig. 13.10. To investi gate the temporal evolution of the patterns, we will use the order parameter a(t) defined by
JVT 5 > - ivr *>1
{13 8)
-
where NA is the number of elements above the threshold, NB is the num ber of elements below the threshold, and AA and Ag represent two in terconnected sublattices with antiphase oscillations that give rise to the chessboard configuration. One can see in Fig. 13.12 that a(t) is periodic. This time series looks like the time series produced by a relaxation oscil-
222
Patterns in Chaotic Media
2S.S
time
Fig. 13.12 Temporal behavior of the order parameter a(t) defined by Eq. (13.8) for g = - 1 . 1 2 in networks of (a) 30 x 30 and (b) 50 x 50 chaotic HR neurons. Taken from (Rabinovich et al., 199b)
Fig. 13.13 Two examples of regular envelope patterns of antiphase oscillation in the discrete 2D complex Ginzburg-Landau equation. Taken from (Gaponov-Grekhov and Rabinovich, 1992).
lator consisting of a combination of slow and fast motions. Calculations have indicated that the time scale for the transition from sub-threshold to super-threshold activity depends on the size of the lattice: large networks have transition times slow compared to that for small networks. In this sit uation, waves of this transition of state emerge which are periodic in time over the entire lattice. These types of regular spatio-temporal patterns are reminiscent of the regular envelope patterns of antiphase oscillation in the discrete variant of the complex Ginzburg-Landau model, an example of which is shown in Fig. 13.13. In concluding this chapter, let us note that spatio-temporal patterns like those discussed above are observed in models of chaotic neural assemblies
Coherent patterns on a chaotic checkerboard
223
4000 Fig. 13.14 (a) Spatio-temporal pattern in a 100 x 100 network of HR elements with excitatory synapses between nearest neighbors; (b) time series showing the regular slow activity of a member neuron. Taken from (Rabinovich et al., 1999c).
with generic synaptic coupling. Let us consider an example. In Fig. 13.14a one can see a pattern in the network of HR neurons coupled by excita tory chemical synapses. Again, the slow single neuron oscillations become regular as may be seen in Fig. 13.14b. In the following final chapter we will discuss briefly the importance of spatio-temporal patterns of neural activity for information processing.
Chapter 14
Epilogue: Living matter and dynamic forms
Will it ever happen that mathematicians will know enough about the physiology of the brain, and neurophysiologists enough of mathematical discovery, for efficient cooperation to be possible? Jacques Hadamard After studying the previous chapters, the interested reader may enquire as to what comes next. What are the most promising avenues of thought to follow now? Answers to this question are, of course, subjective. Neverthe less, we will risk expressing our opinion about possible applications of the dynamical theory of patterns to living systems. At the outset it should be clearly noted that this is a unique new field of knowledge where not only the methods of solutions to problems are vague, but the problems themselves often have not yet been well formulated. On what shall we rely for guid ance? We believe the answer is laboratory experiments. Suitably conceived experiments guide the construction of models to describe, for example, the birth from a set of identical cells having identical genetic codes of very intricate organisms containing a large number of subsystems that possess various functions (Koch and Meinhardt, 1994). Likewise, experiments will be crucial for the formulation of models to describe the reception, repre sentation, and long-term retention of information about, and perception of, the surrounding world. What is the basic difference between living and inanimate systems? We believe it is the availability of the information component, in addition to conventional dynamical variables like electric potential, density of sub stance, concentration of quanta, etc., that is quite unusual for dynamical 225
226
Epilogue: Living matter and dynamic forms
systems. Information influencing subsequent behavior may be contained in each element of an assembly (like genes in the nucleus of a cell), or it may be produced by special cells changing the surrounding medium and modes of interrelation (not interaction) between elements that are the building blocks for future patterns. Manifestations of information may be more ex otic. For example, these may be weak electric signals changing the intrinsic dynamics not only of individual cell neurons but also of neural assemblies, thus transforming them, say, from the regime of reception of information to the regime of remembering. The mechanism of switching on the information clock may be different in different situations. For example, it is supposed (Koch and Meinhardt, 1994) that in morphogenesis information about differentiation of initially identical cells is switched on depending on their geometrical position rel ative to each other. Special signals, often not yet understood, in neural assemblies give rise to new couplings between neurons that actually change the topology of the neural assembly. There is no end to such examples. The switching of information mechanisms into pattern-forming pro cesses resembles a game where the rules change during the course of events, often in an unpredictable manner. The changes are frequently so dramatic that after switching, a new system appears that may be a dynamical one only over a certain portion of its life. It is difficult to conceive of this as a classical dynamical system if there are quiescent periods throughout its living cycle. The concept of phase space (i.e. the space of systems to which the concept of closeness is intrinsic) and the concept of trajectory are not determined, simply because the initial state does not determine the be havior of the system throughout its existence. The concepts of large-time behavior, as well as that of basins of attraction lose rigor here because the system lives according to rules given only for a finite time. Nevertheless, a combination of continuous models with finite cellular models, like multi-stable automata, along with some algorithmic rules may allow for the formulation of dynamical models on a new level: these will be models in which the dynamics depend on information. In the simplest case, when information does not change the dynamical system but only provides its initial conditions, we can say that the real-time life of the nonlinear dynamical system is a form of information processing. Since we are more closely involved with problems related to the behavior of neural assemblies rather than morphogenesis, we will restrict ourselves to the nonlinear theory of pattern formation in neural networks.
Hallucinations
227
Analogies between fast and powerful computers and the brain are still seriously discussed in both popular and scientific milieux. In fact, there exists a certain similarity in the functions of neurons and computer ele ments; either may work as threshold detectors, elements of two-valued and multiple-valued logic, integrators, coincidence detectors, etc. Moreover, the interdependence between 'computer ideas' and the anatomic informa tion about the structure of the cortex with its hidden levels and couplings, formed during the course of learning, has led to the development of essen tially new computational structures, namely neural networks (Amari and Arbib, 1980). Of course it is very risky to make any long-range prognoses today, but there is every reason to believe that the brain uses utterly new and often unknown principles in which, we are sure, the nonlinear dynamics of patterns plays a critical role. 14.1
Hallucinations
Recent experiments with specific neurons have revealed a remarkable phe nomenon: the response of a neuron to a very complicated, even irregular input signal proved to be reproducible. This means that the neuron func tions as a dynamical system and its behavior in the course of transformation of information is independent of noise and other extraneous fluctuations. It is essential, however, that the greatest majority of neurons and neu ral assemblies do not merely transform information (as, for example, the eye retina) but actively generate it as well. Perhaps the most remarkable example is hallucination. The psychiatry literature on hallucinations is replete with accounts of false sensory perceptions resulting from different diseases or drug intoxi cation. We will discuss here the dynamical origin of visual hallucinations (olfactory and auditory hallucinations have also been reported) and show that it will provide an opportunity to understand something about the be havior and organization of the cortex itself (Siegel, 1977; Ermentrout and Cowan, 1979; Cowan, 1987; Theunissen et al., 1996; Tass, 1997). Let us review briefly the results of extensive hallucination observations. The four important messages that can be gleaned from those observations are the following:
228
Epilogue: Living matter and dynamic
forms
1. Hallucination patterns are universal and reproducible — the patterns depend on the type of drug more than on the individual. Virtually the same patterns are found to occur in states of insanity, delirium tremens, drug intoxication, insulin hypoglycemia, dreams, nightmares, ecstases, and fevers; all are characterized by the excitation and production of images from memory or from imagination. 2. The geometry of hallucination patterns very often have cylindrical sym metry reminiscent of the symmetry of the retina. 3. The topology and temporal behavior of the patterns depend on the stage of hallucination and the same patterns occur both when the eyes are open and when they are closed. At the early stage, hallucinations are composed of only four basic static patterns (Theunissen et al., 1996; Ermentrout and Cowan, 1979): tunnels or funnels, cobwebs, the 'white light' pattern, and the 'spiral tunnel' structure; examples of the latter two patterns are given in Fig. 14.1. All of these geometric images have a characteristic space scale. 4. Quite often people see hallucinations with oscillating color or kaleidoscopically changing structures. Analysis of the above observations have persuaded investigators as long as sixty years ago that hallucination phenomena have to be induced at higher levels of the visual system than the retina itself. Today's point of view is that the regular geometric nature of migraine-induced illusions or drug hallucinations, for example, are derived from electrical instabilities of organized groups of neurons in the visual cortex (Ermentrout and Cowan, 1979; Cowan, 1987; Theunissen et al., 1996; Tass, 1997; Ermentrout, 1997). If visual hallucinations are generated in the visual cortex, we have to understand to which geometric image in the visual field (or retina) these patterns correspond. In other words, what picture in the visual field trans forms onto the visual cortex the same geometrical structure that the cortex itself is able to generate? It is now well known that there exists a conformal projection of the visual field onto the visual cortex. Since the retina has approximately radial symmetry, and the cortex possesses translational symmetry, this projection has to be a nonlinear coordinate transformation from retinal polar coordinates z = relff to cortical rectangular coordinates
Hallucinations
229
Fig. 14.1 (a) This 'white light' is seen during the early stages of intoxication with hallucinogenic drugs; (b) the 'spiral tunnel' is another early stage hallucinogenic pattern. The main patterns of movement accompanying the picture are reported as pulsation and rotation. Taken from (Siegel, 1977).
w = x 4- iy. This transformation was first suggested by (Schwartz, 1977); see also (Ermentrout and Cowan, 1979). It has the form of a complex log arithm w = Inz, where w represents a point in the cortical plane and z represents a point in the visual field. In terms of real (x, y) variables, the complex logarithm gives x ~ In r,
y ~ 9.
(14-1)
This mapping exhibits the required logarithmic dependence on retina ra dius and the radial symmetry found in experimental data (Leshley, 1941). Using the results in Eq. (14.1) we are able to understand the relationship between cortical patterns and retinal patterns. Figures 14.2 and 14.3 show examples of this relationship where circles (r = const) and radial straight lines (6 = const) in the z-plane are transformed into straight lines, respec tively parallel to the imaginary and real axes in the w-plane. Logarithmic spirals of the form r = AeK0 in the z-plane transform into straight lines in the «;-plane, making a slope of l//c with the real axis. Circles and ra dial straight lines are limiting cases of logarithmic spirals, with n —► 0 and K —y oo, respectively. Now is the final step of this discussion. Suppose we know the patterns the cortex is able to generate. Then what can one deduce about the cortex itself? We are able to give a reasonable answer, though it is neither unique nor complete. First, the visual cortex should be a two-dimensional excitable media that produces static or oscillatory patterns when a control param-
230
Epilogue: Living matter and dynamic forms
■ae HUH Fig. 14.2 Funnel hallucinations in (a) and (c) and tunnel hallucinations in (b) and (d). According to the visuo-cortical transformation (14.1), the hallucinations in (a) and (b) correspond to cortical activity patterns in (c) and (d), respectively. Taken from (Tass, 1997).
Fig. 14.3 Spiral hallucinations. The hallucinations in (a) and (c) correspond to cortical activity patterns in (b) and (d), respectively. Taken from (Tass, 1997).
eter rises above a threshold. Second, the neural assembly comprising the visual cortex should possess a specific spatial interaction function in order to produce spatio-temporal patterns having the characteristic space scale. It may be a lateral inhibition interaction between neurons (sometimes called a Mexican hat interaction) in a two-dimensional lattice for which there is local excitation and distant inhibition, or it may be a lateral excitation for which there is local inhibition and distant excitation; see Fig. 14.4. One
Hallucinations
231
Fig. 14.4 Intensity W(x) of mutual excitation with x being the distance between ele ments of the medium; (a) lateral inhibition or Mexican hat interaction function and (b) lateral excitation function.
then concludes that real neural assemblies are not randomly or globally connected networks but, fortunately, have some intrinsic spatial order, at least in the lower sensory areas (Tass, 1997). This deduction is based on a rudimentary, yet basic understanding of hallucination phenomena. It is likely that modeling of the visual cortex by partial differential, integro-differential, or other sophisticated types of equations will not be necessary. Rather, we can draw on our experience in the investigation of pattern formation in nonequilibrium media with finite spatial scale in stability, as in Rayleigh-Benard convection or in the Faraday experiment, to explain the transition from one pattern to another using appropriate eigenfunctions. Suppose, for example, that stability is lost at a critical wavenumber ko so that all possible perturbations have wavenumber vectors fixed on the circle |fc| = fen as in Fig. 1.4. Then one has a standard problem of mode interaction with zero frequency if it concerns stationary halluci nation patterns, or with fixed frequency if it concerns oscillating patterns (Cowan, 1987; Tass, 1997). Typical results for instabilities generated from a low number of modes are given in Figs. 14.5 and 14.6. These patterns derive from a competition among a small number of modes where the end result is a single mode pattern (e.g. rolls) or a multi-mode pattern (e.g. squares or hexagons). One can imagine that patterns not only compete with each other, but also excite each other through resonant interactions. In this case there will be no fixed pattern since patterns will alternate periodically or chaotically as in a kaleidoscope. Such behavior is reminiscent of temporally changing
232
Epilogue: Living matter and dynamic forms
Fig. 14.5 (a) Periodically blinking rolls of the four-mode cortical instability and (b) blinking visual spirals corresponding to the cortical patterns of (a). Time proceeds through one period T row by row from the upper left corner to the lower right corner in equidistant steps T/16. Taken from (Tass, 1997).
patterns observed in rotating convection; see Chapter 3, Section 3. It is easy to imagine that the appearance of superlattices in patterns of cortical activity, similar to those found in the Faraday experiment (cf. Fig. 5.8), would lead to several scales of hallucination patterns like the lattice tunnel
Hallucinations
233
Fig. 14.6 Periodically blinking tunnels and funnels: (a) visual patterns and (b) patterns in the cortex. This is the result of an eight mode interaction with time proceeding as in Fig. 14.5. Taken from (Tass, 1997).
shown in Fig. 14.7. Topologically, the lattice tunnel pattern is equivalent to patterns that one can find in the self-similar packing of disks of continuously changing diameter; see (Weidman and Pfendt, 1990).
234
Epilogue: Living matter and dynamic
Fig. 14.7
14.2
forms
Lattice tunnel visual pattern. Taken from (Siegel, 1977).
Spatio-temporal patterns and information processing
The main role of the neural system is to control animal behavior accord ing to existing circumstances in the surrounding environment. To do this the nervous system uses sensory information (visual, olfactory, somatosensory, auditory) and, of course, the genetic information which defines the architecture and main dynamical features of individual neurons and neural ensembles. Thus the creation of new information that controls specific be havior in specific circumstances is the result of an interaction of intrinsic information from within and external information from without. How do neural systems do this job? We are going to discuss here only a few aspects of this problem. It is worthy to note that electrical activity is not the only way through which neurons interact. Like many other cells, neurons secrete various chemical substances that act as neuromodulators: see Chapter 11 on the dynamics of Esherichia coli and Bacillus subtilis. These neuromodulators qualitatively alter the dynamical regime of operation of the neuron or group of neurons. This is effected by a reorganization of neural chains which mod ifies their inner couplings, and by combining individual groups into large assemblies possessing qualitatively new dynamics. Such transformations change the action of the secreted chemical substances, thereby modify ing not only the properties of neurons themselves, but also the properties of the dynamical elements (synapses) through which they are connected
Spatio-temporal
patterns and information
processing
235
(Shepherd, 1990). The rules according to which the neural assemblies are restructured are not predetermined. Instead, they depend on the incoming information. Otherwise, the model could be completely explained by inclu sion of the equations that describe the change of these rules. This is the underlying principle of the numerous models of memory, the first of which was proposed by (Hopfield, 1984). It is known that the transmission and processing of information are effected by the modulation and demodulation of a carrier signal. It can be supposed that the representation, processing, and even the generation of information in the cortex are all spatio-temporal modulations of the activity patterns of neural assemblies. The repetitions and modes of activity may be very diverse. Suppose, for example, that some neurons are generated in an extended population as a representation of the real world; then, of no less importance are the inactive neurons which exist at the same time. In addition to such 'black and white' patterns, patterns of differentiated activity are also possible. For example, neurons may have regular dynamics in one region of the population while there is spatio-temporal chaos in another; or separate groups of neurons may be phase-synchronized, with the possibility that each group has topologically different configurations or even belongs to different populations. Two questions which we touch upon here are: How is information related to the neural dynamics of the cortex and how is information carried to the brain? These conundrums pose exciting challenges in neurophysiology today. We are not sure that evolution dictates the optimal manner by which information is transmitted, but since existing animals are the result of natural selection it is reasonable to suppose that information processing in neural systems has to satisfy to some optimizing principles. One such principle could be maximizing the discrimination ability between different sensory information inputs in minimal time. It is reasonable to hypothesize that a dynamical spatio-temporal representation of information in the brain is a reasonable way to satisfy this principle. Investigations of many brain functions have shown that the recurrence of a process or state will be reflected in repeated spatio-temporal patterns of neural activity. As an example, we note the recent results of Caltech researchers. Working with insects (bees and locusts), they have provided evidence that odor identity is reproducibly represented by a combination of spatial (different neurons) and temporal (different firing sequences) codes.
236
Epilogue: Living matter and dynamic forms
This spatio-temporal coding scheme consists of three main and concurrent odor-induced phenomena: (i) a 20-30 Hz oscillatory mean field activity, (ii) patterned and odor-specific neuronal responses, and (iii) transient synchro nization of odor-specific neural assemblies in different areas of the brain (Laurent, 1996, 1999). Because the anatomical design of the olfactory system (circuit macro- and micro-architecture, dendro-dendritic synaptic arrangements, and neural morphologies) is exceptionally similar across an imal brains, from mollusks to insects and from crustaceans to vertebrates, the functional principles that can be established studying one system might apply to the others as well. Thus we suggest that the spatio-temporal rep resentation of the stimuli is a general principle. Such a conclusion is sup ported by the experimental investigations of other sensory neural systems, in particular the visual sensory system (Singer and Gray, 1995). What are the privileges of the spatio-temporal representation of unpre dictable messages coming from the world around us? Why does 'time' have to be included in a coding space also? We think that new experiments will support the following answers to these questions. Spatio-temporal dynami cal information processing inherently contains many of the general features one desires for an ideal brain: large information capacity, reliability, sensi tivity which translates into a specific response to each specific input, and fast recognition. Returning to the comparison between brains and powerful computers, we are faced with one more interesting question: How does the hierarchical organization of the cortex couple with distributed coding? Recent experi mental results help us reconcile the relationship between two extreme views of information coding in large neural assemblies like the cortex. The first es pouses a systematic organization composed of a hierarchy of logic elements that encode simple stimulus features and complex events by calculations with single neurons (Barlow, 1972; Koch, 1999). A contrasting view relies on a fully distributed spatio-temporal representation as discussed above. We can say now that both methodologies are employed in the brain to different extents at successive stages of information processing. It is well known that the world speaks to us using its own and not our language. Nevertheless, we hope, and our previous experience supports our optimism, that the developing and continuously updated language of non linear dynamics will be adequate for understanding such phenomena of the nervous system and more general problems in biology. Now is the time to look for suitable images of what seem to be rather unusual phenomena. One
Spatio-temporal
patterns and information processing
237
curious phenomenon is the fact that reality is continuous while our percep tion of it is discrete — animals are able to concentrate their attention on only a limited number of fixed subjects and/or external stimuli. Another is the fact that perceptible images like imagination and consciousness, vi sual ones included, may be generated in the complete absence of external stimuli. This list of unfamiliar processing phenomena may be continued, but we simply conclude with a traditional reminder: any understanding of information dynamics in neural systems may be considered a success only when supported by reliable, and often very delicate, neurophysiological ex periments.
Appendix A
A Short Guide to Nonlinear Dynamics
A.l
Dynamical systems
A dynamical system is a mathematical object that corresponds to real sys tems (physical, chemical, biological, and others) whose evolution depends uniquely on the initial state. It is described by a system of equations — differential, difference, integral, etc. — which allow for the existence of a unique solution for each initial condition in an infinite period of time. The state of a dynamical system is described by a set of variables; there are different criteria for choosing a particular set of variables: symmetry and/or simplicity considerations, natural interpretation, etc. The set of states of a dynamical system forms a phase space. Each point in the phase space corresponds to a state of a dynamical system and temporal evolution of the system is depicted by phase trajectories. The location of the states relative to one another in the phase space of a dynamical system is described by the notion distance. The ensemble of states at a fixed moment of time is the phase volume of the system. The possibility to describe the behavior of a system by some determin istic equations (e.g. equations devoid of noisy terms) does not necessarily mean that the system is a dynamical system. For example, the equation dx/dt = x2 does not describe a dynamical system because the solution x(t) = ( l / i ( 0 ) - £) - 1 blows up at too = l / i ( 0 ) and the solution x(t) is not defined for t > too- This phenomenon for which the solution becomes infinite in a finite time is called an explosive instability. From the physical point of view, finite-time blow up indicates that the mathematical model of the system has a very limited range of application. 239
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The dimension n of the phase space is the number of independent vari ables necessary to describe the motion. In this guide, we consider primarily motions governed by a system of ordinary differential equations of arbitrary order where time is the independent variable. If time does not appear ex plicitly in the equations, for example as a variable coefficient or in a forcing term, then the equations are said to be autonomous; otherwise, it is nonautonomous. The equations describing the dynamical system can always be reduced to a system of first-order ordinary differential equations; then for an autonomous system the phase space dimension is n = m, where m is the number of equations. For a nonautonomous system it is n = m + q, where q is the number of incommensurate frequencies (see below) by which the system is forced, either directly or indirectly. In each case, the number of degrees of freedom of the dynamical system is n / 2 . The behavior of phase trajectories depicts the evolution of a state of a dynamical system. For example, a degenerate trajectory (a point in the phase space) corresponds to an equilibrium state and a closed trajectory represents periodic motion. A trajectory of quasi-periodic motion with n incommensurate frequencies Wj, i.e. there are no nonzero integers k{ which satisfy the equality 51 " = 1 ^»u;» = ^ passes arbitrarily close to any point of the n-dimensional torus: it is everywhere dense on this torus. Trajectories that are dense in a certain subset of phase space are expected for steady flows, while trajectories that never return to the vicinity of their starting points characterize a transient process. In our illustrations we will refer to hydrodynamical, chemical, electrical and mechanical systems. Considering general phenomena we will assume there exists a control parameter R which can be elevated to a critical value Rc corresponding to some bifurcation phenomenon, and sometimes we will speak about multiple bifurcations. When considering classical problems, we will employ the familiar control parameters, such as Re for Reynolds number and Ra for Rayleigh number. A.1.1
Types of dynamical
systems
Dynamical systems are classified according to the kinds of governing equa tions and type of methodology used. Deterministic systems are those hav ing no random (noisy) inputs or parameters; thus any irregular behavior that arises in the motion must be the result of nonlinearities in the de scribing equations. We distinguish between finite-dimensional and infinite-
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241
dimensional dynamical systems that are, respectively, systems with finitedimensional and infinite-dimensional phase spaces. Finite-dimensional dy namical systems are divided into conservative and dissipative systems, ones which respectively conserve or do not conserve their phase volumes. The set of conservative systems has a subclass of Hamiltonian systems possess ing a time-independent Hamiltonian function. Dissipative systems with an unbounded phase space often have a bounded region to which an arbi trary trajectory may be attracted and never escape. Dynamical systems are also classified into systems with continuous time (flows) or systems with discrete time (maps). The discreetness of time often represents the actual process: time steps at which a pulse passes through an amplifier in a laser; seasonal fluctuations in ecology; generation changes in genetics; etc. We can also distinguish between structurally stable and structurally unstable dynamical systems. Structural stability is a notion that describes a quali tative invariance of the phase space of the dynamical system with respect to small variations of parameters. The parameter values at which the system changes topological organization of phase trajectories in the phase space are referred to as bifurcation points. The established motion of a dissipative system corresponds to an attractor defined as a set of trajectories to which all neighboring trajectories are attracted. Steady state, periodic, or quasi-periodic motions correspond to simple attractors: an equilibrium state, a periodic trajectory, or a winding on a torus, respectively. As a rule a complex nonperiodic regime is repre sented by a strange attractor, a concept discussed in detail in Section A.3.1. Physically, a dissipative system is one where all motions possessing a sufficiently high energy attenuate. In an infinite-dimensional system it is assumed that the volume of any p-dimensional sphere decreases when p is sufficiently large. For a finite-dimensional system described by a system of ordinary differential equations x = X(x)
(A.l)
the dissipative nature of the system corresponds to the inequality V X < 0; this expresses the contraction of volumes in phase space. A.1.2
Equilibrium
states
The equilibrium state of a dynamical system is the state that does not change in time. Equilibrium states may be stable, unstable, or neutrally
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Fig. A.l The behavior of trajectories in the neighborhood of (a) stable and (b) unstable foci Ajt2 = a ± iu; in (a) a < 0 and in (b) a > 0.
stable. The motion of the system near equilibrium may differ depending on the type of equilibrium state. If the equilibrium state is stable in a system with a two-dimensional phase space (one degree of freedom), then with small perturbations of initial conditions the system always restores its equilibrium either through damped oscillations (a stable focus) as in Fig. A.la or aperiodically (a stable node) as in Fig. A.2a. Near an unstable equilibrium state, small perturbations increase, either oscillating (an unstable focus) as in Fig. A.lb or aperiodi cally (an unstable node) as in Fig. A.2b. Near a saddle equilibrium state as illustrated in Fig. A.3, the system may first approach equilibrium and then deviate from it. Finally, in the case of a neutrally stable equilibrium state (a center) as in Fig. A.4, small perturbations may cause frictionless oscillations near the equilibrium state. The motion near an equilibrium state is more complicated in systems possessing several degrees of freedom and depends significantly on initial conditions. The behavior of the dynamical system described by Eq. (A.l) near any of its equilibrium states is obtained by linearizing these equations about given equilibrium point. Solutions in the vicinity of the equilibrium state x° (which is obtained from equation X(x°) = 0) satisfy the linearized equation Xi
=
dXj (x°)x*. dxk
Dynamical
1
\
a
243
systems
/M
b
Fig. A.2 Trajectories in the neighborhood of (a) stable and (b) unstable nodes; in (a) A2 < Ai < 0 and in (b) 0 < Ai < A 2 .
Fig. A.3 Saddle equilibrium state corresponding to the motion of the inverted pendulum depicted at the upper left.
Solutions of the linearized equations can be represented as a sum of expo nential functions a;e Ajt with generally complex exponents A< that are the roots of the characteristic equation det(A — XI) = 0 where the matrix is A = {dXi{-x.°)ldxj) and X(x) is the right-hand side of Eq. (A.l). If all exponents satisfy Re(Afc) < 0 (Re(Afc) > 0), the equilibrium state is asymp totically stable (unstable), and all trajectories that start in the neighbor-
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Fig. A.4 Closed trajectories in the neighborhood of a center point. Example of systems giving this response are an LC-circuit and a simple undamped pendulum as illustrated.
W.'
w: Fig. A.5 Saddle 0 in a three-dimensional phase space: A2 < Ai < 0, A3 > 0; W$ is a one-dimensional stable manifold and W£ is a two-dimensional unstable manifold.
hood of stable (unstable) nodes tend to x° as t —> +00 (t —► —00), as shown in Figs. A.l and A.2 which are topologically equivalent. If Re(Afc) < 0, k = 1 , . . . , m, and Re(A*) > 0, k = m + 1 , . . . , n, then the equilibrium state is a saddle. The trajectories that tend to this saddle as t -> +00 belong to an m-dimensional stable manifold W0", and those tending to the saddle, as t ->• —00, belong to an (n — m)-dimensional unstable manifold WQ, i.e. each corresponds, respectively, to a multi-dimensional m or (n - m) separatrix like those sketched in Figs. A.5 and A.6. In conservative dynamical systems, particularly in Hamiltonian dynam ical systems, only the equilibrium states with purely imaginary or zero A* may be stable (to be more precise, neutrally stable). For example, the un damped oscillations of a simple pendulum or an LC-circuit are described by the point moving along a closed trajectory in the neighborhood of a center equilibrium state for which Aii2 = ±iu; see Fig. A.4.
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systems
245
Fig. A.6 Two-dimensional stable W£ and one-dimensional unstable VVg separatrices of a saddle equilibrium state 0.
If the dynamical system depends on a parameter then, even in nonconservative cases, Re(At) may become zero through variation of this parame ter. In this case, the equilibrium state may undergo bifurcations caused by a loss or gain of stability, or by a change in the dimension of its separatrices as discussed below. A.1.3
Homoclinic
and heteroclinic
trajectories
The trajectory that lies in W£ and WQ simultaneously and does not coin cide with 0 is referred to as a homoclinic trajectory or a saddle trajectory biasymptotic to 0. For steadily propagating waves, this loop correponds to a localized traveling wave whose amplitude vanishes as t —>• ±oo; some solitons belong to this group of waves. If Re(Ai) = 0 for certain A*, then stability of the equilibrium state is determined by succeeding terms of the Taylor series expansion of the vector field X about 0. This linearization procedure is also used to investigate the behavior of trajectories in the neighborhood of periodic motion L. This behavior of small perturbations against the background motion of period r is deter mined by the system's Floquet multipliers. For brevity, we will often refer to Floquet multipliers as simply multipliers. Mathematically, the multi pliers 7 1 , . . . , 7„ are the eigenvalues of the matrix exp(Qr) describing the solution x = C(t) exp(Qt) of a linearized system in the neighborhood of the periodic motion. Here Q is a constant n x n matrix and C(t) is a periodic
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Fig. A.7 Stable W£ and unstable W£ separatrices of saddle periodic motion L.
matrix C(t + r) = C(t). Further information about Floquet theory may be found in (Cesari, 1971). One of the multipliers, corresponding to motion along the closed tra jectory L, is equal to unity; so without loss of generality we henceforth set 7„ = 1. Then if |7<| < 1 (|-y*I > 1) for all t < n — 1, the periodic motion is stable (unstable). If p multipliers lie inside and q multipliers lie outside a unit circle in the complex 7-plane, then p + q = n — 1 and we have a saddle periodic motion like that displayed in Fig. A.7. In this case, L lies at the intersection of two surfaces: the (p + l)-dimensional stable separatrix W[ and the (q + l)-dimensional unstable separatrix W£. It is convenient to study the behavior of trajectories in the neighborhood of a periodic trajectory L by their traces on a section defined to be any (n - l)-dimensional surface D that is nowhere tangent to the flow. The map of each intersection point mo on D into its succeeding intersection of D is called a Poincare map; see Fig. A.8 where two such intersecting orbits are depicted. In £ = ( f i , . . . ,£n-i) coordinates, defined such that the periodic trajectory L intersects D at a zero point £ = 0, the Poincare map locally has the form £ = A£ H , where £ is an image of point £, the dots denote neglected nonlinear terms, and A is a matrix whose eigenvalues coincide with the Floquet multipliers 7 1 , . . . ,7n-iThere exist systems with a global section defined as the surface that is intersected by each trajectory an infinite number of times. In this case the Poincare map actually determines a dynamical system with discrete time. This class includes all systems that describe the effect of a periodic
Dynamical
247
systems
Fig. A.8 Poincare map for saddle periodic motion L and the nonperiodic trajectory in its neighborhood.
perturbation on a self-contained system and can be described in the form x = X(x, 6), 9 = OJ, where X is a vector function periodic with respect to 9. This system has a cylindrical phase space: the points (x, 6) and (x, 6 + 2TT) are identical. The hyperplane defined by 9 = 9Q is an example of a global section. In particular, the equations x + sin x = A sin 6,
9 = u)
(A.2)
for the motion of a pendulum under periodic external forcing with constant amplitude A determine a dynamical system with a global section. Fig. A.9a shows a phase portrait of the unforced (.4 = 0) system. Stable and unstable separatrices of a periodic motion may also intersect. Trajectories that belong to the intersection of stable and unstable separatri ces of different periodic motions are known as heteroclinic trajectories. The trajectory that belongs to the intersection of stable and unstable separatri ces of periodic motion L, but differs from L, is referred to as a homoclinic trajectory. The neighborhood of this trajectory contains, as a rule, an infi nite number of different trajectories, including a countable number of saddle periodic trajectories that comprise a homoclinic structure; see Fig. A.9b. Homoclinic structures, if any, may be used as a criterion for the existence of complex regimes in the dynamical system, as well as for the explanation of a number of nonlinear effects.
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248
separatrix
Dynamics
,k X
Fig. A.9 Phase plane of an inverted pendulum described by Eq. (A.2); (a) unperturbed motion and (b) a perturbed separatrix showing the homoclinic structure.
A.1.4
Limit
cycles
A limit cycle is an isolated closed trajectory in the phase space of a dy namical system that corresponds to periodic motion. The behavior of tra jectories in the neighborhood of a limit cycle is determined by its Floquet multipliers. If the absolute values of all but one of the multipliers are less than unity, then all the trajectories approach infinitely close to the limit cycle, in this case the limit cycle is stable. A stable limit cycle is the math ematical image of periodic self-excited oscillations. For example, the van der Pol equation H(l - x2)x + x = 0
(A.3)
for positive parameter /x has a single stable limit cycle that takes on differ ent shapes as fi is varied; three //-dependent phase portraits are shown in Fig. A.10. If some of the multipliers have absolute values greater than unity and some of them are smaller then unity, we have a saddle limit cycle that lies at the intersection of two separatrix manifolds — a stable one along
Dynamical
systems
249
Fig. A.10 Phase portraits of the van der Pol equation (A.3) with different values of nonlinearity: (a) /i = 0.1, quasi-harmonic oscillations; (b) /J = 1, strongly nonsinusoidal oscillations; (c) /i = 10, relaxation oscillations.
which trajectories approach the limit cycle, and an unstable one along which trajectories diverge from the limit cycle; see Fig. A.7. The transition with a variation of the parameters of one or several mul tipliers through the unit circle in the complex A-plane indicates a change of stability or the disappearance of a limit cycle; see Section A.2.3. A.1.5
Quasiperiodic
motion
The behavior of a dynamical system known as quasiperiodic motion is mo tion with N > 2 incommensurate frequencies U{. The solution x(t) corre sponding to quasiperiodic motion can be written in the form N
x(r.) = Y, A.*exp
-i
*=1
with phases <j>k{t) = Ukt + /?* periodic with period 27T. If we take the phase
A Short Guide to Nonlinear Dynamics
250
recurring time t - —(a-0i
+2nn)
where n is any integer. At these times the second phase will take on the values 4>2=P2 — ( a - / 3 , + 2 7 r n ) . The different frequencies are incommensurate and therefore u^/t^i is irra tional. If we reduce each value of 02 to a value in the range [0,27r] by subtracting an appropriate integral multiple of 2n, we find that, when n varies from 0 to oo, fa takes on values indefinitely close to any given num ber in that range. That is, in the course of a sufficiently long time, <j>\ and <j>2 simultaneously take on values indefinitely close to any specified state defined by any possible set of simultaneous values of the phases <j>k ■ The time to do so, the Poincare return time, increases very rapidly with N and can become so great that in practice no trace of any periodicity remains. The probability for the system to be in a given small volume near a chosen point in the space of phases <j>\, (fo,... , <J>N is the ratio of this volume (5
A.2
Bifurcations
The term bifurcation (from the Latin word bifurcus meaning forked) denotes the acquisition of a new quality by the behavior of a dynamical system with small changes in its parameters. Bifurcation corresponds to the restructur ing of the motion of a physical system. The foundations for the theory of bifurcations were laid down by A. Poincare and A. M. Lyapunov in the early 1920's and later this theory was further developed by A. A. Andronov and colleagues. Knowledge of basic bifurcation phenomena is a useful tool for the investigation of specific dynamical systems, in particular for the prediction of parameter values corresponding to the transition to new mo tions and for the evaluation of their regions of existence and stability; see (Arnold et al, 1994).
Bifurcations
251
Fig. A. 11 Thermal convection in a plane layer of fluid heated from below: (a) the stationary state 0 for (T+ - T _ ) < (AX) C where the fluid is at rest; the convecting states 1 and 2 for T+ — T- > (AT) C depend on the initial conditions; (b) the corresponding phase portraits.
An example of the restructuring of motion in a real system is thermal convection in a horizontal layer of fluid heated from below: a small increase in the temperature 7+ of the lower surface to a certain temperature dif ference AT above the temperature 7 1 of the upper surface does not cause macroscopic motion of the fluid, depicted as state 0 in Fig. A. 11a; in this case the heat flux between the lower and upper surfaces is due solely to molecular heat transfer. However, when the difference AT reaches a crit ical value (AT) C , cellular convection appears either as state 1 or state 2 in Fig. A.11a. This is an example of a pitchfork bifurcation. In this bifur cation, through the loss of stability of a symmetric equilibrium, two new asymmetric equilibria with cellular structures branch out as illustrated in Fig. A.lib. In this process the symmetric equilibrium position continues to exist, but loses its stability. In typical one-parameter families of asymmetric systems, pitchfork bifurcations do not occur. Mathematically, bifurcation is a change in the topological organization of trajectories in the phase space with small variations of system parame ters. This definition is based on the concept of the topological equivalence of dynamical systems: two systems are topologically equivalent — they have identical structures made by trajectories in phase space — when tra jectories can be mapped one into another by a continuous transformation of coordinates and time. An example of this equivalence is the motion of a pendulum with different values of friction A:; when the friction is small, the trajectories on the phase plane look like spirals as in Fig. A. 12a, and
A Short Guide to Nonlinear
252
Dynamics
when it is large the trajectories are parabolas as in Fig. A.12b. These phase portraits, that would seem different at first sight, can be transformed into each other by introduction of a new coordinate system, i.e. the transition from the phase portrait in Fig. A. 12b to that in Fig. A. 12a is not a bifur cation because it does not represent a transition from a given system to a topologically nonequivalent one.
^
7T\
Fig. A. 12 Phase portraits of the system x + kx + x = 0 for different k: (a) k < 2; (b) k > 2. Here F denotes a focal point and N denotes a nodal point.
Among the great variety of bifurcations encountered in the analysis of models of physical systems, the most important are the so-called local bifurcations. The simplest and most tractable of them are bifurcations of equilibrium states and the birth of periodic motions, each respectively considered in the following two sections. A.2.1
Bifurcations
of equilibrium
states
Our discussion will be aided by the summary results presented in Table A.l. The principal bifurcations of equilibrium states encountered in science and engineering are: (1) The appearance or disappearance of two fixed points. An example of this bifurcation is the motion of a marble in a potential well with two minima as sketched in Fig. A. 13. When the local minimum at D is removed, the equilibrium states, saddle S and center C 2 , merge and disappear as shown in Fig. A. 14. In dissipative systems the typical example of this type of bifurcation is the saddle-node bifurcation illustrated in Fig. A. 15. (2) The Andronov-Hopf bifurcation. This bifurcation is the birth of a limit cycle from an equilibrium state. An example is the tran-
Bifurcations
253
x x
X
Fig. A. 13 (a) Schematic for a marble moving in a potential U(x) with two minima and (b) its phase portrait with two centers C\ and Ci and saddle point S.
sition of a van der Pol oscillator from a stationary state to a selfoscillatory state. An appropriate change of variables transforms the van der Pol equation (A.3) with parameter fi into the equation given in No. 4 of Table A.l with parameter a. When a changes from negative to positive, the stable focus at the origin of (x,x) in the phase plane gives birth to a limit cycle whose amplitude for small Q is 0{a~1/2), while the focus becomes unstable. (3) The pitchfork bifurcation. This bifurcation is the birth of three equilibrium states from a single equilibrium state. It is exemplified by the change in the motion of a ball descending a parabolic chute when a hump appears at the bottom of the chute. Here the equi librium state C in Fig. A.16a gives birth to the three equilibrium states in Fig. A.16b: the saddle S and two centers C\ and C 2 . This provides conditions for the existence of stable asymmetric motions in a completely symmetric system and represents the simplest ex ample of spontaneous symmetry breaking; another example is the onset of thermoconvection illustrated in Fig. A.ll.
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A Short Guide to Nonlinear
Dynamics
*-X
Fig. A. 14 (a) Schematic for the marble moving after bifurcation and (b) its phase portrait with single center C\.
Fig. A. 15 Saddle-node bifurcation showing the birth of two equilibrium states, a sad dle S and a node N: (a) phase portrait before bifurcation; (b) phase portrait after bifurcation.
Local bifurcations can be observed in the evolution of small pertur bations in a system described by linearized equations. In the dynamical system x = X(x;/z), in which x is the vector of physical variables, ft is a parameter, and x°(/z) is the equilibrium state, small perturbations £ are
Bifurcations
255
Fig. A.16 Pitchfork bifurcation: The birth of three equilibrium states from a single stable state with small variation of a parameter (the shape of the chute): (a) chute with a single minimum and its phase portrait with one equilibrium state C of the center type; (b) chute with two minima and its corresponding phase portrait with three equilibrium states: a saddle S and two centers C\ and Ci-
described by the equation £° = A((j,)£, where A(p) = dXi{-x.0{n);^}/dxj. If the roots Afc(^) of the characteristic equation det(.<4(/x) - XI) = 0 do not lie on the imaginary axis of the complex A-plane, then no bifurcation takes place in the neighborhood of the equilibrium state with a small change in Only when one or several roots A(/x) lie on the imaginary axis of the com plex A-plane does a bifurcation occur; this situation is depicted in Fig. A.17 where the real root A = 0 and a pair of complex conjugate roots takes on the pure imaginary values ±iu at p. = pc. All bifurcations of the death or birth of equilibrium states correspond to one or several roots crossing the imaginary A-axis. The birth of an equilibrium state consisting of a saddle S and node N is depicted in Fig. A.18. This type of bifurcation is encoun tered, for example, in the problem of the competition of populations x\ and X2 feeding from a single source. The kinetic equations modeling the populations has the form ±i = [1 - (xi + pix2)]xl x2 = [1 - (x2 + P2Xi)]x2.
(A.4)
When both pi and p2 are greater than unity, either of the species may win in the struggle for existence; this state is shown in Fig. A. 18a. On the other hand, when p\ or p^. is less than unity, only one of the species will survive under arbitrary initial conditions; this state is shown in Fig. A.18b. Similar equations describe mode competition in lasers and the competition of different spatial structures that appear in thermally convecting fluids.
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A Short Guide to Nonlinear Dynamics
lm A
a>
Re A
-co Fig. A. 17 One real and two complex conjugate roots lie on the imaginary axis of the complex A-plane, each corresponding a bifurcation at some value p. = p,c.
N x,
0
N x,
Fig. A.18 Phase portraits of Eqs. (A.4) for (a) p\, pz > 1 and (b) p\ < 1, p2 > 1
When two roots of a characteristic equation become purely imaginary, a limit cycle either appears from the equilibrium state or dies in it (Table A.l, No. 4). Again with reference to Fig. A.17, this implies that for all values of the parameter fi larger than but close to the critical value // c , there exists a periodic solution that tends for fi —> fic to the stationary solution x°(/i). The limit cycle inherits the stability of the equilibrium state when /x = (ic. We recall that this is an Andronov-Hopf bifurcation.
Equation of a self-excited generator with hard excitation: x + /i(l - I 2 + orx1)x + X = 0 .
van der Pol equation: x - (a - x2)x + x = 0 .
3. Hard with regard to am plitude and fre quency.
4. Soft with re gard to ampli tude and hard with regard to frequency.
5. Soft with regard to am plitude and fre quency.
2. As item 1.
Model Amplitude equations for the van der Pol generator under the influence of periodic forcing: a = a[l - (a 2 + b2)} - Au>b - A/2 b = o[l - (a 2 + 62)] - A M where A u is the frequency detuning parameter. van der Pol - Duffing equation: x - /i(l - x 2 ) i + x - i 3 = 0 .
No. Type 1. With re gard to ampli tude hard and soft with regard to frequency.
Such a bifurcation is realized when varying two or more parameters. Situations of this type arise in the equations of hydrodynamics.
The Andronov-Hopf bifurcation is found in many different branches of science.
This is one of the most typical bifurcations for the emergence or disappearance of periodic motions.
For stationary waves in nonequilibrium media such a bifurcation corresponds to the transition from a quasiharmonic wave to a soliton and then to a cnoidal wave.
Comments In terms of the initial (not averaged) equations £ - /i(l - x2)x + x = j4sin8, © = to this bifurcation depicts the birth of a torus. Ex perimentally that corresponds to the transition of a nouautonomous oscillator from synchroniza tion to beats.
«1 m
WWW
§§§
A##
€)VD
Table A.I. Birth of periodic motions. The first column specifies the type or character of the arising periodic motions (self-excited oscillations); the second column makes reference to equations modeling such systems; the third column gives additional comments, and the sketches in the last column illustrate, from left to right, the phase portraits before bifurcation, at the moment of bifurcation and after bifurcation.
S3 OI
Go
o' 3
|
to
258
l
n+\
A Short Guide to Nonlinear Dynamics
•Xfl+1
Fig. A.19 Poincar6 map of the section i = 0 for a self-oscillatory with hard excitation: (a) no stable oscillations, no limit cycles; (b) the instant of bifurcation, the curve is tangent to the bisector; (c) stable 1 and unstable 2 limit cylces.
A.2.2
Birth of periodic
motions
Table A.l summarizes the basic types of bifurcations to the birth (if the phase portraits are viewed from left to right) or the disappearance (if viewed from right to left) of periodic motions. They are classified into four groups. If we consider the disappearance of periodic motions, the first bifurcation group (Table A.l, No. 1 and No. 2) includes the bifurcations in which the period of the motion tends to infinity as fi —> fic, while the average amplitude of oscillations does not tend to zero. An example of such a bifurcation in self-oscillating systems is the onset of modulation caused by an external periodic forcing. In an amplitude equation, a limit cycle (the image of modulated oscillations) is born from a separatrix saddle-node loop when two equilibrium states, a saddle and a node, merge and disappear (Table A.l, No. 1). Knowledge of this bifurcation allows a determination of the properties of the regime born after transition through the critical point: the established modulation will have finite amplitude and a nearly zero modulation frequency. The second bifurcation group is represented by the disappearance of a stable periodic motion at the instant it merges with an unstable periodic motion (Table A.l, No. 3); this is referred to as a tangential bifurcation. Such a bifurcation for a self-oscillator is presented in Fig. A. 19 as a Poincare map. Figure A.19a depicts the state of the system in the absence of sta-
van der Pol generator under the influence of periodic forcing: x — fi(l — x2)x + x = j4sin0 6=u;
Forced oscillations of an elas tic rod under the influence of small periodic forces.
2. Birth of twofrequency oscilla tions.
3. Birth of a pair of stable periodic motions.
For a = nn/q (with integers n and q and a ^ 0, 27r/3, n/2) a torus emerges with stable and unstable periodic motions. For a = 0, TC, 27r/3,7r/2 no smooth torus arises and the situation is more complicated. Such a bifurcation is typical for nonlinear systems (with the po tential energy exhibiting two min ima) under the influence of exter nal forces.
An infinite chain of bifurcations of period doubling is one of the most general paths towards chaotic in realistic systems.
Parametrically excited nonlinear oscillator with periodic forcing: x+fci+(l+6cos©):E-l-a: 3 = 0
e = u>
Comments
Model
No. Type 1. Bifur cation of period dou bling.
\
/ l \X/'y
\^
f 11iIf itC-
> )
n u 01 w tta
[\ O/flla (_-/£• t!V
Table A.2. Change of stability of periodic motion. The first column specifies the type of bifurcation and/or motion; the second column refers to equations modeling such systems; the third column gives additional comments; and the illustrations show (from left to right) the respective phase portraits before and after bifurcation, and the respective multipliers which (from top to bottom) are, respertively: A = —1; A — exp(ia) with a ^ nn/q; and A = + 1 .
to
3
S'
to
260
A Short Guide to Nonlinear Dynamics
ble oscillations, i.e. there are no limit cycles. Figure A. 19b illustrates the moment of bifurcation: the graph xn+\ versus xn plotted is tangent to the bisector of the first quadrant. This tangency point marks the birth of two periodic motions illustrated in Fig. A. 19c; that labeled 1 is stable and that labeled 2 is unstable. The previously discussed Andronov-Hopf bifurcation (Table A.l, No. 4) is the third bifurcation group. It is similar to the second group, but is represented by the disappearance of a stable periodic motion at the instant it merges with an unstable equilibrium state. Bifurcations of the fourth group (Table A.l, No. 5) are usually encountered in systems that depend on two or more parameters.
A.2.3
Change of stability
in periodic
motions
An important feature of bifurcations exhibiting a change of stability in periodic motion is the value of the Floquet multipliers 7 1 , . . . , 7 n - i a t a critical value fi = /xc. Recall in Section A. 1.3. that have we set 7„ = 1. If the multipliers 7 1 , . . . , 7 n - i have absolute values less than unity, the initial periodic motion will be stable. Bifurcations related to the loss of stability occur at parameter values of the system for which one or several multipliers have absolute values equal to unity. These ideas are summarized in Table A.2. If one of the multipliers is equal to negative unity, a bifurcation referred to as a period-doubling bifurcation occurs (Table A.2, No. 1). At the mo ment of bifurcation a small absolute perturbation simply changes its sign over one period and then, on the next period, the trajectory will close on itself in the linear approximation. The original periodic motion gives birth to a stable periodic motion with approximately twice the period, while the original motion will be unstable. Bifurcations of the birth of a twodimensional torus from a periodic trajectory (Table A.2, No. 2) correspond to the appearance of oscillations with two incommensurate frequencies in a physical system. A bifurcation in which two stable limit cycles are born simultaneously (Table A.2, No. 3) is encountered in systems that depend on two parameters or in systems with a specific type of symmetry. Bifurcations that result in the disappearance of stationary or periodic regimes (equilibrium states or limit cycles) may cause the dynamical system to enter a regime of chaotic oscillations.
Chaotic
A.3
Oscillations
261
Chaotic Oscillations
Chaotic oscillations are irregular oscillations of a fully deterministic non linear system that are almost indistinguishable from random noise. Complex behavior of nonlinear oscillatory systems was observed long before dynamical chaos was understood to be a possible inherent response in such systems; see, for example, the experiments of (van der Pol and van der Mark, 1927) and the double-disk model experiments of the magnetic dynamo (Rikitake, 1958). Moreover, quite a few mathematical tools were available to aid in the description of the nontrivial behavior of dynamical systems in phase space, such as homoclinic Poincare structures. However, neither physicists nor mathematicians realized at the time that determin istic systems may behave chaotically. It was only in the 1960's that the understanding of randomness was revolutionized as a result of discover ies made in mathematics and in computer investigations (Lorenz, 1963) of model physical systems.
A.3.1
Characteristics
of chaos and the strange
attractor
Chaotic oscillations, like white noise, have a continuous Fourier spectrum and a decaying self-correlation function. They can be produced by a dy namical system having a finite number of degrees of freedom, in contrast to noise which needs a system in which an infinite number of degrees of freedom are excited. The origin of complex irregular behavior of a finite-dimensional system is related to the instability of all (or nearly all) individual motions. A chaotic set is a collection of unstable trajectories in a bounded region of phase space that become entangled and produce chaotic motion. The quantities called Lyapunov exponents provide a computable mea sure of the degree of chaoticity for a trajectory. The Lyapunov exponents Xj of a given trajectory describe the mean exponential rate of divergence (if Xj > 0) or convergence (if Xj < 0) of neighboring trajectories. Let us define the Lyapunov exponents for the flow x(t) generated by the au tonomous system (A.l) with initial condition Xo- Linearizing Eq. (A.l) about this solution gives, for small perturbations w = Ax, the governing equation
Tif = M W W
A Short Guide to Nonlinear Dynamics
262
where M — dX/dx is the Jacobian of X. The mean exponential divergence rate of two initially close trajectories with separation distance /(xo,t) is A(xo) = lim
lim
*-+oo Ax(0)->0
GM£8)1-
«**
In the phase space there exists an n-dimensional basis {ej} of w such that for any w, A takes on n values Ai(xo) = A(xo,ei). These characteristic Lyapunov exponents can be ordered in size: Ai > A2 > . . . > A„. The sum of positive Lyapunov exponents is equal to the Kolmogorov-Sinai entropy m
K2 = '£,\j>0
(A.6)
in which m is the number of positive Aj (K2 = 0, when m = 0). The set of trajectories characterized by a positive Kolmogorov-Sinai entropy is a chaotic set. The value of K2 is a quantitative measure of the complexity of the flow x(t); see (Lichtenberg and Leberman, 1992) for further details. In a dissipative system, a region in phase space may attract all neigh boring trajectories. Such a region containing an attracting set of unstable trajectories is called a strange attractoi it contains neither stable fixed points nor stable limit cycles. In analogy with the limit cycle being the image of periodic oscillations in dissipative systems, the strange attractor is the image of chaotic oscillations. Let us choose in a chaotic set an ensemble of pieces (elements) of tra jectories of duration T that maintain a distance t from each other. Assume that any duration T of an arbitrary trajectory on the attractor lies in the eneighborhood of at least one of the elements. We will use C{T, e) to denote the number of elements in the ensemble. The number C(T, e) grows with decreasing e or with increasing T. The growth of C{T, e) with decreasing e is naturally related to the geometrical complexity of the attractor, while its growth with increasing T is a consequence of the instability of trajectories on the attractor. Consider the two following characteristics of motion on the attractor fc=lim
l i m f ^ f ^ V e-+0 T-»oo \
T
)
c=Um
l i m f ^ n
T-+00 e-»0 \
ln(l/e)
/
The quantity h is referred to as the topological entropy of the system and c is its fractal dimension. The observable corresponds in the effective phase space (infinite-dimensional, most probably) of the system to a limiting set
Chaotic Oscillations
263
Fig. A.20 Cartoon of the evolution of a drop of 'phase liquid' for a Hamiltonian system.
of trajectories. The dimension of this set is referred to as the dimension of the space series and the topological entropy of the system, taken only on this limiting set, is the topological entropy of the space series. These characteristic measures are important features used to describe the complex behavior of a dynamical system. A.3.2
Chaotic Hamiltonian
systems
Chaotic properties are encountered even in very simple Hamiltonian sys tems. Consider, for instance, an oscillator under the action of periodic forcing described by the equation x + kx + (1 - b cos 6)x + x3 = 0,
9 = ui.
(A.7)
When k — 0, this is a Hamiltonian system. The system has a threedimensional phase space and the initial phase volume is apparently con served. Let us see what happens in a definite parameter region of the system if we insert a drop of 'phase liquid' into the space (x, x, 9). In a certain time, the drop will be mixed and deformed in a complicated man ner, as shown in Fig. A.20, and will fill a definite region of phase space corresponding to chaotic motion. However, within this phase space will be islands of regular behavior; these result from initial conditions to which regular periodic or quasiperiodic behavior corresponds. This is well pronounced on the section 9 = 80; see Fig. A.21 where traces of phase trajectories are plotted. Reg ular motions are represented by invariant two-dimensional tori on which trajectories corresponding to quasi-periodic motion appear as white islands in Fig. A.21a. Such tori are destroyed in the region of chaos. Apparently, regions of regular and chaotic behavior are separated in three-dimensional phase
264
A Short Guide to Nonlinear Dynamics
Fig. A.21 Phase portraits of the chaotic sets on a section of the system (A.5): (a) Hamiltonian system k — 0 with 6 = 25; (b) dissipative system k = 0.12 with 6 = 25. Taken from (Izrailev et al., 1981).
space (as well as in four-dimensional phase space on constant energy threedimensional surfaces). These are referred to as systems with a separated phase space (Lichtenberg and Lieberman, 1992). There are no geometrical (topological) rules that would ensure the sep aration of chaotic and regular motions in a phase space with dimension higher than four. It turns out that regions of chaotic behavior in different parts of phase space may be interconnected by pieces of the same trajec tory, and such an interconnection usually occurs along separatrices. This phenomenon in a special type of diffusion called Arnold diffusion. The onset of chaos in Hamiltonian systems like that in Eq. (A.7) depends dramatically on the amplitude of external forcing. Physically, it is very simple. At sufficiently high amplitudes, there appear a great number of fundamental harmonics, with nonlinear resonance possible at any of the harmonic frequencies. A.3.3
Chaotic self-excited
oscillations
In systems with dissipation k, such as the damped, parametrically-forced oscillator given in Eq. (A.7), energy is no longer conserved; it decreases and thus one would expect a simple motion. Nevertheless, chaotic behav ior persists in such dissipative systems; see Fig. A.21b. Of course, chaotic self-oscillations are realized not only within the simple model (A.7) of a nonautonomous oscillator, but in nearly every nonlinear oscillatory dissi pative system with periodic forcing of sufficient amplitude. Even when
Chaotic
Oscillations
265
Fig. A.22 Phase portrait of the strange attractor on a section of the forced damped system (A.8) with w = l,0 = O,k = 0.05, and A = 7.5. Taken from (Ueda, 1991).
the oscillator's potential possesses only one minimum, i.e. one equilibrium state in the phase space of the unperturbed system, chaotic oscillations are likely to be interpreted as a random exchange of different simple motions whose images are periodic trajectories. Systems described by the classical equations x + kx +Px + x* = AsinO, x-kx(l
2
3
-x ) +0x + x = As\n9,
9=w
(A.8)
9=u
(A.9)
behave exactly in this way. Equation (A.8) is the forced Duffing equation describing nonlinear resonance with linear damping, while Eq. (A.9) is the forced van der Pol-Duffing equation describing a synchronization of oscil lations. Solutions of Eq. (A.9) exhibit not only beats, but also complex oscillations almost indistinguishable from random disturbances. The strange attractor for Eq. (A.8) is shown in Fig. A.22 at the param eter values indicated in the figure caption. The motion on this strange at tractor is established, chaotic self-oscillations. As with periodic self-excited oscillations whose mathematical image is a limit cycle, basic characteristics (oscillation spectrum, dimension, and entropy) of the motion established on a strange attractor do not depend on initial conditions. Initial conditions influence only the behavior of a transient process. In a system with many degrees of freedom, the dimension of a strange attractor may be much smaller than that of its phase space; in this event there is partial synchronization of the system's degrees of freedom.
266
A.4
A Short Guide to Nonlinear Dynamics
Synchronization of oscillations
Synchronization of oscillations is the matching of frequencies, phases, or other characteristics of signals generated by interacting oscillatory systems. We can distinguish between mutual synchronization, where partial subsys tems affect one another to produce synchronized oscillations, and forced or external synchronization where a regime of oscillations changes under the action of an external force. Forced frequency locking — the onset of oscilla tions at the frequency of the external forcing — is referred to as frequency trapping. Frequency trapping is the simplest example of synchronization phenomenon and was described back in the 17"1 century by Huygens who considered the change in acceleration of a pendulum clock suspended on a horizontal beam undergoing transverse oscillations. Synchronization theory is most well developed for quasi-harmonic oscil lations in weakly nonlinear systems. By way of illustration, consider, an equation (Landau and Lifshitz, 1987, Section 30) for the complex amplitude a of a nonlinear oscillator to which we add a term describing weak harmonic external forcing. This equation averaged over a period of external forcing / = ooe1' + c.c. takes the form d = n[a(l - \a\2) + i0\a\2a + a0 + i£a]
(A.10)
where the parameter £ measures a detuning between the frequency of selfexcited oscillations and the external forcing frequency (here normalized to unity), (i is the amplification coefficient for an autonomous oscillator, and /? is the nonlinear frequency shift. The synchronization regime corresponds to the stable equilibrium state of Eq. (A.10), and in the original threedimensional phase space this regime corresponds to a stable limit cycle. As f increases, the synchronization regime either ceases (for weak external signals) or loses its stability (for strong signals). The region of detuning in which the synchronization regime is established is referred to as an entrainment band. The boundary of the entrainment band is found from Eq. (A. 10) as follows: first, a resonance curve p = |a°(£)| 2 = p(£) of the intensity of synchronous self-excited oscillations is obtained from the condition defining the existence of synchronization (d = 0); then the stability of this regime is determined from the linearized form of Eq. (A.10). Entrainment bands for weak and strong signals, along with the variations of amplitude and frequency with detuning, are shown in Figs. A.23a and A.23b, respectively.
Synchronization
of oscillations
267
M
-*,
n
0
*,
4
n
4,
§
a Fig. A.23 Resonance curves (solid lines), boundaries of the stability region of the steady state regimes (dashed lines), and synchronization region — £i < £ < £ (cross-hatched) on the (p, £)-plane for (a) weak and (b) strong signals. The vertical dashed lines exhibit the boundary of instability and M and SI are the amplitude and frequency of modulation, respectively. Taken from (Rabinovich and Trubetskov, 1989).
In the phase plane of the complex amplitude [Im(a), Re(a)] of Eq. (A.10), Fig. A.24 shows a sequence of phase portraits at different values of detuning. Outside the trapping band the regime of synchronized oscillations becomes a regime of beats: the Andronov-Hopf bifurcation (for strong signals) or the bifurcation of the birth of a limit cycle out of a saddle-node separatrix loop (for weak signals) are observed; the latter scenario is that depicted in Fig. A.24. In three-dimensional phase space, the transition to the regime of beats corresponds to an attracting two-dimensional torus with a quasiperiodic winding on its surface. The synchronized oscillations of an ensemble of oscillators under the action of the same external harmonic force can be investigated in a similar
268
A Short Guide to Nonlinear Dynamics
Fig. A.24 Bifurcation of merging and mutual disappearance of node and saddle equi librium states resulting in a stable limit cycle corresponding to beats; (a) — £1 < £ < £1, (b)£ = 6 , a n d ( c ) { > £ i .
fashion. The mutual synchronization of quasi-harmonic oscillators in the simplest case of biharmonic resonance (u>2 = 2UJI + £) may be analyzed in the framework of the following system of equations for complex amplitudes ai,2 of coupled modes in a self-excited oscillator possessing two degrees of freedom di = / i i [ l -(|a!| 2 -|-/9i 2 |a2| 2 )]-l-aia2ate i 4 a2 = A*2[l - (|a 2 | 2 + P21 |ai | 2 )] - <x2a2e*.
(A.ll)
Here f is the frequency detuning from exact resonance, p.\ and y.i are the linear growth rates of the coupled modes, p\i and p%\ are the coefficients of nonlinear damping due to mode competition, and <j\ and 02 determine the strength of resonant interaction. Again, the synchronization regime corresponds to a stable equilibrium state. Mutual synchronization is ob served in systems possessing two and more degrees of freedom and very often chaotic self-excited oscillations appear after destruction of the syn chronization regime. Synchronization occurs when the individual oscillators have frequencies which are near multiples of each other. A practical ap plication of this phenomenon is the mutual synchronization of modes in optical cavities used to generate ultrashort pulses in lasers. The averaged description giving rise to equations of the form (A.10) and (A.ll) is not adequate for strongly nonlinear responses. In this case it is expedient to use a qualitative theory of dynamical systems. Within this theory, the synchronization of periodic oscillations of two self-oscillatory systems may be described as follows.
Synchronization
of oscillations
'■°l—i—i—I—I—i—i—i
1
'-
ot_l 0
I
I
r~,
rJr :
. I
269
I
1 I
O.S
I
l_l 1.0
r Fig. A.25
Plot of the Cantor function [i(-y) — the devil's staircase.
Suppose periodic self-excited oscillations are intrinsic to each of the systems x = /(x) y = g(y)
(A.12)
i.e. a stable limit cycle L\ or L2 exists in the phase space of the system. Then, for 7 = 0, the system x = /(x)+7/i(x,y) y = g(y) + v(x,y)
(A.13)
will have an attracting two-dimensional torus To = L\ x L 2 for which each system oscillates independently. As the coupling parameter 7 increases, the motions of partial subsystems of Eqs. (A.13) cease to be independent; this corresponds to a bifurcation to the torus T 7 that persists to be an attractor for Eqs. (A.13). In particular, synchronization corresponds to the birth of a stable limit cycle on this torus. The plot of the rotation number /J. of the system on the torus T 7 as a function of the coupling parameter 7 gives more detailed information on restructuring in the system as 7 is varied. The rotation number is a limit of the ratio of phases >(£) and ip(t) of the oscillations of the individual generators: \i = limt_+00[(/?(t)/T/)(t)]. The dependence of rotation number on the coupling parameter appears as a sequence of continuously diminishing steps — the devil's staircase — as seen in Fig. A.25. To be more precise, the
270
A Short Guide to Nonlinear
Dynamics
7 Fig. A.26 Arnold's tongues. The values of p/q for the rotation number corresponding to synchronization for a stable limit cycle on a torus are given inside each tongue. Here e is the parameter measuring supercriticality.
function /i(7) grows on a Cantor set. The rotation number takes on each value equal to the ratio of the integers p/q (synchronization) on a certain interval, with p and q corresponding to the harmonic mode numbers at which mutual synchronization occurs. Following the variation of 7 together with another parameter, for example Hi = \x2 = t in Eqs. (A.11), one can isolate synchronization regions on a plane. These regions are usually shaped as tongues, called Arnold tongues, as shown in Fig. A.26. Mutual synchronization of motions is intrinsic to generators of both periodic and chaotic self-oscillations. In contrast to the case of periodic oscillations, the motions of interacting nonidentical chaotic subsystems are matched only on average in time. The subsystems may have equal di mensions, energy spectra of partial oscillations, and topologies of strange attractor projections onto partial subspaces, even though the space series may not coincide locally in time. Figure A.27 shows strange attractors of the individual subsystems in the autonomous regime (c = 0) and attractor projections onto partial subspaces in the regime of chaotic synchronization (c = 10) for the system described by the coupled equations xi + kii\ + (1 + qcosflt)xi
+ Xj = c(±2 — i\)
x2 + k2x2 + (1 + qcosQt)x2
+ x\ — c(±i - x 2 )
(A.14)
at the parameter values indicated in the figure caption. The levels of chaotic synchronization may be different. In particular, for the interaction of iden tical subsystems, oscillations of the subsystems may coincide completely.
Dynamical chaos and turbulence
271
Fig. A.27 For parameter values fi = 2, *i = 0.48, k2 = 0.45 in Eqs. (A. 14) one observes (a) attractors of noninteracting (c = 0) generators projected onto the (xi,ii)plani (i — 1,2), and (b) projections of the attractor onto partial subspaces ( i i , i i ) or (12,3-2) at chaotic synchronization (c = 10), the projections being identical. Taken from (Afraimovich et a/., 1986).
A.5
Dynamical chaos and turbulence
There hardly exists a phenomenon that could compete with turbulence for breadth of interest accompanied by impassioned discussions and con tradictory statements of physicists, mathematicians, and engineers. This is connected with the exceptional complexity of the problem, as well as with significant differences in the way it is understood by scientists. In terms of an applied problem, turbulence models reduce to the formation of some effective equations, much simpler than the Navier-Stokes equa tions, for the calculation of drag, average heat and mass transfer, and other characteristics of turbulent flows. Such equations can be written based on intuitive knowledge as well as on various semi-empirical hypotheses. The main purpose of these equations is to predict real experimental observa tions and some remarkable results have been obtained (FVost and Moulden, 1977; Deissler, 1998). However, from the viewpoint of a physicist, such an approach to turbulence leaves aside critical fundamental questions. For example, how does a laminar flow become disordered? It is essential to un-
272
A Short Guide to Nonlinear Dynamics
derstand whether the mixing of fluid elements results from uncontrollable fluctuations or is due to instabilities inherent in the flow. Noise and inher ent fluctuations play an important role, no doubt, but only as a triggering mechanism to initiate instability. These are key considerations for any type of turbulence, be it hydrodynamical, plasma, chemical, or atmospheric. In other words, we deal with a fundamental problem — how a nonlinear field of arbitrary origin transforms to a disordered, random motion and how this motion is to be described. A review of this subject may be found in (Bohr et al, 1998). The evolution of viewpoints on the origin of randomness in the theory of turbulence and in statistical physics have much in common. The concepts formulated by (Reynolds, 1883) for small perturbations growing as a result of a developing linear instability cannot explain the transition to turbulence in simple hydrodynamical flow situations such as Poiseuille and Couette flow between plane parallel walls. According to the linear theory, there exists no critical Reynolds number within the framework of the NavierStokes equations for laminar flow in a pipe to become unstable. However, experimentalists know all too well that such a flow becomes turbulent at Reynolds number Re = O(10 3 ) in common experimental facilities. Many pioneers of turbulence investigations, for example T. von Karman and G. I. Taylor, believed that similar to the motion of gases, turbulence could be understood and interpreted only using a statistical approach. Their standpoint was supported by the fact that, at rather large Reynolds numbers, the number of degrees of freedom involved in the motion is so large that only a mean description of the flow is possible. The discovery of the phenomenon of dynamical chaos, the random be havior of completely deterministic systems which gained respect in all fields of research, cardinally changed accepted concepts on the origin of turbu lence and the origin of randomness in general. A chaotic set of trajectories in the corresponding phase space is the mathematical image of the random motion of a dynamical system. When we deal with turbulence at finite Reynolds numbers, the main point of interest is the established turbulent motion. The image of such a motion in the phase space could be an at tracting chaotic set — a strange attractor. Flow in a closed system is one in which fluid particles continuously recirculate past points previously visited. Examples include circular Cou ette flow between rotating cylinders and thermoconvection in a differen tially heated fluid confined between two horizontal plates. Experiments
Dynamical chaos and turbulence
273
in closed systems have shown that the most common scenarios for tran sition to chaos are the destruction of quasiperiodic motion (Ruelle and Takens, 1971; Newhouse et al, 1978), period-doubling sequences (Feigenbaum, 1978), and intermittency (Pomeau and Manneville, 1980). Some experimental results exhibiting these scenarios of transition to chaos are reproduced in Fig. A.28. In each case the motion evolves from a steady or periodic flow with a discrete spectrum through a series of bifurcations to a chaotic motion characterized by a continuous spectrum. Still more complex combined scenarios (Gollub and Benson, 1980) are possible in real situations, but the above canonical scenarios for particular flows proved the validity of the concepts of dynamical processes in the region of transition to turbulence, at least for flow in closed systems. The nature of the flow bifurcations preceding the appearance of disor der is not the only criterion for the applicability of chaotic dynamics to the description of the onset of turbulence. It is possible to reconstruct a strange attractor, prior to transition to turbulence, directly from obser vational data. This technique was first employed for thermoconvection in a closed cavity (Dubois and Berge, 1981), for baroclinic flow in a rotating fluid system (Farmer et al, 1982), and for Taylor-Couette flow (Brandstater et al, 1983). Some model dynamical systems have been formulated using several or dinary differential equations (Galerkin modes) rather than through direct analysis of the Navier-Stokes equations; see (Rabinovich, 1978) for a review. This avenue for modeling seemed quite reasonable in the early stages of de velopment of the dynamical theory of turbulence. However, real turbulent flow is a much more complex phenomena than motion described by a finite number of modes which produce dynamical chaos. First of all, developed turbulence is characterized by not only chaotic temporal fluctuations, but also by spatial disorder. This means that turbulence has to be associated with spatio-temporal dynamical chaos. Another important point is the existence of coherent structures in turbulent flows. This became obvious in the definitive mixing layer experiments of (Brown and Roshko, 1971), (Winant and Browand, 1974) and (Browand and Weidman, 1976). These structures have their own dynamics and the model equations for turbulent flow may have two interactive parts: equations which describe the dynamics of localized coherent structures and equations for nonlocalized modes. When considering the onset of turbulence, the number of degrees of freedom of the corresponding dynamical system can be estimated by direct
274
A Short Guide to Nonlinear
Dynamics
3,
ST 1S.1 R.
42.7 R c
?i
W,/4
3 Zitf.-
d^/2
i
to,
I
¥ 43.0 R c 3
0>,/4
eo,/8
0 12 008 0-04
_
100.4 Rc
\WAK
-004 1600
2000
1600
2000
Fig. A.28 Experimental results exhibiting the three classic transitions to chaotic mo tion. Above are frequency spectra showing: (a) the disruption of a quasi-periodic regime in Couette-Taylor flow, taken from (Fenstermacher et al., 1979); (b) a period-doubling se quence in Rayleigh-Benard convection, taken from (Libchaber and Maurer, 1978). The time series in (c) illustrates an intermittency scenario in Rayleigh-Benard convection, taken from (Gollub and Benson, 1980).
275
Dynamical chaos and turbulence
4
D
3
2
10
12
14
16
Re Rec Fig. A.29 Reynolds number variation of the correlation dimension D of a strange attractor corresponding to turbulent Taylor-Couette flow; Rec is the critical value of Re for onset of Taylor vortices. Taken from (Brandstater et al., 1983).
measurement of the correlation dimension D. It is important to remember that determination of the flow dimension does not necessitate a reconstruc tion of the attractor. (Grassberger and Procaccia, 1983) proposed a tech nique for measuring the correlation dimension D of motion directly from time series at different locations in the flow; see Chapter 12, Section 3. One set of measurements obtained for the Taylor-Couette flow system is given in Fig. A.29. This idea, along with experimental results obtained in other closed flow systems, encouraged the next step: to establish the origin of the development of turbulence in open shear flows (Kozlov et al., 1988). Flow in an open system is one for which particles do not recirculate past positions previously visited. Direct measurements of the dimension of turbulent mo tion in a boundary layer as a function of the streamwise coordinate showed, inter alia, that new degrees of freedom gradually become excited to produce turbulent motion along the direction of flow; see the experimental results given in Fig. A.30. An experiment (Libchaber, 1987) was performed where the complica tion of Rayleigh-Benard flow dynamics in a closed test cell was traced from the appearance of regular macroscopic convection, through a transition to chaos, to fully developed soft and then hard turbulence. The experiments were carried out using gaseous helium contained in a cylinder at 4°K. Re sults for the variation of axial heat transfer across a cylinder, measured by the Nusselt number Nu (the ratio of the total heat flow to the heat conduction flow), are reproduced in Fig. A.31.
A Short Guide to Nonlinear Dynamics
276
"wr * - * « -
«j—rira—ton—7bo—7tr
x[mm]
Fig. A.30 Variation of the correlation dimension D with downstream distance x of the boundary layer flow over a flat plate. Taken from (Kozlov et al., 1988).
10s
TTT
U=
Hard Turbulence
Soft Turbulence
10*
"
& ,
Nu 10'
-
",i ■
•• 10° 101
11 11
in 10 s
10*
10'
10"
Ra Fig. A.31 Variation of the Nusselt number Nu as a function of Rayleigh number Ra for Rayleigh-Benard convection in gaseous Helium. The insert shows the experimental cell and the position of bolometers used to characterize the dynamical states. Taken from (Libchaber, 1987).
The Rayleigh number Ra was varied over eight orders of magnitude. Such a variation in Ra is possible, at practically constant Prandtl number, by proper adjustment of the static pressure in the cylinder. Local tempera-
Dynamical chaos and turbulence
277
ture measurements were taken at two different points in the cylinder which permitted a determination of the boundary for the onset of spatial disor der. After the appearance of an oscillatory instability, chaos was born by standard scenarios, and then a low-dimensional strange attractor appeared with increasing values of Ra. For Ra ~ 1.5 • 105, the correlation be tween the time series at the two points was destroyed and spatio-temporal chaos (turbulence) was established. We can speculate that this regime also corresponds to a dynamical behavior, but one of a very high dimension. Thus it was not long ago that three directions in the theory of turbu lence were developing in parallel and practically independently: statistical, structural, and dynamical theories. The results obtained in each thrust were related to different problems and answered specific questions arising in qualitatively different experimental situations. It is surprising that the autonomy existed for such a long time: only in recent years has there ap peared the tendency and possibility to construct a unified theory.
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Appendix B
Key Experiments in Pattern Formation
B.l B.l.l
Parametrically excited patterns Experiments
with
liquids
On July 1, 1831 Faraday wrote in his diary "Mercury on a tin plate being vibrated in sunshine gave very beautiful effects of reflection." That's how Faraday described experiments on the excitation of capillary ripples appear ing on the surface of a liquid layer vibrating vertically. Wave excitation on a liquid surface is in this case analogous to the excitation of oscillations of a pendulum whose suspension point is vibrating vertically. It is the socalled parametric excitation where one of the governing parameters — the acceleration due to gravity — changes periodically in time. The experi ments on the parametric excitation of capillary ripples on a liquid surface carried out by (Faraday, 1831) are, evidently, the first purposeful investiga tions into the dynamics of fluid patterns. In any case, they were conducted seventy years before Benard's experiments on thermal convection that are broadly cited as the earliest experiments on the formation of spatially pe riodic structures in extended fluid systems. As one can see from Faraday's diaries (Martin, 1932) in which the experiments were described in ample de tail, Faraday did not focus on the mechanism of parametric capillary wave excitation; instead, he investigated the characteristics of observed patterns noting their topological properties, the influence of the liquids used, the effect of boundary conditions, and so forth. Faraday used very simple devices to produce vibrations. He mounted glass or wooden plates as horizontal beams with one or both ends rigidly fixed. A small container partially filled with liquid was secured at the crest 279
Key Experiments
280
w
in Pattern
Formation
JZLA_Q.
Fig. B.l Figures copied from Faraday's diary: (a) a cell fixed on a vibrating plate; (b) substrates of different shapes; (c) time evolution of structures in the course of formation of a tetragonal lattice; (d) possible patterns of parametrically excited capillary ripples. Taken from (Martin, 1932).
of flexural beam oscillation as sketched in Fig. B.la. In some experiments Faraday simply spread the liquid on substrates that were pieces of glass of different shapes as shown in Fig. B.lb. The liquid layer covered the entire surface of the glass repeating the shape of the substrate. In this variation of the experiment, Faraday produced oscillations by rubbing his finger against the bottom wetted glass surface, thereby exciting oscillations of the plate at some resonant frequency. In other experiments Faraday used a tuning fork as a source of external periodic oscillations. The sustained oscillations of the tuning fork enabled patterns on the liquid surface to be viewed for a relatively long period of time. Faraday excited capillary ripples on the surface of many liquids: water, milk, alcohol, oil, ink, white of egg, and mercury. The planform structure of the ripples was described as follows: "Crispations almost always quadrangular, always were well formed, but modified by edge of water or liquid." Faraday examined the process of emerging structures. In his notes of June 23, 1831 one can find the follow ing: "The system generally begins by circular waves or heaps concentric
Parametrically
excited
patterns
281
to glass rod...; at the same time there is a tendency to form linear heaps across the plate and parallel to each other...; these break up the circles and are themselves broken up, producing the independent heaps, at first rather irregular but almost instantly assuming the quadrangular arrangement..." Thus, Faraday was the first to establish that a tetragonal pattern occur ring at parametric excitation on a liquid surface is due not to boundary conditions, but owed its origin to its "preferableness" over other patterns. Faraday wrote: "It is very evident that the quadrate form of the crispations or waves is the natural one, for in that form the distance which the particles have to move through is the shortest possible distance, i.e. considered in relation to the courses which they would probably take and the number of these courses. In that arrangement the dividing lines are the shortest possible also, and other physical reasons suggest themselves to be consid ered hereafter." Faraday gave no further detailed evidence of the fact that quadrangular lattices are the most preferable. The picture illustrating the above quotation is given in Fig. B.ld. It is apparent that the dividing lines, the perimeter of the cell for the tiling in Fig. B.ld, are the shortest possi ble for hexagons and not for squares. Moreover, at parametric excitation, the frequency determines the wavelength of capillary waves; therefore, the structures should be compared for a fixed wavelength rather than for a fixed area of the elementary cell. Thus, Faraday's qualitative reasoning cannot be regarded to be satisfactory, but he formulated precisely the problem of pattern selection: he understood why squares were stable in his experi ments, and under what conditions a system of rolls, triangles, or hexagons may be realized. The frequency of vibration of liquid in a container was observed by (Faraday, 1831) to be only half of the vibration frequency imposed on the container. The problem was next investigated by (Matthiessen, 1868, 1870), who found that the liquid vibrations in his experiments were syn chronous with that of the container. The discrepancy between Faraday's and Matthiessen's observations led Lord Rayleigh (Rayleigh, 1883b) to make a further series of experiments which supported Faraday's observa tions. Matthiessen had written about a good agreement between his theory and experimental data. However, (Rayleigh, 1883b) pointed out two mis takes which compensated one another to a certain extent: the speed of wave propagation was determined incorrectly, and the waves were supposed to be absolutely gravitational ones, the effect of surface tension having been ne glected. Furthermore, (Rayleigh, 1883a) suggested a theory of "maintained
282
Key Experiments in Pattern Formation
vibrations" which might be extended to explain Faraday's The above controversy was not resolved until (Benjamin and Ursell, 1954) put the analysis of parametrically excited free surface waves on firm ground. Their theory leads to a governing Mathieu equation which reveals that the liquid layer will become unstable at all integer half-multiples of the natural frequency of liquid vibration: thus Faraday and Rayleigh observed a subharmonic resonance while Matthiessen observed synchronous resonance. If there are no complicating factors such as a magnetic field for fer romagnetic liquids (Wirtz and Fermigier, 1994), an electrostatic field, the presence of suspended particles, or heating a liquid close to its critical point (Fauve et al., 1992), then one can distinguish in the problem of para metric wave excitation on the surface of a liquid layer several governing parameters for a spatially extended system: supercriticality, frequency, dis sipation, and liquid layer thickness. Although the full partitioning of this four-dimensional space into regions corresponding to qualitatively different regimes has not been completed, researchers have obtained an insight into many aspects of pattern formation and some of their bifurcations. For ex ample, conditions for the existence of lattice patterns consisting of pairs of standing waves uniformly distributed about around the azimuth and having different symmetries has been clarified. The angle between the di rections of propagation of neighboring pairs of waves was found to be n/N, where TV" is an integer. It has been shown that a square lattice consist ing of two mutually orthogonal standing waves (N = 2) is formed in an infinitely deep liquid at moderate dissipation and supercriticality. As the supercriticality is increased against the background of a perfect lattice, a wave modulation appears that stimulates a transition from a regular regime to spatio-temporal chaos. The origin of these waves was analyzed in (Ezersky et al, 1985, 1986; Tufallaro et al., 1989; Daudet et al., 1995). It was established that transversal amplitude modulation (TAM) occurs in each of mutually orthogonal standing waves forming a square lattice. (Gluckman et al., 1993) experimentally demonstrated that the irregular modulation of a regular lattice exists; by averaging over a great number of space series such as that shown in Fig. B.2a, they obtained the strictly spatially periodic pattern in Fig. B.2b. Besides square lattices, standing waves in the form of roll structures (N — 1) and hexagons (N = 3) are regularly observed in parametrically excited capillary waves. The necessary condition for the formation of roll structures is a mode competition that appears at strong nonlinear damp-
Parametrically
excited patterns
283
Fig. B.2 A single snapshot of the transverse chaotic modulation (a) of the average spatially periodic pattern shown in (b). Taken from (Gluckman et a/., 1993).
Fig. B.3 ripples.
Bound state of two topological charges in parametrically excited capillary
ing. Such a damping may be due, for instance, to the absorption of forced higher harmonics; see Chapter 3 where this mechanism of nonlinear damp ing is considered. Therefore, roll structures are the most preferred mode of parametric instability in a liquid with large viscosity (Kudrolly and Gollub, 1996). In this case, dislocations are likely to form against the background of roll and tetragonal patterns (Ezersky et ai, 1994). It was further shown by (Ezersky et ai, 1995) that at the onset of parametric excitation, each dislocation is a coupled state of two topological defects having like charges; see Fig. B.3. The defects are spaced apart along the direction of propagation of cap-
284
Key Experiments in Pattern Formation
illary waves. The distance between the topological charges in a dislocation depends on such governing parameters as the degree of supercriticality, the magnitude of dissipation, and the thickness of the liquid layer. Experi ments verified that a dislocation may transform into a modulation wave when, for example, the depth of the liquid layer is increased. With in creasing depth, the distance between the topological charges increases, the charge strength increases rapidly, and a solitary envelope wave replaces the dislocation. Bound states of topological charges possess certain particle-like properties (Ezersky et a/., 1995). Two dislocations with topological charges of opposite sign may self-annihilate, leaving behind a perfect set of aligned rolls. Dislocations having like signs may unite to form quasi-stable states such as a domain wall (Ezersky et al., 1994). Dislocations belonging to mutually orthogonal modes making a tetragonal lattice cannot annihilate or form long-lived structures; however, one dislocation may be scattered by another. At large supercriticalities, an ensemble of interacting dislocations produces spatio-temporal chaos. Experiments have established that not only roll and tetragonal struc tures, but hexagonal lattices may be formed at instability. The existence of hexagons in Faraday ripples was justified theoretically in the paper of (Zhang and Viiials, 1997). Roll, tetragonal, and hexagonal patterns are re placed by spatio-temporal chaos when the supercriticality is increased. As was shown recently in (Kudrolly and Gollub, 1996), the transition to a reg ular state may occur nonuniformly in space: regions with spatio-temporal chaos are able to co-exist with regions in which the instability patterns are absolutely regular as seen in Fig. B.4. With increasing supercriticality, the regions of the irregular field occupy an increasingly larger area in a test cell. As demonstrated in (Ezersky, 1991), the transition to spatio-temporal chaos also may occur through temporal intermittency: regular and chaotic patterns formed by capillary-gravity waves replace one another uniformly in space and quasi-periodically in time. The time interval between these alter nating patterns grows with increasing supercriticality and chaos eventually appears in the system of parametrically excited waves. The transition to temporal intermittency takes place only at the liquid layer depth for which the group velocity of capillary waves is close to the phase velocity of lowfrequency gravitational waves. Low-frequency, large-scale perturbations in this case are an additional degree of freedom that allows for the appearance of temporal intermittency.
Parametrically excited patterns
285
Fig. B.4 Regular and chaotic regions in parametrically excited capillary ripples. Taken from (Kudrolly and Gollub, 1996).
Fig. B.5 Tenth-order quasicrystals formed on the surface of parametrically excited cap illary ripples. Taken from (Binks and van de Water, 1997).
Experiments indicate that a structure of N modes may be generated; when N > 3 the pattern is called a quasicrystal. Eighth-order quasicrystals consisting of two square lattices at an angle of 45° (N = 4) (Christiansen et a/., 1992), and tenth-order quasicrystals (N = 5) as shown in Fig. B.5, can appear with monochromatic forcing. When ripples are excited by an exter nal field of the form acos(4wt) +bcos(ou)t + <j>), twelfth-order quasicrystals (N = 6) may be found. As noted in (Edwards and Fauve, 1993, 1994), the breaking of quasicrystal symmetry and the transition to chaos may oc cur through the appearance of topological defects. In this case, structures with absolutely nontrivial topology are formed that are difficult to identify without a priori information.
286
Fig. B.6 1995).
Key Experiments
in Pattern
Formation
Spiral waves parametrically excited in a square cavity. Taken from (Kiyashko,
Multi-armed spirals may also be formed during parametric excitation of waves (Kiyashko, 1995; Kiyashko et al., 1996). Boundary conditions turn out to be a significant factor for the initiation of such spirals. The above studies show that multi-armed spirals evolve from a target pattern (circular rolls) when its symmetry is broken. Such symmetry breaking occurs in the neighborhood of a sidewall where topological defects are born and accumulate in the center of the target. If the topological defects have charges of opposite sign, they annihilate each other and no spiral structure is formed. The propagation of several topological defects having like signs towards the target center results in the manifestation of a spiral structure. In particular, (Kiyashko et al, 1996) demonstrated that the movement of like-signed defects to the center of a target is caused by the mean meridional circulation induced by capillary ripples. Although the cylindrical geometry of the container wall significantly influences the formation of spirals, it is not the decisive factor for their inception. Spirals are evidently the most preferable structure for the region of control parameters of the experiment. This is confirmed by the observation of spirals in a square cell like that shown in Fig. B.6. Solitary waves analogous to an 'oscillon' (c/. Fig. (8.3)) may also ap pear as the result of parametric excitation of a liquid layer; an example is shown in Fig. B.7. Soli ton-like patterns appear at small supercriticality on the surface of a liquid layer having a depth of the order of the viscous penetration thickness 6 ~ (v/u)1!2 or smaller, where v is the kinematic vis cosity of the liquid and ui is the frequency of wave oscillation. Additional
Parametrically
excited
patterns
287
Fig. B.7 A fluid 'oscillon' on parametrically excited ripples. Taken from (Lioubashevski et ai, 1996).
information about solitary states in the Faraday experiment may be found in (Fineberg and Lioubashevski, 1998). For realization of the dissipative patterns discussed above, it is not necessary to use pure Newtonian fluids. (Lioubashevski et ai, 1999) ex perimented with a colloidal suspension for an investigation of oscillons and propagating solitary waves in parametrically excited dissipative media. Sus pensions provide a bridge between Newtonian fluids and dry granular ma terial since the interactions between tiny suspended particles are mediated by the fluid between them. Experiments to produce parametrically excited patterns in dry granular material are discussed in the following section.
B.1.2
Experiments
with granular
material
Granular materials are large collections of discrete macroscopic particles like sand, salt, or small spheres. Such materials exhibit a unique mixture of properties of liquids, solids, and even gases, depending on the rates of energy injection and dissipation. If the particles are noncohesive, then the forces between them are only repulsive so that the equilibrium shape of nonexcited material is determined by external boundaries and gravity only. If the grains are dry, the interstitial gas often can be neglected in determining the properties of the granular system. In fact, granular materials are an additional state of matter in their own right. The science of granular media has a long history; interested readers should consult the review by (Jager et ai, 1996). In the same paper in which he reported his classic vibrating fluid experiments, (Faraday, 1831) discovered a convective instability in vibrated powder. Over a half century later (Reynolds, 1885) introduced the notion of dilatancy, which implies that a compacted granular material must expand in order for it to undergo any shear. Usually, granular material refers to a particle system in which the
288
Key Experiments
in Pattern
Formation
Fig. B.8 Side view of a parametric wave obtained in a thin layer of granular material. The horizontal white line indicates the vertically moving plate; (a) and (b) show the wavy pattern immediately before particles collide with the plate; (c) after the collision a lateral transfer of grains occurs. Taken from (Cerda et al., 1997).
size of the particle is larger than one micron; otherwise thermal agitation becomes important and Brownian motion can be observed. Above one micron thermal agitation is negligible. Patterns in granular systems can be found most anywhere in Nature: dunes in a desert, sand ripples in riverbeds and littoral zones. All kinds of defects, domain walls, spatial disorder, and other phenomena can be encountered. We are going to discuss here some key experiments related to the dynamics of patterns in granular material. Owing to the large frictional forces experienced during particle collisions, the existence of steady dynamical patterns needs an energy source to overcome dissipation. There are many different ways to parametrically introduce energy into a granular system. To date, the most popular laboratory setup by which patterns may be observed is the modern variant of the Faraday experi ment: a vertically vibrated container partially filled with a large number (~ 104) of small spherical particles (Umbanhowar et al., 1998). However,
Parametrically
excited
patterns
Fig. B.9 Patterns in a vertically oscillating granular layer at / = 67Hz: (a) T = 3.3, (b) T = 4, (c) T = 5.8, (d) T = 6, (e) T = 7.4, (f) T = 8.5. Taken from (Melo et at., 1995).
(Pouliquen et al., 1997) have observed well-organized packings in laterally excited granular layers of large horizontal extent, which in some cases re sult in packing ratios <j> well in excess of the random close packing limit <j> = 0.64. In both horizontally and vertically vibrated systems, the onset of pattern formation is found to depend crucially on the depth of the granular layer and on the dimensionless acceleration T = Aw2/g, where g is the the acceleration of gravity, A is the amplitude of container vibration, and u is its frequency; T is often referred to as the granular temperature (Jaeger et a/., 1996). For vertically vibrated systems, the primary instability gives rise to a pattern oscillating with frequency CJ/2. Three side-view snapshots of a vi brating layer, taken during a period of oscillation of the primary instability,
290
Key Experiments in Pattern Formation
is shown in Fig. B.8; both vertical and lateral particle movement may be observed. It is possible to excite rolls, squares, or hexagonal patterns and also domains separated by kinks. When kinks are present, each domain is oscillating with frequency w/2, but the temporal phases in neighboring domains differ by n radians, as seen in Fig. B.9. In some cases when in stability gives rise to rolls or squares, granular solitary waves or oscillons are observed (c/. Fig 8.3). For oscillons to exist, the primary bifurcation must be hysteretic so that both a flat layer and disturbance waves can coexist at the same parameter value. At large supercriticality, observed spatio-temporal patterns often appear similar to capillary wave patterns that develop in the Faraday experiment. As mentioned above, the layer thickness, measured in particle diameters, is an important control parameter in experiments on vibrating granular material. In the study by (Olafsen and Urbach, 1998) the particles were uniform 1 mm diameter stainless steel spheres forming a monolayer. In thick granular layers the particle-particle collision rate is much higher than the forcing frequency u>. On the other hand, the collision rate in monolayers is of the order of the forcing frequency. This difference in collision rates gives rise to different patterns observed in thick and thin layers; compare, for example, the thick layer patterns in Fig. B.9 with the thin layer patterns in Fig. B.10. In the monolayer at large acceleration T, the particles are in a gas-like phase (Fig. B.lOa). When the acceleration amplitude is slowly decreased, the average kinetic energy of the particles decreases and localized transient clusters of low-velocity particles appear. The bright peaks in Fig. B.lOd correspond to regions of two or three particle diameters in size. As the acceleration amplitude is further decreased, the typical cluster size increases to 10-12 particle diameters as in Figs. B.10e,f. Within a few minutes at this acceleration there appears a condensate of particles that comes to rest on the plate. In this situation the pattern contains stationary particles in one phase and constantly moving particles in the other. As previously noted, granular matter takes on, in different situations, the behavior of liquids, gases, or solids. For example, (Mujica et al., 1999) revealed the existence of a solid-liquid transition that precedes a subharmonic wave instability. Using measurements of the pressure of the grains colliding with the container, it was shown that there exists a critical value T p for which the layer undergoes a phase transition; this value is smaller than the critical value T/, at which hydrodynamic waves appear. Depend-
Parametrically
excited
patterns
291
Fig. B.10 Instantaneous (left column) and time-averaged (right column) photographs of patterns in a vibrating granular monolayer. In (a) and (b) at T = 1.01 are uniform particle distributions typical for a gas phase; in (c) and (d) at T = 0.8 clusters are visible at higher intensity points in a time-averaged image; in (e) and (f) at T = 0.76 a region of collapse is visible and the time-averaged image shows that the particles in the collapsed region are stationary while the surrounding gas particles continue to move; in (g) and (h) an ordered phase is observed. Taken from (Olafsen and Urbach, 1998).
ing on the excitation frequency, two kinds of modes can be distinguished: hydrodynamical waves and dilational waves. The hydrodynamic surface waves can be considered as the natural mode of excitation that exists in a fluidized granular layer. It involves a lateral transfer of grains within the layer. By contrast, in very low amplitude surface displacement dilation
292
Key Experiments in Pattern Formation
waves, the layer slightly dilates and compacts alternatively in space and time. At high values of u and for the same value T/, where hydrodynamic waves appear, dilation waves are detected instead. Since the stresses in a nearly static granular layer are highly inhomogeneous, localized stress chains are formed. Such chains are very specific and unlike anything seen in ordinary liquids or solids; see (Gollub and Langer, 1999). A comparison of patterns observed in vertically vibrated liquid and gran ular layers shows that dissipative media forced in the same manner often display universal properties independent of the media. However, as noted above, it is possible to observe in granular material new phenomena con nected with phase transitions between different states of vibrating particles and the excitation of new types of propagating wave modes. This clearly stems from the fact that granular materials can exhibit the behavior of liquids, gases, or solids which provides an additional degree of freedom for parametrically forced systems.
B.2 B.2.1
Thermal convection Rayleigh-Bdnard
convection
Thermal convection — the buoyant motion induced by a temperature gradi ent in a fluid layer — is observed both in Nature and in different engineering applications. We present here a short review of key experiments in a liquid layer uniformly heated from below in test cells so large that the sidewalls do not affect significantly the structure of patterns that may appear. Early systematic investigations of the conditions under which convec tion occurs in such systems were made by Benard at the beginning of the 20"1 century (Benard, 1900, 1901); his experiments have been described in detail by (Koschmieder, 1992). Attempts to give a clear understanding of experiments on convection had been made by some scientists before B6nard, but their observations are only of historical interest because they were not performed under controlled conditions and, consequently, were not repro ducible. In spite of its simplicity, the experimental apparatus shown in Fig. B . l l enabled Benard to reveal the underlying phenomenon associated with convectively unstable flow — the formation of cellular fluid structures — that strongly influenced the subsequent development of theoretical and experimental research in thermoconvection. In his experiments, Benard used a thin layer of whale's spermaceti, a
Thermal convection
'JSs-
Fig. B.ll 1901).
Schematic diagram of B6nard'sexperimental apparatus. Taken from (Benard
waxy, rigid substance with melting temperature 46° C. For investigation of thermal convection, the spermaceti was placed on a cast iron plate (PP in Fig. B.ll) heated by steam from boiling water up to 100°C. The depth of molten spermaceti was about 1 mm, the diameter of the cell was 20 cm, and its upper surface was in contact with the ambient air. Using modern termi nology, this large aspect ratio of cell width to 'fluid' depth (~ 200) meant that Benard was investigating thermal convection in a spatially extended system. The convective motion was visualized with the aid of various fine grained reflective particles such as aluminum flakes, graphite powder, and sometimes lykopodium. Principal attention was given to the dependence of the horizontal wavelength A of the structure on layer depth d. Benard found that as the depth of the layer is varied, the ratio X/d remained nearly constant. More accurate measurements have verified that as d increases X/d also increases: X/d = 3.378 for d = 0.440 mm and X/d = 4.049 for d = 0.835 mm. Benard also investigated the influence of differential heating across the fluid layer on the instability wavelength. With Benard's appa ratus, the vertical temperature contrast AT across the fluid layer could be varied by allowing the plate to cool down naturally after the steam heating was terminated; this resulted in a quasi-steady decrease in the tempera ture contrast, owing to the large thermal capacity of the cast iron plate.
294
Key Experiments in Pattern Formation
z d
h
* JL „
° hi. ^-7r±*Z?
t .,
i-^rf ,' 1
....
=
'».
Fig. B.12 Schematic of forces acting on a fluid element vertically displaced between uniformly heated plates with temperatures T\ > X2.
It was found that decreasing the vertical temperature contrast across the fluid layer reduces the horizontal instability wavelength A. This result has been corroborated by more recent and accurate experimental data. Early attempts to explain the physical mechanism responsible for the formation of cellular structures in Benard's experiments were made by (Rayleigh, 1916) who attributed it to an instability associated with the temperature dependence of density. It is important to point out that sur face tension was neglected in Rayleigh's analysis. We omit details of his calculations and only summarize his qualitative speculations. Rayleigh con sidered a layer of liquid with fixed temperatures at its upper (T2) and lower (T\) boundaries as depicted in Fig. B.12. In the absence of convection, a regime of heat conduction is realized, with the temperature linearly dependent on the vertical ^-coordinate, antiparallel gravity. If in this environment an element of liquid moves from the bottom upwards, it will be surrounded by a cooler, more dense liquid and thus cool down by conduction. The higher the viscosity, the slower the element moves and the higher the thermal conductivity, the faster it cools down. Apparently, therefore, the convection appears when the buoy ancy force driving the fluid parcel upwards dominates over the retarding effects of viscous friction and thermal cooling. If the liquid element has characteristic size L, then the buoyancy force F), acting on this element may be estimated from Archimedes' principle as Ft, ~ pg0(AT)'L3, where g is acceleration due to gravity, /3 is the temperature coefficient of volume expansion, p is a characteristic {e.g. mean) fluid density, and (AT)' is the
Thermal
convection
295
local temperature contrast between the fluid element and the ambient liq uid. The time r ~ W/d for the element to rise the distance between the plates may be estimated by equating the buoyancy force to the Stokes drag Fs ~ pvLW on the moving liquid element; here W is the characteristic rise velocity and v is the fluid kinematic viscosity. Equating these forces and elimination of W gives the rise time r ~ i>d/g(3(AT)'L2. For convec tion to appear, the rise time certainly should be smaller than the time T'K during which the temperature of the element equilibrates with the temper ature of the ambient liquid. From the thermal conduction equation, one readily finds the estimate T'K ~ L2/K, where K is the thermal diffusivity of the liquid. The condition r < T'K for the appearance of convection gives L2/K > vd/gfi(AT)'L2. Shifting from local to global estimates of particle size L ~ d and temperature contrast (AT)' ~ AT = T\ — T-2, one arrives at a criterion for the onset of convection g/3ATd3 Ra = — > 1. UK
This dimensionless parameter has come to be known as the Rayleigh num ber. It may also be written as the product of two time scale ratios, viz.
where TV = d2 /u and TK = d2 /K are respectively the vertical viscous and thermal diffusion time scales, and T& = (d/gflAT)1/2 is the buoyancy time scale obtained by solving the equation Fj = ma ~ md/r2. Writing the Rayleigh number in this form shows that the effective time scale for diffusion of momentum and heat, TJ = y/rurK, must be larger than the buoyancy time scale Tj, for fluid convection to occur. B.2.2
Patterns
in Rayleigh-Benard
convection
Couette-Taylor flow, Rayleigh-Benard convection, and the Faraday problem are the most thoroughly investigated phenomena among pattern-forming systems. Convective patterns in spatially extended systems (cf. Chap ter 1) usually occur when a sufficiently large temperature gradient is applied across a fluid layer. In traditional experiments, the patterns appear in a fluid layer of height d confined between two thermally conducting horizon tal plates; see Figs. 2.5 and 2.6. In standard conditions (Boussinesq fluid,
Key Experiments
296
in Pattern
Formation
Ra
Pr Fig. B.13 Flow regime diagram exhibiting various possible types of motion in RayleighB6nard convection; the horizontal line corresponds to Ra = Rac. Taken from (Krishnamurti, 1973).
no rotation, homogeneous heating from below, a spatially extended system, etc.) only two dimensionless control parameters, the Rayleigh number Ra and the Prandtl number Pr, are necessary to describe pattern formation in convective layers. The Prandtl number Pr
v K
may be viewed as the ratio of the vertical thermal diffusion time TK to the vertical viscous relaxation time T„. The value of Pr varies widely for different experimental fluids, ranging from O(10 - 2 ) for liquid helium and liquid metals, to 0(1) for air, 0(10) for water, and O(10 3 ) for some silicone oils. The spatio-temporal patterns and the transition to turbulence are, in general, strongly dependent on the value of Pr; see the flow regime di agram of (Krishnamurti, 1973) displayed in Fig. B.13. Only for slightly supercritical Rayleigh numbers Ra > Rac in the vicinity of the the critical Rayleigh number Rac are the instability patterns independent of Pr. With increasing Ra, secondary instabilities that appear for small and large Pr are different. The ideal spatially periodic patterns like rolls can be observed in real experiments only for very specific experimental conditions. Usually the convective patterns are slightly disordered (c/. Fig. 12.1) and exhibit com-
Thermal
convection
297
plex spatio-temporal dynamics as (Ra — Rac) increases. For liquids with Pr ~ 1 the deformation of rolls (roll curvature) induces slowly varying, long-range pressure gradients (Siggia and Zippelius, 1981) that generate a mean flow, which in turn alters the roll curvature; see Chapter 9 and also (Croquette et al, 1986; Pocheau and Daviaud, 1997). The most important experimental method for the visualization of convective flow patterns is the shadowgraph technique (Croquette, 1989; de Bruyn et al., 1996). Typical experimental test cells are constructed with a reflective bottom plate and a transparent sapphire top plate. Many ex periments have been carried out using both quasi-Boussinesq fluids and non-Boussinesq fluids. A Boussinesq fluid is one for which the density vari ation due to temperature is entirely responsible for fluid buoyancy, but does not significantly alter the fluid inertia nor the viscous diffusion forces. Since visualization of convection in liquid helium and in liquid metals is extremely difficult, but see (Pool and Koster, 1994) and (Koster, 1997), there is little information on convective patterns at small Pr. For Pr > 1, however, much is known about pattern formation in Rayleigh-Benard con vection. At slightly supercritical Rayleigh numbers, roll structures are gen erally observed. For sufficiently large Pr (water or silicone oil) rolls exist for Ra < 2 • 104. At larger Ra before transition to unsteady convection, a regime of two-mode convection exists. Patterns in this region often ap pear as two mutually orthogonal roll systems. For Pr ~ 1 roll structures may exist only in a small range of supercriticality; with increasing Ra they are transformed into an ensemble of interacting spirals (Hu et al., 1995). Non-Boussinesq features, such as the dependence of the viscosity on the temperature, are responsible for the formation of many crystalline patterns like hexagons (cf. Chapter 5) that occur both at small (air) and large (sil icone oil) Prandtl numbers. The validity of the Boussinesq approximation depends not only on the type of liquid, but also on the geometry of the cell; see the interesting review by (Bodenschatz et al., 2000). B.2.3
Benard-Marangoni
convection
Consider again the differentially heated horizontal liquid layer of thickness d with temperature contrast AT = T\ - T2 > 0. Benard-Marangoni con vection takes place when there is no gravitational force, in which case the onset of convection is due solely to surface tension gradient forces. For most liquids the coefficient of surface tension a decreases with increasing
298
Key Experiments in Pattern Formation
1
^
¥ f
° ^
l_ *~^x
-^*
1 Tl
T
'\ Fig. B.14 Schematic used to discuss surface tension gradient driven convection in a differentially heated liquid layer with a lower rigid boundary and an upper free surface; Ti > T 2 .
temperature which, for small temperature contrasts (T - T 0 ), gives rise to the approximation o =tfoCTo)- a(T - T0) where a > 0 is the local variation of surface tension with temperature. Again, for a small temperature contrast, there will be a linear tempera ture profile between the plates due to thermal conduction. If a spot having higher temperature (lower surface tension) appears on the surface of a liq uid layer, it will spread under the action of a surface traction force Fa as depicted in Fig. B.14. Once the surface layer begins to move, viscosity will induce motion in the liquid bulk and, by continuity, uplift the heated fluid from below. If the warmer liquid at the bottom does not cool down quickly enough, an overturning instability occurs in the liquid layer. As in Rayleigh-Benard convection, the time for a fluid particle to rise from the lower to the upper plate is r ~ d/W and the cooling time of liquid elements is TK . Now, however, the driving mechanism is the surface traction force F„ ~ (Aer)Z, ~ a(AT)'L wrought by the temperature dependence of the surface tension coefficient. Here (AT)' is the local temperature contrast between the local hot spot and the surrounding fluid surface, and L is the perimeter of the expanding spot in Fig. B.14 which, by the equation of continuity, also characterizes the diameter of rising parcels replacing the fluid spreading across the liquid surface. Equating Fa to the viscous
Thermal
convection
299
drag Fs ~ pvLW on the rising fluid element gives, after elimination of the vertical velocity W, the estimated rise time r ~ pvd/a(AT)'. Then changing from local scales L and (AT)' to global scales d and AT analogous to our discussion for Rayleigh-Benard convection, the criterion r < TK for overturning fluid motion yields Ma=
a
A
T
d
i > 1.
pvK
This dimensionless parameter, called the Marangoni number, may also be written as the product of two time scale ratios, viz.
where r„ and TK are as previously defined, and ra = (pd3/aATy/2 is the surface tension time scale obtained by solving the equation F„ — ma ~ md/T%. Writing the Marangoni number in this form shows that overturning fluid motion will occur when the effective heat and momentum diffusion time scale TJ — yJrvTK is larger than the surface tension time scale T„. By calculating the ratio
Ma _ (n\
_
a
one can determine when the destabilizing effect of surface tension gradients dominates over that due to buoyancy. In the low gravity limit (g —> 0) as may be achieved in experiments performed on an orbiting spacecraft, or in the thin layer limit (d —> 0), destabilization by surface tension gradients dominates over that due to gravity. Note that the ratio Ma/Ra contains neither viscosity nor thermal diffusivity. It is also apparent that, for any given fluid, one can pass continuously from one type of thermoconvection to another by simply changing the depth of the liquid layer. Surface tension gradient convection, sometimes referred to as thermocapillary convection or Benard-Marangoni convection, was realized in Benard's experiments: the spermaceti layer had a rather small depth (~ 1 mm). The observed weak dependence of the ratio X/d on layer depth was caused by weak buoyancy effects. Although surface tension was mentioned in subse quent discussions of Benard's experiments, the thermogravitational mecha nism proposed by Rayleigh — the temperature dependence of liquid density — was for a long time regarded to be the cause of instability leading to the
300
Key Experiments in Pattern Formation
formation of convection cells. The horizontal scale of the instability pattern in Rayleigh's theory was very close to that observed in experiments. In a spatially extended system for which there is only one controllable length scale — the liquid layer thickness d — the planform wavelength of insta bility modes is 0(d). Calculations for constant temperature upper (freesurface) and lower (rigid) boundaries yield A = 2.68 d. Of course, this value did not coincide exactly with data from Benard's experiment, but the im perfect experimental boundary conditions for the temperature was used to explain the discrepancy. It was argued that the finite thermal conductivity of the boundaries increases the horizontal instability wavelength predicted by linear theory, which then gave a better correlation between theory and experiment. Only much later did experiments conducted by (Block, 1956) reveal that, in rather thin liquid layers where the Rayleigh number is small and insufficient for the onset of instability, convective motion is caused by the temperature dependence of surface tension. After Block's definitive exper iments, the mechanism for the onset of convection was studied in great detail; primary attention was given to the types of the structures formed when the destabilizing effects of gravity and surface tension compete. A linear analysis shows that modes having arbitrary orientation of wavenumber vectors amplify exponentially at the onset of instability. When nonlinear effects come into play, however, the mode of instability depends on the physical properties of the liquid in which a convective flow is realized. For a Boussinesq fluid wherein only density is temperature dependent, an interaction of the amplified modes gives rise to roll structures and to struc tures consisting of two mutually orthogonal roll systems (Le Gal et a/., 1985), or to roll systems making an arbitrary angle with each other. If the Boussinesq approximation does not hold, for example if the viscosity of the liquid strongly depends on temperature, then mode coupling may cause phase synchronization of three modes in which the angle between neighbor ing wave vectors is 120°, and a hexagonal structure is formed (Koschmieder, 1992; Eckert et ai, 1998). The behavior of rolls appearing at the onset of thermal convection was studied in the laboratory by (Clever and Busse, 1979). To investigate the time evolution of rolls in an experiment, a controlled perturbation was in troduced: the liquid was heated periodically in space so that a regular distribution of convective rolls could be initiated at a given instant of time. They observed that the roll structure breaks down as a result of a develop-
Thermal
convection
301
ing varicose instability — a superposition of Eckhaus and zigzag instabili ties. These types of instabilities have been investigated theoretically using model equations such as those presented in Chapter 4. Transitions between hexagonal and tetragonal structures, as supercriticality is increased, were observed only recently in experiments at Pr = O(10 3 ) (Eckert et a/., 1998). It should be emphasized that perfect patterns — rolls, squares, or hexag onal cells — are rather seldom formed in a casual experiment unless they are artificially initiated as in the experiment of (Clever and Busse, 1979) discussed above. More typical is the formation of patterns with broken order. Topological defects play a particular role in the breaking of order. The simplest defects are linear defects or edge dislocations (c/. Fig. 7.1a). Perfect patterns may be stable to defect excitation. This was illustrated by (Whitehead, 1983) who investigated the artificial generation of topological defects in a square cell by local heating of a region smaller than the size of an instability cell. He showed that the local heating power must exceed a critical value for defects to be formed. If the power is below critical, heating can only exert reversible changes in the convective structure, and spatial periodicity is recovered when the heating ceases. The interaction of edge dislocations emerging out of controlled ini tial conditions in a hexagonal lattice was studied by (Afenchenko et a/., 1996). This experiment revealed that edge dislocations belonging to differ ent modes, having opposite topological charges, attract each other to form a penta-hepta defect. This defect appears when intability is initiated from uncontrolled noise. A penta-hepta defect in a hexagonal lattice represents, evidently, the most stable breaking of order. In patterns having a domain structure, penta-hepta defects tend to arrange themselves at boundaries separating regions of perfect structures. The statistics of penta-hepta de fects in hexagonal structures was studied by (Perez-Garcia et a/., 1988). It was found that, as the supercriticality e = (Ra — Rac)/(Rac) increases from zero to five, the number of defects — cells having five or seven contiguous neighbors — grows by 40% as e~Bl€ where B is a constant, analogous to the activation energy for defect density in real crystals and for which e plays the role of temperature. While patterns possessing translational symmetry (rolls, square lattices, or hexagons) are common at large Prandtl numbers, spirals are generally observed in experiments at small Prandtl number. Spiral structures at the onset of convection in carbon dioxide (Pr ~ 1), stable in a finite domain of the governing parameters, were reported by (Hu et a/., 1993). In this
302
Key Experiments in Pattern Formation
Fig. B.15 One can find edge dislocations of opposite topological charges in roll patterns on zebra skins. Photograph taken at the San Diego Zoo in California, USA.
situation, the dynamics of interacting spirals may be an example of spatiotemporal chaos which is observed at increasing supercriticality.
B.3 B.3.1
Diffusive chemical reactions Turing
patterns
Nearly fifty years ago the British mathematician Alan Turing wrote a paper (Turing, 1952) in which he suggested that chemical reactions with appro priate nonlinear kinetics coupled to diffusion could lead to the formation of stationary patterns of the type encountered in living organisms. Stripes on a zebra, hexagons on a giraffe, and spots on a panther, for instance, could be the result of the development of a Turing instability. A similar concentration instability could be involved in the differentiation between various cell types in the early stage of embryonic development (Koch and Meinhardt, 1982; Segel, 1984; Murray, 1989). The Turing patterns emerge spontaneously from a uniform background without any specific external in terference. They distinguish themselves by the active role that diffusion coefficients of activator Du and inhibitor Dv play in the destabilization of spatial homogeneity in chemical reactions. We must emphasize that Turing patterns are patterns formed by dif-
Diffusive chemical
reactions
303
fusion even when the spatially homogeneous steady state is stable in the absence of diffusive effects. As pointed out in Chapter 2, an important restriction on the diffusion constants of the activator and inhibitor in any system that produces Turing patterns is that the inhibitor must diffuse more rapidly than the activator, a condition often described as local ac tivation and lateral inhibition; see Chapter 14. The intuitive explanation of the Turing instability is the following. The local activation is essential for the amplification of small local inhomogeneities (Koch and Meinhardt, 1994): a substance u is said to be autocatalytic or self-enhancing if a small increase of u over its homogeneous steady-state concentration induces a further increase in u. If u begins to increase at a given position, its positive feedback would lead to an overall activation. Thus the self-enhancement of u has to be complemented by the action of a fast-diffusing antagonist, called the inhibitor. Two types of reactions are typical: (i) an inhibitory substance v is produced by the activator that, in turn, slows down the acti vator production; or (ii) a catalyzer is consumed during autocatalysis and its depletion slows down the self-enhancing process. The first laboratory observation of (non-oscillatory) Turing structures was made in 1989 by (Castets et ai, 1990). Using the chlorite-iodinemalonic acid, the so-called CIMA reaction, they discovered the formation of stationary stripes and hexagons with characteristic wavelengths of the order of 0.2 mm. A subsequent study by (Ouyang et ai, 1991) showed similar patterns and also led to the first experimental determination of the bifurcation diagram for the Turing instability. In many cases the patterns remained stable for twenty hours or more. More recently, several other observations of Turing structures in a reactor consisting of a thick layer of gel have been reported; a detailed description of these experiments is given in the excellent book by (Epstein and Pojman, 1998). It is not an easy task to realize all conditions of the Turing instability in experiments: spatial homogeneity, nonoscillatory instability, and a large ra tio of the diffusion constants of the activator relative to the inhibitor. This latter requirement is the most difficult to realize under laboratory condi tions; in an aqueous solution, all small molecules diffuse at nearly the same rate since their diffusion constants vary only slightly with their molecular mass. The most successful reactor to date for observing Turing patterns is the CIMA disc gel reactor schematically illustrated in Fig. B.16. CIMA is a redox reaction characterized by positive feedback due to autocatalysis, on one hand, and inhibition on the other.
304
Key Experiments in Pattern Formation VIDEO CAMERA
GEL
>x
RESERVOIR A
T
RESERVOIR B
z.
UQHT
Fig. B.16 Schematic representation of the gel reactor used by (Castets et al., 1990) and (Ouyang et al., 1991) for observation of T\iring patterns.
(Castets et al., 1990) used starch as a color indicator that switches be tween yellow and deep blue with changes in the iodine concentration. The key point here is that starch is able to interact with iodine (the activator) and make a complex compound which is practically immobile in the gel, thus fulfilling the requirements for Turing pattern formation. Model equa tions describing the CIMA reaction can be written in the dimensionless form (Lengyel and Epstein, 1992; Lengyel, et al., 1993) du
Auv 2 u — 1 + u2 + V u
~di dv = a
K-IT?)
+ cV 2 u
(B.l)
These equations are reminiscent of the Brusselator equations discussed in Chapter 2 — only the nonlinearity is different. Here u represents the con centration of iodide ions, v is the concentration of chlorite ions, a and b are parameters respectively related to the feed concentrations and the experi mentally determined rate constants, c is the diffusion rate ratio of chlorite to that of uncomplexed iodide, and a is a rescaling parameter which pri marily depends on the concentration of the complexing agent (starch). It is clear from Eqs. (B.l) that the effective ratio of the diffusion coefficients is ac. The bifurcation diagram for the Lengyel-Epstein model obtained from a
Diffusive chemical
reactions
305
30
20
b 10
0 -. 20
.
. 40
, 60
,
, 80
a Fig. B.17 Stability boundaries in (b,a) parameter space at several values of a in the model described by Eqs. (B.l) in the range of chemically accessible parameters; c = 1.5. The region of diffusion-induced instability indicated by the cross-hatching. Taken from (Lengyel and Epstein, 1992).
Fig. B.18 Turing structures generated in a batch reactor: (a) mixed spots, targets, and stripes of wavelength 0.25 mm; (b) network-like structures of average wavelength 0.3 mm. Taken from (Lengyel et al., 1993).
linear stability analysis of Eqs. (B.l) for c = 1.5 is given in Fig. B.17. The 'Turing space' lies in the dashed region 8 < cr < 10. This model describes all basic Turing patterns which have been observed in the experiments of (Lengyel et al, 1993); see Figs. B.18 and B.19. Recently it has been shown that the chlorine dioxide-iodine-malonic acid reaction is photosensitive to visible light and spatially uniform illumi nation of Turing patterns at large intensities eliminates them (Munuzuri et al, 1999). Patterns in this system are sensitive not only to illumina tion intensity, but also to the frequency of periodic illumination. (Horvath et al, 1999) observed the fastest suppression of patterns at a frequency
Key Experiments in Pattern Formation
306
Fig. B.19 Comparison of (a) experimental and (b) calculated Turing patterns (left) and corresponding Fourier spectra (right) using the model description given by Eqs. (B.l). Taken from (Lengyel et at., 1993).
of illumination equal to the frequency of autonomous oscillations in the corresponding well-stirred reaction system. B.3.2
Oscillating
chemical
reactions
It is our understanding that oscillating inhomogeneous reactions were first observed by (Bray, 1921). He studied the decomposition of hydrogen per oxide into water and oxygen, with iodine as a catalyst. Unfortunately, his results encountered general mistrust: it was supposed that they contra dicted the second law of thermodynamics. Some thirty years later, thanks to the works of Onsager and Prigogine, it became apparent that oscillating chemical reactions have nothing to do with the second law of thermody namics simply because nonequilibrium processes are involved here. (L. On sager was awarded the Nobel Prize in chemistry in 1968 and I. Prigogine in 1977 for his development of the foundation of the thermodynamics of irreversible processes.) It is well known that the principle of continuously increasing entropy that follows from the second law does not hold far from thermodynamic equilibrium. The proof is the birth from initial disorder of various regular spatio-temporal structures about which we now have a good understanding. Nevertheless, after publication of the works by Onsager and Prigogine, the story repeated itself. B. P. Belousov observed an oscillatory chem ical reaction during which bromate ions oxidized brommalon acid. The reaction was catalyzed by complex ions of ferrum. Again the scientific community regarded the results to be strange and they were published in an obscure specialized document entitled "Collected Theses on Radiation Medicine" (Belousov, 1959). Only thanks to (Zhabotinsky, 1967, 1974) did
Diffusive chemical
reactions
307
this chemical reaction become a paradigm for the birth of various dissipative spatio-temporal structures like targets, spirals, chemical turbulence, etc. The principal feature of the modern version of the Belousov-Zhabotinsky reaction is the presence of bromate ions in the acid solution that oxidizes an organic substance. In a two-dimensional reactor filled with a solution having equal initial concentrations of reagents, the developing concentration instability leads to a spontaneous breaking of symmetry. This results in the formation of patterns possessing cylindrical symmetry around individual points, i.e. the translational symmetry is broken as may be seen in Fig. 9.2. If one wants to observe spirals and not targets, one only needs to shake the reactor or otherwise disrupt the cylindrical wave front. The result will be the formation of topological dislocations and spiral waves will be born as in Fig. 9.1.
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Index
action potential, 210 activator, 302 aggregation, 174 anisotropic, 102 antiphase regularization, 221 Arnold diffusion, 264 Arnold tongues, 270 attractor, 241 strange, 241, 262 autocatalytic chemical reaction, 302
boundary layer, Brusselator, 17 bursts, 210, 221 Cantor set, 76, 195, 270, center, 242 chaos, dynamical, 261, 271 in conservative systems, 263 in dissipative systems, 261, 264 regularization of, 218, 220 routes to, 273 spatio-temporal, 202, 273, 277 spiral, 11, 149, 192 target, 147 chaotic scattering, 118 chaotic set, 261, 262 characteristic equation, 243 chemical gel reactor, 19, 303 chemotaxis, 173 chirality, 186 competition, modal, 24, 63, 65, 81, 83, 231, 282, of populations, 255 spatio-temporal, 39 spiral, 178 structural, 28, 34 'climbing', 93 coarse grain, 214
Belousov-Zhabotinsky reaction, 133, 306 bifurcation, 250 Andronov-Hopf, 216, 252, 256, 260 period-doubling, 260 pitchfork, 251, 253 point, 241 Ruelle-Takens scenario, 273 saddle-node, 252, 258 tangential, 258 Turing, 15 Turing-Hopf, 25 bion, 175 bistability, 107 black film, 154 bound states, dynamical, 118 of structures, 113 of topological defects, 93,105, 284 320
Index coherent structures, 107, 202, 205 convection, Benard-Marangoni, 99, 297 Raleigh-B&iard, 19, 292 correlational integral, 198 crystals, hard, 80 soft, 82 turbulent, 82 defects, 88, 92 in a hexagonal lattice, 96 dendrite-type structures, 184, 185, 236 deterministic system, 240 devil's staircase, 269 dimension, correlational, 198 fractal, 194, 262 Lyapunov, 204 of spatial series, 194 director field, 93, 191 disclination, 87 dislocation, 87, 96 bound state of, 93 disorder, 11 dynamical, 12, 193 evolutionary, 201 finite-dimensional, 12, 196 'frozen', 58 nondeterministic, 193 spiral, 135, 192 dispersion, in the Faraday experiment, 26 for waves on deep water, 25 domain wall, 9, 87, 89 Duffing equation. 265 dynamical system, 239 autonomous, 240 conservative, 241, 261 dissipative, 241, 263 finite-dimensional, 240 infinite-dimensional, 241 nonautonomous, 240
321
spatial, 193 entrainment band, 266 entropy, Kolmogorov-Sinai, 196, 198, 262 topological, 262 Faraday experiment, 25, 77, 102, 143, 279 flexural waves, 160 Floquet multipliers, 245 focus, 242 fractals, Sierpinski gasket, 195 Vicsek snow-flake, 195 free energy functional, 30 gel reactor, 304 geometrical complexity, 262 Ginzburg-Landau equation, complex, 45, 133 discrete, 50 vector, 51 conservative, 46 dissipative, 46 'gliding', 93 gradient system, 30, 37 granular material, 111, 287 temperature, 289 Gross-Pitaevsky equation, 46 hallucination patterns, 228 spirals, 228, 230 tunnels and funnels, 230, 233 white light, 228 heteroclinic, 247 hexagons, 66 hole solution, 53, 54, 135 homoclinic, structure, 247 trajectory, 245 ideal brain, 236
322
incommensurate frequencies, 240, 249 information component, 225 inhibitor, 302 inhibitory connection, 221 instability, Benard-Marangoni, 297, 299 Benjamin-Feir, 53 cortical, 232 Eckhaus, 43 explosive, 108, 239 Kiippers-Lortz, 40 long-wavelength, 57 parametric, 25, 279 Rayleigh-Benard, 19, 292, 295 Rayleigh-Taylor, 21 Turing, 15 zig-zag, 43 Kolmogorov-Petrovskii-PiskunovFisher equation, 174, 181 Kuramoto-Sivashinsky equation, 56 lateral excitation, 230 inhibition, 230, 303 Lengyel-Epstein model, 304 life cycle, 174 limit cycle, 248 limiting capacity, 194 localized structures, 107, 118 Lyapunov exponent, 196, 261 functional, 31 manifold, stable, 244 unstable, 249 Marangoni number Ma, 160, 166, 299 Maxwell-Bloch equations, 50 mean flow, 144, 146, 297 media, bistable, 107 isotropic, 45, 64
Index randomly inhomogeneous, 210 two-component, 17 Navier-Stokes equations, 23, 162 neuronal models, FitzHugh-Nagumo, 48, 137 Hindmarsh-Rose, 209 neutral stability, 19, 21 Newell-Whitehead-Segel equation, 35,93 node, 242 nonlinear damping, 37 Nozaki-Bekki solution, 53 Nusselt number Nu, 275 order, long-range, 12, 182 parameter, 11, 23, 25, 29, 35, 64, 89, 92, 144, 190, 192, 221 complex, 50, 51, 93 short-range, 12 oscillon, in fluids, 286 in granular material, 111, 290 parametric excitation, 25, 279 particle-like structures, 111, 118 patterns, chaos-suppressed, 214 in chemical reactions, 131 in colonies of microorganisms, 173 in convection, 295 in granular material, 287 in Nature, 173, 225, 302 in neural-like systems, 48, 136, 209, 214 in soap films, 151 in the Faraday experiment, 279 on a checkerboard, 220 Penrose tiling, 75 penta-hepta defect, 96 perfect structure, 63 period-doubling, 260 phase,
Index
equation, 42, 55 space, 239 synchronization patterns, 213 trajectory, 239 volume, 239 Poincar^ map, 246 return time, 250 Prandtl number Pr, 296 quasicrystals, 4, 63 quasiperiodic motion, 82, 249, 273 quasi-steady vortex patterns, 152, 170 motion, 151, 154, 162 Raleigh number Ra, 20, 99, 276, 295 reaction-diffusion system, 16 resonance, biharmonic, 268 in the Faraday experiment, 25, 145 subharmonic, 282 synchronous, 282 nonlinear, 265 Reynolds number Re, 162, 272 rolls, 66 rotation number, 269 rotational invariance, 134 saddle, 242, 244 saddle-node, 252, 258 Schrodinger equation, 46 self-organization of microorganisms, 175 separatrix, 246 singularity, of amplitude, 35 of order parameter, 11 of phase, 35, 47, 97, 133 topological, 87, 88 vortex, 47 spikes, 210 spirals, 47, 54, 129 active, 131, 133
323
in the Faraday experiment, 143, 286 in chemical reaction, 129, 307 in colonies of microorganisms, 174 in convection, 145, 297, 301 in neural-like media, 136 passive, 131, 142 square pattern, 70 steady-streaming, 144, 161 Stokes drag, 295 structural stability, 241 Stuart-Landau equation, 45 superlattice, 73 Swift-Hoenberg equation, 29, 107 generalized, 35, 81, 147 symmetry breaking, 15, 253, 286 synchronization, 266 chaotic, 208, 270 clusters of, 210, 213 global, 210 external, 266 mutual, 266 systems, closed, 272 open, 275 Taylor-Couette flow, 275 target, 87, 130, 174 topological, charge, 93 defect, 88, 92 turbulence, 271 chemical, 307 defect-mediated, 202 phase, 57 spiral, 58 transition to, 273 Turing patterns, 19, 302 van der Pol equation, 248 van der Pol-Duffing equation, 265 vortex, Abrikosov, 109 in the Faraday experiment, 144
324
in microorganisms 186, in vibrating soap films 151, 152, 170 pairs, 154, 169 quadrupole, 152 singularity, 47 walker model, 186 Williamowski-Rossler model, 211
Index