Glassy dynamics in arrays of chaotic oscillators

Normand Mousseau∗, S. I. Simdyankin∗, Marc-André Brière∗ and E. R.Hunt†

∗Département de physique et GCM, Université de Montréal, C.P. 6128, succ. centre-ville,Montréal (Québec) Canada H3C 3J7

†Department of Physics and Astronomy, Ohio University, OH, USA 45701

Abstract. Arrays of coupled chaotic elements have been used as models for studying a widerange of phenomena such as synchronization and pattern formation in biological and physicalsystems. We present experimental and numerical results showing that these arrays can also help usunderstand the origin of the stretched exponential dynamics observed in glasses and other complexsystems. Stretched exponential behavior has been measured over many decades in a 1D array ofcoupled diode-resonators, just above a crisis-induced intermittency transition. Similar results areobtained numerically in an array of identical chaotic oscillators, confirming the chaotic origin ofthis universal behavior. In these systems, we find that the fundamental physical quantity associatedwith stretched exponentials is not the auto-correlation function but, rather, the distribution of timesspent in dynamical traps. Here, we review these results and discuss their relation with other systems.We will also present results obtained on higher dimensional networks.

1. INTRODUCTION

Coupled chaotic systems show a great richness of dynamical behaviors and, as such,can serve as toy-model to study many physical phenomena. This was the case forsynchronization and pattern formation, for example. Numerous researchers have alsofound that arrays of coupled non-linear elements could behave in a way reminiscent ofthat of glasses [1, 2, 3], with very slow dynamics and freezing in disordered states. Thisis an important discovery because the problem of the glass transition is one of the oldestproblems in condensed-matter physics and remains a topic of high interest.

Fragile glasses are characterized by an increase of many orders of magnitude inviscosity as the temperature drops by a few degrees. Near this region, the dynamicsof relaxation of various quantities can be fitted by a stretched exponential,

C(t) = C0e−(t/τ0)β

, (1)

with 0 < β < 1. Glasses also undergo aging and many properties depend on the historyof preparation, and on the waiting time before a measure is taken or an external strainapplied. It is not clear, at this point, how these two properties are related.

In the last few years, much experimental, numerical and theoretical efforts have beentargeted at trying to identify a length-scale associated with the dynamical slowing downand characterize the dynamical heterogeneities found in glasses.

We discuss here some recent results obtained in the course of a detailed study of theglassy dynamics in an array of coupled diode resonators. The experimental results can

FIGURE 1. Schematic diagram of a single diode-resonator and its coupling to its nearest neighbors ina one-dimensional arrangement.

be reproduced using coupled logistic maps, suggesting that these results are universal.In the next section, we describe the experimental set-up and the model used to

reproduce some of the results numerically. We then present and discuss the resultsobtained both experimentally and numerically.

2. METHODOLOGY

2.1. Experimental set-up

We consider a one-dimensional chain of 256 coupled diode-resonators connected asa ring. A schematic diagram showing its basic elements is presented in Figure 1. Thediode resonator circuit is simply a series combination of a 30-mHy inductor and arectifier type pn-junction diode (1N1004) driven by a sinusoidal source at 100 kHz.This circuit has been studied extensively over the last 20 years and is known to followthe period-doubling route to chaos as the ac drive is increased [4, 5, 6]. These elementsare coupled diffusively through resistors, as shown in the diagram, and period-boundaryconditions are used. In spite of a careful selection, the drive voltage for a given feature,such as the period-3 fixed points, varies by up to 20% between basic elements. Thisone-dimensional setup has been previously studied with unidirectional coupling [7] andunder the conditions of stochastic resonance [8, 9, 10].

The overall phase diagram of this system, as the drive voltage is increased, representsa kink-forming route to spatio-temporal chaos [11]. At low drive voltage, all elements

follow the period-doubling route to chaos in a nearly perfect synchronous manner, withno or little current in the coupling resistors. As the drive is increased, the lattice reachesa two-band chaotic orbit with all elements, again, being in the same band at the sametime. Further increasing the voltage leads to the formation of spatial structures as coarse-grained period-2 phase kinks (two-band structures with elements alternating bands at agiven time) start to appear [7]. It is interesting to note that each time a new kink isformed, the orbit stabilizes itself. At it turns out, there is appreciable power dissipated inthe coupling resistors at the kink position, reducing the available phase space. Increasingthe driving, the number of kinks increases until all elements are on a kink, i.e., withneighbors in the opposite band. Further increasing the drive voltage, the system goesthrough a high-dimensional version of the well-studied attractor-merging crisis [12]. Wedefine the critical drive voltage Vc as the point where the two attractors just touch. Justabove Vc, the system oscillates between full spatio-temporal chaos and coarse-grainedperiodicity, being trapped for long periods in the latter state. In the case of a singlechaotic element, this crisis-induced intermittency was shown to display exponentiallydistributed switching times [12]. As discussed here, we also find a form of crisis-inducedintermittency in coupled systems but the trap time distribution is now dominated bystretched exponentials.

2.2. Coupled logistic maps

The experimental results can be reproduced qualitatively using an array of coupledlogistic maps as introduced by Kaneko [13] and Johnson et al. [7] a few years ago:

xi(tn+1) = (1−α) fi(tn)+α

2

[

fi−1(tn)+ fi+1(tn)]

(2)

where fi(tn) = rxi(tn)[

1− xi(tn)]

and α is the coupling between oscillators. Althoughthis map is sufficient for the problem discussed here, it does not describe the fulldynamical behavior of the experimental system, which includes some memory effectwhich is best reproduced by a two-dimensional map [6].

Simulation are typically run on lattices with periodic boundary condition varying insize between 100 and 10000 sites. Initial conditions are set at random and the first millionstep discarded to avoid transient effects. Simulation length vary between 107 to 1010 timesteps.

Results below are generally presented for α = 0.25 and r = 3.83 or 3.8888, two setsof parameters showing a glassy behavior.

2.3. Analysis

Figure 2 shows a 4096-cycle time series for the 256 elements. The two phases of thetwo-band attractor are represented as black and white, and only every other time stepis displayed. Note the creation, annihilation and diffusion of the boundaries of the two-band attractor. With the coupling resistors used, 150 kOhms, the coarse-grained spatialperiodicity of the lattice is about 4 sites.

FIGURE 2. Time series of the 256 diodes in the intermittent state, just above Vc. Since the systemdisplays a fundamental period-two oscillation in time below Vc, we show the state of the diodes at eventime-steps in a binary representation.

We are interested in the statistical distribution of trap time in a period-two cycle.This quantity is formally equivalent to the distribution of time intervals between zerocrossings of renewal processes such as random walks [14] and has the advantage that itcan be measured experimentally and numerically to a high degree of accuracy for thissystem [15].

Generally, however, experimentally measurable relaxation response correspond toauto-correlation functions which are given by

C(t) = 〈σ(t ′)σ(t ′+ t)〉t ′. (3)

A general framework describing the direct relation between this quantity and the distri-bution of trap time was recently introduced by Luck and Godrèche [14]. Applying thisframework to our problem, we can show that a stretched exponential trap time distribu-tion implies also a stretched exponential decay in the auto-correlation function, albeitwith a different exponent [16].

There is therefore a one to one correspondence between the trap time distributionpresented here and the more standard auto-correlation function.

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0 1 2 3 4 50.05

0.1

0.2

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1Global

t0.25

C(t

)FIGURE 3. Experimental trap-time distribution for a single site measured over 1010 cycles. Inset: Auto-correlation function measure over 4096 cycles and averaged over 256 sites (global average).

3. RESULTS

3.1. Stretched exponential dynamics

As already mentioned, just above what would be called crisis-induced intermittancyin a single chaotic oscillator, the trap-time distribution presents an unusual stretchedexponential dynamics, reminiscent of glasses. However, while for most experimentalmeasures of stretched exponential relaxation in these systems, the results span fewenough decades that they can be fitted by a number of distributions, experimental resultsfor the coupled system presented here, taken over a full day at a rate of 10 kHz, shows awell-fitted curve over 6 decades in the distribution.

For completeness, we also plot the auto-correlation function, which also follows astretched exponential form. Due to the way we measure the auto-correlation function,we can only get a fraction of the statistics obtained for the trap distribution. Even withless statistics, however, the auto-correlation function follows a stretched exponentialover about a decade, in agreement with the theoretical analysis.

Because the set-up is disordered, with diode-resonators differing by at most 20 % intheir critical voltage, after a careful selection, each site displays a different dynamics.Although all sites show a stretched exponential trap-time distribution, the stretchingexponent can vary considerably. For example, Fig. 4 show the trap-time distribution forthree sites on a 256-diode lattice at the same driving with β varying by more than afactor of two.

Similar effects could be produced numerically when site disorder is introduced in thevariable r.

The stretching exponent β varies continuously with the external drive voltage. Fromabout 0.35 to close to 1.0 as the system is close to fully chaotic. Closely following thisbehavior, the time factor, τ0, also decreases, resulting in significantly longer traps as the

0 50 100 150 2001

10

100

1000

10000

100000

t

P(t

) Fast

Slow

1 10 100

0.5

1

2

5

10

20

t

-log(

P(t

))FIGURE 4. Trap-time distribution at 3 different sites on a 256-diode lattice. The main figure shows alog-normal plot of the distribution. In inset, we show a log-log vs. log plot for the three sites. The curvesare translated vertically to ease viewing. The corresponding β are 0.8, 0.4 and 0.3, respectively.

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2.9 3 3.1 3.20

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Drive

τ 0 (1

00 D

rive

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es)

FIGURE 5. β and τ as a function of the drive voltage for the 256-diode array. These values are takenfor a generic site, with an average activity.

voltage goes down. This is similar to what is seen numerically, for the model above,with 0.33 < β < 0.70. In this case, the longer traps at the low β are more than 250 timeslonger than for β = 0.70, reaching time scale of more than 30,000 time steps.

By contrast, the increase in the length scale of the exponentially-decreasing spatialcorrelation function goes up from 2 to 8 sites. Moreover, as can be seen in Fig. 2, itappears that there is little correlation between the width of a trap and its duration. Aparallel can be drawn between these surprising features and the absence of any diverginglength scale in glasses as the relaxation times increase rapidly, near the glass-transitiontemperature.

Act

ivity

0.1

0.2 Experiment

0 100 2000.04

0.06

0.08

0.1

0.12 Simulation

Site position

FIGURE 6. Top: Activity as a function of site in a 256-site experimental setup. Bottom: Activity in thecoupled map lattice as a function of the coupling α and the driving parameter r.

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avel

engt

h

FIGURE 7. Spatial wavelength as a function of the coupling strength for a 256-element network.Results presented in this paper are for a coupling,Rc, of 10−5 Ohms−1

The absence of long-range spatial correlations is also revealed in Fig. 6 which plots thenumber of times a site has flipped from one phase to another normalized by the numberof time steps in the simulation. A low activity indicates a site that remains trapped forlong time. Although the experiment is taken on a relatively short period of time, 2048time steps sampled at a frequency 10 times that of the basic clock tick, it is clear thatspatial correlation is minimal.

While spatial correlation increases only slightly with decreasing driving, the spatialperiodicity remains unchanged. As it turns out, this quantity depends only on the cou-pling strength and not on the driving. Figure 7 shows that the spatial wavelength followslinearly the coupling strength. The stretched exponential behavior is mostly constrained

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N=1000a=0.25

(d)

x

r=3.83 r=3.8888r=4

FIGURE 8. Natural invariant density for the external perturbation(left-hand panels) and an isolated sitesubmitted to this external perturbation (right-hand panels), in a numerical simulation, as described by Eq.(4). Panels (a) and (b) show the variation of the density, ρ(a) and ρ(x) as a function of the coupling αwhile panels (c) and (d) show the variation of the density for various driving, r.

to a spatial periodicity between 4 and 5 sites.

3.2. Origin of the stretched exponential

In order to better understand the origin of the stretched exponential, we turn to thenumerical model. The coupled-map model introduced above reproduces closely thefeatures seen experimentally. This demonstrates that the stretched exponential behavioris a generic property of coupled chaotic systems and not caused by frozen disorder [15].

In order to investigate the origin of the stretched exponential behavior in this system,we rewrite Eq. (2) as an external perturbation, at on an isolated site:

x(t +1) = (1−α) f [x(t)]+α a(t). (4)

Within this picture, it is convenient to relax the coarse-grained picture and to introducethe natural invariant density, ρ(x). This density is defined such that ρ(x)dx gives thefraction of the time the orbit spends in the interval dx around x. A similar quantity canbe defined for the external perturbation, ρ(a).

For the uncoupled map in the fully chaotic, the natural invariant ρ(x) has the well-known “U” shape [12]. ρ(a), representing the distribution for

[

f (xi−1)+ f (xi+1)]

isalready very Gaussian-like and peaked at 0.5.

These densities are modified significantly with the coupling. For the single site naturalinvariant, ρ(x), the peaks at 0 and 1 have moved inside. The competition between the

3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

r

x

FIGURE 9. Bifurcation diagrams for a single site on the coupled map lattice with N = 16 (smallerdots) and for the map x(t + 1) = 0.75 f [x(t)]+ 0.1625 (larger dots). The perturbation in the second mapcorresponds to the contribution of the neighbors in a spatial period-four regime [the peak at 0.67 inFig. 8(d)].

period oscillations and the chaotic behavior are clearly seen there, by a non-vanishingdensity between the peaks at x = 0.45 and x = 0.85. The central peak for ρ(a) isshifted to the right and a shoulder appears near 1. The central peak arises when the twoneighbors of site i are in opposite band, i.e., when the lattice is in a spatial period-four.The effect of this perturbation on site i in Eq. (4) is, in effect, to shift the bifurcation mapfrom a chaotic to a temporal period-two regime, setting the site into a trap (see Fig. 9).

While Eq. (4) can help understand the origin of the traps in these coupled systems, italso underlines the need for local spatial organization. For example, selecting the per-turbation a(t) from the distribution ρ(a) presented in Fig. 8 generates only an exponen-tial trap-time distribution. If we introduce a bias in the distribution, by favoring with astretched exponential probability, a value of a(t) corresponding to the peak in ρ(a), thenthe map also shows stretched exponential distributions. This demonstrates that once astretched exponential is present in the dynamics, it will force it on the rest of the lattice.

This is confirmed by constructing a(t) by adding the state of two sites selected atrandom on a large lattice and using this perturbation on an isolated map. The trap-distribution on this map also displays a stretched-exponential trap distribution, with astretched exponent β larger than that of the lattice.

Since there is no direct communication between the two sites selected at random (thespatial correlation is very short range — less than 10 sites), it is even sufficient thateach neighbor be locked into a stretched-exponential trap-distribution independently toimpact similarly the central site.

00.2

0.40.6

0.81 3.7

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3.8

3.85

3.9

3.95

4

0

0.1

0.2

0.3

0.4

0.5

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activ

ity

00.2

0.40.6

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0.1

0.2

0.3

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activ

ity

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0.2

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3.9

3.95

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ra

activ

ity

FIGURE 10. Activity in the coupled map lattice as a function of the coupling α and the drivingparameter r for the 1D, 2D and 3D numerical models.

3.3. Higher-dimensional lattices

In view of the existence of 3D configurational glasses, it is interesting to assesswhether the stretched-exponential trap-time distribution persists in higher dimension.In view of the discussion of the previous section, one could expect that the stretchedexponential behavior disappear rapidly with increasing dimensionality. How fast exactlyhas to be established experimentally.

Fig. 10 shows the phase diagram of the coupled map, in terms of the level of activityas function of the driving r and the coupling parameter α . While the overall level ofactivity, save for very small couplings, decreases significantly with the dimensionalityof the lattice, the qualitative structure of phase diagram remains similar.

In all dimension, there is a significant frozen region above α ' 0.1, where the latticelocks into a spatial and temporal period two. At very high coupling, the activity increasesagain as the whole lattice starts behaving as a single, perfectly in step, oscillator. Theinteresting region of the phase space, for our purposes is therefore in the mediumcoupling, 0.2 < α < 0.4, where stretched exponential dynamics occurs in 1D.

Changes are significant enough, however, that we might wonder whether or notthe close relation between experiment and simulation will survive higher dimensions.While a 3D experimental set-up is currently being put together, simulations indicatethat the parameter window for stretched exponential behavior decreases in size. Thereare, however, some regions of the phase diagram where stretched exponential trap-distribution occur in both 2D and 3D but a full characterization of those has not beencompleted at this point.

4. CONCLUSION

Coupled arrays of chaotic elements display are remarkably rich range of dynamics.We have found that for a wide range of parameters, these coupled lattices behavein ways very reminiscent of glasses, with frustration inducing stretched exponentialdistributions. These results, which are seen both experimentally and numerically appearto be universal. Because the dynamical systems can be much more easy to measureand simulate, they can provide us with a new way to study the origin of the stretchedexponential relaxation in glasses.

Spatial organization appears critical, here, to generate the stretched exponential trap-time distribution. The stability of traps is ensured by the presence of a spatial period-fourregion. Nevertheless, the short range spatial correlations indicate that there is not muchstability gained by extending greatly the period-four region. As β decreases from 0.50 to0.33 and the longer trap increase in time 250-fold, the spatial correlation length goes upfrom 2 to 8. This can be compared with the absence of a diverging length scale observedin glasses as the dynamics slows down to a halt.

Although we have made much progress in understanding this new dynamics in arraysof coupled chaotic elements, there are still a few questions that remain to be answeredregarding the effects of dimension and the links between these systems and configura-tional glasses. We are hopeful that studying further these phases will help us identify the

origin of stretched exponential relaxation, a question which has been with us for morethan 150 years by now.

ACKNOWLEDGMENTS

This work is supported in part by the ONR (EH), NSERC and NATEQ (NM). NM is aCottrell Scholar of the Research Corporation.

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and Bulsara, A. R., Chaos, 8, 604 (1998).9. Löcher, M., Cigna, D., and Hunt, E. R., Phys. Rev. Lett., 80, 5212 (1998).10. Löcher, M., Chatterjee, N., Marchesoni, F., Ditto, W. L., and Hunt, E., Phys. Rev. E, 61, 4954–4961

(2000).11. Gade, P. M., Chatterjee, N., Mousseau, N., and Hunt, E. R., N/A (2002), in preparation.12. Ott, E., Chaos in Dynamical Systems, Cambridge University Press, Cambridge, 1993.13. Kaneko, K., editor, Theory and Applications of Coupled Map Lattices, Wiley, Chichister, 1993.14. Godrèche, C., and Luck, J. M., J. Stat. Phys., 104, 489–524 (2001).15. Hunt, E., Gade, P., and Mousseau, N., cond–mat/0204179 (2002).16. Simdyankin, S., and Mousseau, N., in preparation.