Stretched-exponential dynamics in a chain of coupled chaotic oscillators

Abstract

Abstract. -We measure stretched exponential behavior, exp −(t/τ0) β , over many decades in a one-dimensional array of coupled chaotic electronic elements just above a crisis-induced intermittency transition. There is strong spatial heterogeneity and individual sites display a dynamics ranging from near power law (β = 0) to near exponential (β = 1) while the global dynamics, given by a spatial average, remains stretched exponential. These results can be reproduced quantitatively with a one-dimensional coupled-map lattice and thus appear to be system independent. In this model, local stretched exponential dynamics is achieved without frozen disorder and is a fundamental property of the coupled system. The heterogeneity of the experimental system can be reproduced by introducing quenched disorder in the model. This suggests that the stretched exponential dynamics can arise as a purely chaotic phenomenon.Stretched exponential relaxation is almost ubiquitous in complex systems. It is found in glasses [1], spin glasses [2], polymers, high-dimensional cellular automata [3], random networks [4] coupled chaotic oscillators [5,6] and others [7]. For many of these problems, however, the microscopic origin of this dynamics is difficult to establish both experimentally and theoretically. Understanding the underlying causes for stretched exponential relaxation remains, therefore, a central goal in the study of the dynamics of complex systems [8][9][10], as witnessed by the flurry of theoretical, experimental and numerical results that have appeared on the subject in the last few years [10][11][12][13].In the last 20 years, many physical processes have been understood in terms of nonlinear dynamics and chaos, these concepts often going beyond the reach of standard equilibrium statistical physics. Here, we present experimental results showing a stretched exponential dynamics, exp −(t/τ 0 ) β , in a simple one-dimensional chaotic system comprised of a chain of chaotic electronic oscillators. In particular, we find that in the vicinity of the transition to spatiotemporal chaos, individual site dynamics shows this behavior over six orders of magnitude and that the β's associated with individual site range from 0.8 (nearly exponential) to 0.2

Universal persistence exponent in transition to antiferromagnetic order in coupled logistic maps

“…It is possible that having the same persistence exponents points towards a certain "subclass" within a given universality class. We note that such antiferromagnetic states are also obtained in coupled diode resonators [20] for some parameter value. It would be interesting to explore the possibility of obtaining the persistence exponent in this experimental setup.…”

Relationship between dynamical heterogeneities and stretched exponential relaxation

“…Simulations on a related model of diffusively coupled via short range interactions nonlinear maps show that the fit can be extended to more than 9 orders of magnitude, ruling out any other power law or simple combination of pure exponentials [10,22]. The quality of the fit in this system is sufficient to distinguish the region of the parameter space where the dynamics is described by power-law distribution coupled with an exponential cut-off from the really stretched exponential distributions.…”

“…The existence of these inhomogeneities is difficult to detect experimentally, but they readily account for SER, and predict correctly the values of beta in many experiments [1]. Dynamical heterogeneities have been seen experimentally and numerically in glasses and supercooled liquids [5,6], spin-glass models [7,8], LennardJones alloy mixtures with ellipsoidal probes [9] as well as in coupled-chaotic systems such as diode-resonator arrays [10]. In particular, the concept of dynamical heterogeneities has emerged as critical for the description of the microscopic dynamics associated with the dramatic slowing down of the relaxation and diffusion processes as the temperature approaches T g , the glass-transition temperature [11,12,13].…”

“…Recently, Hunt et al found that the distribution of the renewal time of a certain stochastic process measured in a one-dimensional lattice of coupled diode resonators could be fitted to a stretched exponential function over 6 orders of magnitude [10]. Simulations on a related model of diffusively coupled via short range interactions nonlinear maps show that the fit can be extended to more than 9 orders of magnitude, ruling out any other power law or simple combination of pure exponentials [10,22].…”

Glassy dynamics in arrays of chaotic oscillators

“…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 zero crossings of renewal processes such as random walks [14] and has the advantage that it can be measured experimentally and numerically to a high degree of accuracy for this system [15].…”