Synchronization of Spatiotemporal Chaos: The Regime of Coupled Spatiotemporal Intermittency

Abstract

Synchronization of spatiotemporally chaotic extended systems is considered in the context of coupled one-dimensional complex Ginzburg-Landau equations (CGLE). A regime of coupled spatiotemporal intermittency (STI) is identified and described in terms of the space-time synchronized chaotic motion of localized structures. A quantitative measure of synchronization as a function of coupling parameter is given through distribution functions and information measures. The coupled STI regime is shown to disappear into regular dynamics for situations of strong coupling when localized structures become unstable, hence a description in terms of a single CGLE is not appropriate. [S0031-9007(97) [2]. The characterization of low-dimensional chaos is now a mature subject with well established techniques, including techniques of chaos control. In this context, the demonstration that the familiar phenomenon of synchronization of two regular oscillators [3] by a weak coupling can also be displayed by chaotic oscillators is an important new idea. This conceptual development has opened a new avenue of research with interesting practical implications. Chaos in extended systems is a much less mature subject, and many investigations are still at the level of classifying different types of behavior. Concepts and methods of statistical mechanics are commonly invoked in terms of "phase diagrams" and transitions among different "phases" of behavior [4][5][6][7]. Still, the possibility of a synchronized behavior of spatially extended systems in a spatiotemporal disordered phase is an appealing idea that we address in this Letter. More specifically, we will consider an extended one-dimensional system in a chaotic regime known as spatiotemporal intermittency (STI) [5], and we will characterize a coupled STI regime.By synchronization of two chaotic oscillators O 1 and O 2 , it is meant in a strict sense that plotting the time series O 1 ͑t i ͒ vs O 2 ͑t i ͒ one obtains a straight line of unit slope. For many practical applications, synchronization of chaotic oscillations calls for an expanded framework and the concept of "generalized synchronization" has been introduced [8,9] as the appearance of a functional dependence between the two time series. In this context, we understand here by synchronization the situation in which O 1 ͑t i ͒ becomes a given known function of O 2 ͑t i ͒. Transferring these concepts to spatially extended systems, we search for correlations between the space͑x i ͒-time͑t j ͒ series of two variables O 1 ͑x i , t j ͒ and O 2 ͑x i , t j ͒. The synchronization of O 1 and O 2 occurs when these two space-time series become functionally dependent. This idea is different from the one much studied in the context of coupled map models in which the coupling and emerging correlations are among spatially coupled oscillators. Here we search for correlations of two variables at the same space-time point.Our study has been carried out in the context of complex Ginzburg-Landau Equations (CGLE) which give a prototype example of chaoti...

Dynamics of defects in the vector complex Ginzburg–Landau equation

“…In both cases one can approximate |k S | ≈ 2k H . For small k R , the dominant first-order derivative terms in (18) indicate that small perturbations on the waves travel at a group velocity v g = 2(α − β)k H .…”

“…The behavior in the glass and in the gas phase can be interpreted in terms of synchronization and generalized synchronization, respectively, of spatiotemporally chaotic configurations of the two field components [18,14]. In this context, another interesting quantity that gives information about the transition from the glassy to the gas phase is the mutual information between field components, that can be computed from the individual and joint probability densities.…”

“…Since we are interested mostly in the dynamics of topological defects, we consider here just the twodimensional VCGL case, for which point defects are topologically stable. For VCGL dynamics in one spatial dimension we refer the reader to [18,14,19]. Section 2 presents the basic equations and properties.…”

Stochastic Spatiotemporal Intermittency and Noise-Induced Transition to an Absorbing Phase

“…A particular instance of STC is a regime called spatiotemporal intermittency (STI) which is present in a large variety of systems [9][10][11][12]. Generally speaking, this regime is a chaotic spatiotemporal evolution (the turbulent phase) irregularly and continuously interrupted by the spontaneous formation of domains with a wide range of sizes and lifetimes, where the behavior is ordered (laminar).…”

“…Among the available scenarios, few of them consider the framework of stochastic partial differential equations (SPDE) [5][6][7] to describe STC. A successful example is, however, the mapping of the Kuramoto-Shivashinsky equation (describing a STC regime named phase turbulence) to the stochastic model of surface growth known as KardarParisi-Zhang equation [8].A particular instance of STC is a regime called spatiotemporal intermittency (STI) which is present in a large variety of systems [9][10][11][12]. Generally speaking, this regime is a chaotic spatiotemporal evolution (the turbulent phase) irregularly and continuously interrupted by the spontaneous formation of domains with a wide range of sizes and lifetimes, where the behavior is ordered (laminar).…”