Elsevier

Physics Reports

Volume 366, Issues 1–2, August 2002, Pages 1-101
Physics Reports

The synchronization of chaotic systems

https://doi.org/10.1016/S0370-1573(02)00137-0Get rights and content

Abstract

Synchronization of chaos refers to a process wherein two (or many) chaotic systems (either equivalent or nonequivalent) adjust a given property of their motion to a common behavior due to a coupling or to a forcing (periodical or noisy). We review major ideas involved in the field of synchronization of chaotic systems, and present in detail several types of synchronization features: complete synchronization, lag synchronization, generalized synchronization, phase and imperfect phase synchronization. We also discuss problems connected with characterizing synchronized states in extended pattern forming systems. Finally, we point out the relevance of chaos synchronization, especially in physiology, nonlinear optics and fluid dynamics, and give a review of relevant experimental applications of these ideas and techniques.

Introduction

The origin of the word synchronization is a greek root ὺγχρóνoς which means “to share the common time”). The original meaning of synchronization has been maintained up to now in the colloquial use of this word, as agreement or correlation in time of different processes [1].

Historically, the analysis of synchronization phenomena in the evolution of dynamical systems has been a subject of active investigation since the earlier days of physics. It started in the 17th century with the finding of Huygens that two very weakly coupled pendulum clocks (hanging at the same beam) become synchronized in phase [2]. Other early found examples are the synchronized lightning of fireflies, or the peculiarities of adjacent organ pipes which can almost reduce one another to silence or speak in absolute unison. For an exhaustive overview of the classic examples of synchronization of periodic systems we address the reader to Ref. [3].

Recently, the search for synchronization has moved to chaotic systems. In this latter framework, the appearance of collective (synchronized) dynamics is, in general, not trivial. Indeed, a dynamical system is called chaotic whenever its evolution sensitively depends on the initial conditions. The above said implies that two trajectories emerging from two different closeby initial conditions separate exponentially in the course of the time. As a result, chaotic systems intrinsically defy synchronization, because even two identical systems starting from slightly different initial conditions would evolve in time in an unsynchronized manner (the differences in the systems’ states would grow exponentially). This is a relevant practical problem, insofar as experimental initial conditions are never known perfectly. The setting of some collective (synchronized) behavior in coupled chaotic systems has therefore a great importance and interest.

The subject of the present report is to summarize the recent discoveries involving the study of synchronization in coupled chaotic systems. As we will see, not always the word synchronization will be taken as having the same colloquial meaning, and we will need to specify what synchrony means in all particular contexts in which we will describe its emergence.

As a preliminary definition, we will refer to synchronization of chaos as a process wherein two (or many) chaotic systems (either equivalent or nonequivalent) adjust a given property of their motion to a common behavior, due to coupling or forcing. This ranges from complete agreement of trajectories to locking of phases.

The first thing to be highlighted is that there is a great difference in the process leading to synchronized states, depending upon the particular coupling configuration. Namely, one should distinguish two main cases: unidirectional coupling and bidirectional coupling. In the former case, a global system is formed by two subsystems, that realize a drive–response (or master–slave) configuration. This implies that one subsystem evolves freely and drives the evolution of the other. As a result, the response system is slaved to follow the dynamics (or a proper function of the dynamics) of the drive system, which, instead, purely acts as an external but chaotic forcing for the response system. In such a case external synchronization is produced. Typical examples are communication with chaos. A very different situation is the one described by a bidirectional coupling. Here both subsystems are coupled with each other, and the coupling factor induces an adjustment of the rhythms onto a common synchronized manifold, thus inducing a mutual synchronization behavior. This situation typically occurs in physiology, e.g. between cardiac and respiratory systems or between interacting neurons or in nonlinear optics, e.g. coupled laser systems with feedback. These two processes are very different not only from a philosophical point of view: up to now no way has been discovered to reduce one process to another, or to link formally the two cases. Therefore, along this report, we will summarize the major results in both situations, trying to emphasize the different dynamical mechanisms which rule the emergence of synchronized features.

In the context of coupled chaotic elements, many different synchronization states have been studied in the past 10 years, namely complete or identical synchronization (CS) [4], [5], [6], phase (PS) [7], [8] and lag (LS) synchronization [9], generalized synchronization (GS) [10], [11], intermittent lag synchronization (ILS) [9], [12], imperfect phase synchronization (IPS) [13], and almost synchronization (AS) [14]. All these phenomena will be referred to in this report, along with the most relevant examples in which their occurrence was found.

CS was the first discovered and is the simplest form of synchronization in chaotic systems. It consists in a perfect hooking of the chaotic trajectories of two systems which is achieved by means of a coupling signal, in such a way that they remain in step with each other in the course of the time. This mechanism was first shown to occur when two identical chaotic systems are coupled unidirectionally, provided that the conditional Lyapunov exponents of the subsystem to be synchronized are all negative [6].

GS goes further in using completely different systems and associating the output of one system to a given function of the output of the other system [10], [11].

Coupled nonidentical oscillatory or rotatory systems can reach an intermediate regime (PS), wherein a locking of the phases is produced, while correlation in the amplitudes remain weak [7]. The transition to PS for two coupled oscillators has been firstly characterized with reference to the Rössler system [7].

LS is a step between PS and CS. It implies the asymptotic boundedness of the difference between the output of one system at time t and the output of the other shifted in time of a lag time τlag [9]. This implies that the two outputs lock their phases and amplitudes, but with the presence of a time lag [9].

ILS implies that the two systems are most of the time verifying LS, but intermittent bursts of local nonsynchronous behavior may occur [9], [12] in correspondence with the passage of the system trajectory in particular attractor regions wherein the local Lyapunov exponent along a globally contracting direction is positive [9], [12].

Analogously, IPS is a situation where phase slips occur within a PS regime [13].

Finally, AS results in the asymptotic boundedness of the difference between a subset of the variables of one system and the corresponding subset of variables of the other system [14].

The first scenario of transition among different types of synchronization was described for symmetrically coupled nonidentical systems and consisted in successive transitions between PS, LS and a regime similar to CS when increasing the strength of the coupling [9].

The natural continuation of these pioneering works was to investigate synchronization phenomena in spatially extended or infinite dimensional systems [15], [16], [17], [18], [19], [20], to test synchronization in experiments or natural systems [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], and to study the mechanisms leading to desynchronization [33], [34]. These topics will be treated specifically in different sections of this report.

The present report is organized as follows.

In Section 2 we describe the complete synchronization phenomenon, both for low and for high-dimensional situations, and illustrate possible applications of the introduced techniques in the field of communicating with chaos.

In Section 3 we move from identical to nonidentical systems, and summarize the concepts of phase synchronization, lag synchronization, imperfect phase synchronization, and generalized synchronization. We also describe a general transition scenario between a hierarchy of different types of synchronization for chaotic oscillators.

In Section 4 we further extend to the case of structurally different systems. Here, collective dynamics may emerge in the case of a coupling between systems which are confined onto chaotic attractors with different structural properties. Analogies and differences with structurally equivalent systems are pointed out.

In Section 5 we discuss the situation of uncoupled systems subjected to a common external source, and we summarize the main results related to noise-induced synchronization. Furthermore, we analyze the case of weakly coupled systems, where synchronization is enhanced by the action of external noise.

Section 6 is devoted to the discussion of synchronization between space-extended systems. We first describe the situation of a large ensemble of coupled chaotic elements, and then move to the case of continuous space-extended systems, i.e. systems extended in space whose evolution is ruled by partial differential equations.

Finally, in Section 7 we summarize the main synchronization features observed so far in laboratory experiments and in natural phenomena, with a particular attention to the data analysis tools which are nowadays used to detect epochs of synchronization in practical situations.

Section snippets

Complete synchronization

As said in the Section 1, chaotic systems are dynamical systems that defy synchronization, due to their essential feature of displaying high sensitivity to initial conditions. As a result, two identical chaotic systems starting at nearly the same initial points in phase space develop onto trajectories which become uncorrelated in the course of the time. Nevertheless, it has been shown that it is possible to synchronize these kinds of systems, to make them evolving on the same chaotic trajectory

Synchronization in nonidentical low-dimensional systems

In the last section, it has been shown that when identical chaotic systems are coupled properly with strong-enough coupling strength, they can achieve complete synchronization by following the same chaotic trajectory. Synchronization in this case is associated with the transition of the largest transverse Lyapunov exponents of the synchronization manifold from positive to negative values.

However, experimental and even more real systems are often not fully identical, especially there are

Synchronization of structurally nonequivalent systems

All what is said above describes the situation of coupled structurally equivalent systems, i.e. either equivalent systems or systems where the nonidentity resulted in a rather small parameter mismatch. Only recently the study of synchronization phenomena for large parameter mismatches has been addressed [130], by description of the appearance of synchronized collective motion in systems with more than 50% mismatch in parameters.

However, in nature one cannot expect to cope with coupled

Noise-induced synchronization of chaotic systems

In the first four sections we have presented a review of synchronization behavior of two coupled chaotic systems. In this section, we consider the situation that two chaotic systems are not coupled directly or only weakly coupled, but subject to a common fluctuating driving signal which is assumed to be noise in many contexts.

Counterintuitive effects of noise in nonlinear systems have been a subject of great interest in the context of stochastic resonance [142], [143], [144], doubly stochastic

Cluster synchronization in ensembles of coupled identical systems

Synchronization effects in large populations of coupled chaotic dynamical units has also become a subject of active investigations [15], [191], [192].

When passing to space extended systems, a first approach consists in connecting a set of concentrated chaotic systems by means of a given coupling (local or global) between the individuals constituting the set. Space–time chaos synchronization for this kind of systems has been studied for populations of coupled dynamical systems [108], for systems

Experimental synchronization of chaos

In the last sections, we have shown that when nonlinear dynamical systems are coupled or influenced by a common driving signal, synchronization can be established among the systems. The type and the degree of synchronization depend on the coupling strength as well as on the structures of the systems. When the subsystems are nearly identical and the coupling is strong enough, complete synchronization may be observed between the states x(t) and y(t) of two coupled system, or with an appreciable

Acknowledgements

The authors are grateful to V. Anishchenko, F.T. Arecchi, E. Barreto, V. Belykh, J. Bragard, S. Bressler, R. Brown, M.J. Bünner, T. Carroll, H. Chaté, A. Farini, A. Giaquinta, B. Gluckman, C. Grebogi, H. Kantz, L. Kocarev, K. Josić, Y.-C. Lai, A. Lemaitre, I. Leyva, R. Livi, H. Mancini, D. Maza, A.S. Mikhailov, E. Ott, U. Parlitz, L.M. Pecora, A. Pikovsky, A. Politi, I. Procaccia, M. Rabinovich, M.G. Rosenblum, R. Roy, N. Rulkov, S. de San Roman, T. Sauer, S. Schiff, V. Shalfeev, K. Showalter,

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