Elsevier

Chaos, Solitons & Fractals

Volume 9, Issue 10, 1 October 1998, Pages 1703-1707
Chaos, Solitons & Fractals

Generalized Synchronization in Chaotic Systems

https://doi.org/10.1016/S0960-0779(97)00149-5Get rights and content

Abstract

A general approach for constructing a response system to implement generalized chaos synchronization with drive systems is proposed. Possible applications of generalized chaos synchronization to communication are discussed, i.e., that encoding communication realized through generalized synchronization is more reliable than the usual synchronization methods.

Introduction

Synchronization in chaotic systems has been an extensively investigated topic since 1990 1, 2, 3, 4 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and many possible applications, especially in communications have been discussed 14, 15, 16. Recently, a generalization of synchronization in chaotic (drive–response) systems was proposed, where two systems are said to synchronize if a (static) functional relation exists between the states of both systems. This kind of synchronization was called generalized synchronization (GS). Some numerical schemes are proposed for detecting the presence of the functional relation between the states of the coupled systems. A more general theory for GS was developed in [17], where a theorem was given on the necessary and sufficient conditions of the presence of functional relation between the states of drive and response systems (therein called unidirectionally coupled systems). Nevertheless, even if we know of the existence of functional relation between the states of two given systems, it is still a hard problem for us to know how they are related to each other. For some certain, practical purposes, it is of significance to know how the state of one system varies with the state of the other system.

In this paper, we deal with GS from another (especially control theory) standpoint of view. The problem is formulated as follows. Consider a system which is taken as drive systemẋ=f(x)ẏ=g[y,u(x)],y=H(x).limt→∞∥y(t,y0)−yH(t,x0)∥=0g[H(x),u(x)]=DH(x)f(x),Δẏ=Dyg[H(x(t,x0)),u(x(t,x0))]Δy,Δẏ=ẏ(t,y0)−Ḣ(x(t,x0))=g(y(t,y0))−DH(x(t,x0))ẋ(t,x0)=g[H(x(t,x0))+Δy,u(x(t,x0))]−DH(x(t,x0))f(x(t,x0))=g[H(x(t,x0)),u(x(t,x0))]+Dyg[H(x(t,x0)),u(x(t,x0))]Δy−DH(x(t,x0))f(x(t,x0)+o(∥y∥2)=Dyg[H(x(t,x0)),u(x(t,x0))]Δy+o(∥y∥2).A basic technique for proving asymptotical stability is the well-known Lyapunovs direct method, which may be difficult to use in proving stability, but is useful in constructing asymptotically stable systems.

Another technique is to compute the conditional Lyapunov exponents [1] of system (2) with respect to y = H(x(t,x0)), which is based on the fact that, if all conditional Lyapunov exponents of (2) with respect to y = H(x(t,x0)) are negative, then y = H(x(t,x0)) is asymptotically stable.

Section snippets

An approach for constructing response systems

For a given system ẋ=f(x) and functional relation y = H(x):RnRn ( for simplicity, we assume this relation is diffeomorphic), we want to construct a new response system ẏ=g(u(x),y) such that GS occurs between x(t) and y(t), i.e., limt→∞|y(t)−H(x(t))|=0 where u(t) is the driving function.

To this end, let g(u(x),y) be of the formg(u(x),y)=g1(y)+g2[u(x)−u(H−1(y)),y]Now if GS occurs between x(t) and y(t), then we have, by virtue of (5),g1(y)≡DH[H−1(y)]f[H−1(y)].Take y = H(x)+Δy, we get from (4)

Speculation on applications to communication

Applications of drive-response type synchronization to communication has been discussed extensively in the literature. The usual approach for synchronization is to construct a response system, which is a copy of the drive system or a copy of a subsystem of the drive system, so that the copy (sub)system can synchronize with the original one. Nevertheless, it is not always the case that a copy of the drive system, or copy of the subsystem of it, can be synchronized with the original system. On

Summary

The central point of this paper is to show how to construct a response system to H-synchronize with the drive system for a given functional relation H. Furthermore, our arguments indicate that GS is more general and practical and, in many cases, can be realized by a scalar driving signal, which is a scalar function of the variables of the drive system, as the hyperchaotic Rossler systems shows. This observation is of significance in the applications of GS to communications.

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