Generalized synchronization of chaotic systems by pure error dynamics and elaborate Lyapunov function

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Abstract

The generalized synchronization is studied by applying pure error dynamics and elaborate Lyapunov function in this paper. Generalized synchronization can be obtained by pure error dynamics without auxiliary numerical simulation, instead of current mixed error dynamics in which master state variables and slave state variables are presented. The elaborate Lyapunov function is applied rather than the current plain square sum Lyapunov function, deeply weakening the power of Lyapunov direct method. The scheme is successfully applied to both autonomous and nonautonomous double Mathieu systems with numerical simulations.

Introduction

Chaos synchronization has been applied in secure communication [1], [2], biological systems [3], [4], and many other fields [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. One of the intricate types of chaos synchronization is generalized synchronization, which has been extensively investigated recently [26], [27], [28], [29], [30], [31], [32], [33]. The generalized synchronization is studied by applying pure error dynamics and elaborate Lyapunov function in this paper.

The pure error dynamics can be analyzed theoretically without auxiliary numerical simulation, whereas the aid of additional numerical simulation is unavoidable for current mixed error dynamics in which master state variables and slave state variables are presented, while their maximum values must be determined by simulation [34], [35], [36], [37], [38]. Besides, the elaborate Lyapunov function is applied rather than current plain square sum Lyapunov function, V(e)=12eTe, which is currently used for convenience. However, the Lyapunov function can be chosen in a variety of forms for different systems. Restricting Lyapunov function to square sum makes a long river brook-like, and greatly weakens the power of Lyapunov direct method.

Based on the Lyapunov direct method [39], generalized synchronization is achieved and a systematic method of designing Lyapunov function is proposed. The technique is successfully applied to both autonomous and nonautonomous double Mathieu systems. This paper is organized as follows. In Section 2, the method of designing Lyapunov function is presented, and generalized synchronization is obtained. Section 3 contains the examples of autonomous and nonautonomous double Mathieu systems, and numerical simulations show the feasibility of the proposed method. Finally, the conclusions are drawn.

Section snippets

Design of Lyapunov function

Consider the master and slave nonlinear dynamic systems described by ẋ=f(t,x)ẏ=f(t,y)+u(t,x,y) where x,yRn are master and slave state vectors, f:R+×RnRn is a nonlinear vector function, and u:R+×Rn×RnRn is controller vector.

Generalized synchronization means that there is a functional relation y=g(t,x) between master and slave states as time goes to infinity, where g:R+×RnRn is a continuously differentiable vector function. Define e=yg(t,x) as the generalized synchronization error vector,

Generalized synchronization of double Mathieu systems

In this section, the functional relation between master and slave states is yi=gi(t,xi)=α(t)xi+β(t)(i=1,2,,n). To demonstrate the use of the proposed method, two examples of autonomous and nonautonomous double Mathieu systems are presented.

Conclusions

The generalized synchronization is studied by applying pure error dynamics and elaborate Lyapunov function in this paper. By classification of the forms of V̇(t,e), the complexity of designing suitable Lyapunov function is reduced greatly. The proposed method is effectively applied to both autonomous and nonautonomous double Mathieu systems.

Acknowledgment

This research was supported by the National Science Council, Republic of China, under Grant Number NSC 94-2212-E-009-013.

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