^{1, 2}

^{2}

^{3}

^{1}

^{2}

^{3}

The generalized synchronization problem is studied in this paper for different chaotic systems with the aid of the direct design method. Based on Lyapunov stability theory and matrix theory, some sufficient conditions guaranteeing the stability of a nonlinear system with nonnegative off-diagonal structure are obtained. Then the control scheme is designed from the stable system by the direct design method. Finally, two numerical simulations are provided to verify the effectiveness and feasibility of the proposed method.

As a very important topic in nonlinear science, chaos synchronization has been studied extensively in both theory and applications, such as mathematics, physics, biology, and engineering community. Since the pioneering work of [

Generalized synchronization can be regarded as an extension of complete synchronization, antisynchronization, or projective synchronization and it means that there exists a functional relation between the states of two systems. It is noted that the two systems may have different dimensions. The generalized synchronization problem between two complex networks with nonlinear coupling and time-varying delay was developed in [

Recently, the direct design method [

The rest of this paper is organized as follows. In Section

The following notations are used throughout this paper.

In this section, generalized synchronization between two coupled chaotic systems is introduced. The concept of generalized synchronization was first proposed for unidirectionally coupled systems, that is, systems coupled in a drive response configuration, by Rulkov et al. [

Schematic diagram of a drive-response configuration. Here, the system

Given a vector map

Define the error variable

In fact, system (

Some necessary results [

Matrix

the eigenvalues of

there exists a positive diagonal matrix

For any matrix

there exists a symmetric positive definite matrix

If

In this section, we focus on the stability of a nonlinear system with nonnegative off-diagonal structure. Then a control scheme is proposed by the direct design method.

Consider the following system with state dependent coefficients described by

If the

Construct a Lyapunov function

If the

there exists a vector

Construct a Lyapunov function

Our main purpose is to design a control scheme

If

Substituting (

The results obtained in the above discussion can be applied to complete synchronization [

It is worth noticing that the structure of matrix

The control scheme (

In this section, to illustrate the effectiveness of the proposed method, we discuss two numerical simulations for two cases: (i) for identical systems, generalized projective synchronization between two Chua’s circuits; (ii) for different dimensional systems, generalized synchronization between HyperRössler system and Chen system.

Consider the following Chua’s circuit

Generalized projective synchronization of systems (

Next, the two following controllers are designed from Theorems

At first, design the controller as

Secondly, design the controller

The initial states of drive and response systems (

Antisynchronization errors and state trajectories of systems (

Antisynchronization errors and state trajectories of systems (

Consider the HyperRössler system

Set

Chaotic attractors of HyperRössler system and Chen system with initial states

Chaotic attractor in

Chaotic attractor of Chen system with parameters

Generalized synchronization errors and state trajectories of systems (

This paper has studied generalized synchronization of different chaotic systems. By using Lyapunov stability theory and matrix theory, the stability of a nonlinear system with nonnegative off-diagonal structure is obtained. Then the control scheme can be designed from transforming the error system into this stable system by using the direct design method. It is worth noticing that the structure of this stable system is significantly different from existing results of the direct design method. Numerical simulations are presented for identical systems and nonidentical systems with different dimension, respectively.

This work is supported by the National Natural Science Foundation of China (61075065, 60774045, U1134108), the Ph.D. Programs Foundation of Ministry of Education of China (20110162110041), and the Talent Introduction Scientific Research Foundation of Northwest University for Nationalities (Grant no. xbmuyjrc201304).