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This paper considers the complete synchronization problem for

Since the pioneering work of Pecora and Carroll [

Nowadays, the synchronization of multiple chaotic systems has attracted increasing attention. It has been widely used in secure communication area in order to reduce the synchronizing cost of multiple chaotic communication and make simultaneous multiparty communications possible. Therefore, the synchronization in multiple chaotic systems has more advantages and deserves to be deeply investigated comparing with the conventional chaotic synchronization. Several types of synchronization in an array of chaotic systems have been investigated in the past few years, for example, the global synchronization in [

However, it is notable that the realization of synchronization of

The paper is organized as follows. In Section

Consider

Now the above simple coupling form is applied to investigate the synchronization of

Let the state error be

Our purpose is to design the controllers

Here a direct design control method [

First, we choose the control input

Consider the systems (

Choose a Lyapunov function to be

The derivative of

Since

From Lyapunov stability theory, we know that the equilibrium

The error dynamic system (

There are many possible choices for

The antisymmetric structures in Theorem

Since the antisymmetric structure is related to the coefficient matrices and the states of the original system, the selecting of the coefficient matrices with antisymmetric structure is an important and difficult task. In the next section, we will demonstrate the proposed approaches for the special structure through numerical examples.

In this section, we use two simulation examples to illustrate the effectiveness of the proposed schemes. The synchronization is simulated for the non-identical and identical chaotic systems, respectively.

When the drive system and response systems are identical chaotic systems, the drive system and response systems are all the Lorenz chaotic system. They are described as follows:

Let the synchronization error state be

We design the controllers

The error systems (

If the conditions

Fourth order Runge-Kutta integration method is used to numerical simulation with time step size 0.001. Let the initial conditions of the drive system and the response systems be

Dynamics of the variables

The state trajectories

The state trajectories

The state trajectories

When the drive system and response systems are non-identical chaotic systems, the Chen system, Lü system, and Lorenz system are considered as drive system and response systems, respectively. They are described as follows:

Let the synchronization error state be

The controllers

The error systems (

From Theorem

Similar to Case

Dynamics of the variables

The state trajectories

The state trajectories

The state trajectories

This paper concerns the synchronization of

The author’s work was supported in part by the Applied Mathematics Enhancement Program (AMEP) of Linyi University and the National Natural Science Foundation of China, under Grants 61273012 and 61273218, by a project supported by the Scientific Research Fund of Zhejiang Provincial Education Department under Grant Y201223649, by a Project of Shandong Province Higher Educational Science and Technology Program under Grants J13LI11 and J12LI58.