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This paper investigates the synchronization and antisynchronization problems of a hyperchaotic complex Chen system with unknown parameters based on the properties of a passive system. The essential conditions are derived under which the synchronization or antisynchronization error dynamical system could be equivalent to a passive system and be globally asymptotically stabilized at a zero equilibrium point via smooth state feedback. Corresponding parameter estimation update laws are obtained to estimate the unknown parameters as well. Numerical simulations verify the effectiveness of the theoretical analysis.

Hyperchaos [

In recent years, chaos synchronization has attracted increasing attention among scientists due to its potential applications in the fields of secure communications; optical, chemical, physical, and biological systems; neural networks; and so forth [

Recently, many researchers have begun to give their attention to the concept of passivity of nonlinear systems. The passivity theory is considered to be an alternative tool for analyzing the stability of nonlinear systems. The main idea of passivity theory is that the passive properties of a system can keep the system internally stable. In order to make a system stable, one can design a controller which renders the closed-loop system passive with the help of passivity theory. For the past decade, the passivity theory has played an important role in designing an asymptotically stabilizing controller for control and synchronization of chaotic and hyperchaotic systems with real variables [

This paper is organized as follows. Section

Consider the following nonlinear affine system:

System (

System (

If system (

Suppose the system (

Recently, the authors [

In this section, we study the synchronization problem of the hyperchaotic complex Chen system (

By subtracting the drive system (

Separating the real and imaginary parts of error dynamical system (

Let

which is the following normal form:

Then we can arrive at the following result.

If the passive controllers are designed as

Construct the following storage function:

The zero dynamics of system (

Differentiating

Then

Furthermore, taking the time derivative of

Since the error dynamical system (

Substituting (

Then, taking integration on both sides of (

For

Letting

This completes the proof.

The controllers (

To investigate the antisynchronization of the hyperchaotic complex Chen system, we need to add system (

Let

If the passive controllers are designed as

The proof of Theorem

In this section, we perform two numerical simulations to demonstrate the effectiveness of the above synchronization and antisynchronization schemes. In the following numerical simulations, the fourth-order Runge-Kutta method is used to solve the systems with time step size 0.001. The system parameters are selected as

For the synchronization of the hyperchaotic complex Chen system, we consider the drive system (

The time response of states for the drive system (

The time response of synchronization error states

For the antisynchronization of the hyperchaotic complex Chen system, we also consider the drive system (

The time response of states for the drive system (

The time response of antisynchronization error states

Hyperchaotic systems with real variables have been investigated extensively over the past three decades. But hyperchaotic complex systems have attracted increasing attention due to the fact that they have much wider applications. So we investigate the synchronization and antisynchronization problems of a hyperchaotic complex Chen system by applying the passive control technique. Based on the fact that once a system is passive, there exists a control law that makes the passive system stable, then the passivity-based controller can be proposed to asymptotically stabilize the error dynamical system. Then corresponding passive controllers and update laws of the parameters are proposed to achieve synchronization and antisynchronization between two hyperchaotic complex Chen systems with different initial conditions, respectively. Furthermore, this work can be extended to achieve synchronization and antisynchronization of other versions of the hyperchaotic complex Chen system, even other types of hyperchaotic complex systems, such as Lorenz system [

This work was supported by the Youth Foundation of Yunnan University of Nationalities under Grant no. 11QN07, the Natural Science Foundation of Yunnan Province under Grants no. 2009CD019 and no. 2011FZ172, and the Natural Science Foundation of China under Grants no. 61065008, no. 61005087, and no. 61263042.