The problem of the dual synchronization of two different fractional-order chaotic systems is studied. By a linear controller, we realize the dual synchronization of fractional-order chaotic systems. Finally, the proposed method is applied for dual synchronization of Van der Pol-Willis systems and Van der Pol-Duffing systems. The numerical simulation shows the accuracy of the theory.

In recent years, the topic of chaos synchronization has attracted increasing attention in many fields. The result of synchronization of chaotic oscillators is used in nonlinear oscillators [

Dual synchronization is a special circumstance in synchronization of chaotic oscillators. The first idea of multiplexing chaos using synchronization was investigated in a small map and an electronic circuit model by Tsimring and Sushchik in 1996 in [

Nowadays, there are many dual synchronization methods, such as in 2000 Liu and Davids introduce the dual synchronization of 1-D discrete chaotic systems via specific classes of piecewise-linear maps with conditional linear coupling in [

The rest of this paper is organized as follows: in Section

We define the following two systems as two master systems.

The error signal for dual synchronization is

The main goal is to synchronize the master systems and the slave systems is equivalent to

Considering the fractional-order system

The dual synchronization of fractional-order chaotic systems between the master systems and the slave systems is achieved if and only if the following condition satisfies

We can rewrite (

where

Equation (

In the first example, we can use the proposed method to achieve the dual synchronization of the Van der Pol system and the Willis system.

The

We should choose the appropriate parameters so that all the eigenvalues of the Jacobian matrix of (

Dual synchronization of the Van der Pol system and the Willis system is simulated. The system parameters are set to be

Error signals between the pair of Van der Pol system.

Error signals between the pair of Willis system.

For Example

So the corresponding slave systems are

The

So the corresponding error matrix are as follows:

The eigenvalue equation of the equilibrium point is locally asymptotically stable. Because

According to what we have studied above, parameters are set to

Error signals between the pair of Van der Pol system.

Error signals between the pair of Duffing system.

In this work, we construct a theory frame about dual synchronization of two different fractional-order chaotic systems and propose a method of dual synchronization. In addition, this method is used for designing a synchronization controller to achieve the dual synchronization of two different fractional-order chaotic systems. Finally, the proposed method is applied for dual synchronization of the Van der Pol-Willis systems and the Van der Pol-Duffing systems. The numerical simulations proves the accuracy of the theory.

This work is supported by the Fundamental Research Funds for the Central Universities of China under Grant no. CQDXWL-2012-007.