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A hybrid projective synchronization scheme for two identical fractional-order chaotic systems is proposed in this paper. Based on the stability theory of fractional-order systems, a controller for the synchronization of two identical fractional-order chaotic systems is designed. This synchronization scheme needs not to absorb all the nonlinear terms of response system. Hybrid projective synchronization for the fractional-order Chen chaotic system and hybrid projective synchronization for the fractional-order hyperchaotic Lu system are used to demonstrate the validity and feasibility of the proposed scheme.

Most recently, many authors begin to investigate the chaotic dynamics and synchronization for fractional-order dynamical systems [

However, most of projective synchronizations for the fractional-order systems have concentrated on studying the same scaling factor [

To illustrate the effectiveness of the proposed HPS scheme in this paper, the HPS for the fractional-order Chen system and HPS for the fractional-order hyperchaotic Lu system are investigated. Numerical simulations are used to demonstrate the effectiveness of the proposed schemes. The organization of this paper is as follows. In Section

There are several definitions of fractional derivatives. In this paper, the Caputo-type fractional derivative defined will be used. The Caputo definition of the fractional derivative, which is sometimes called smooth fractional derivative, is described as

The fractional-order chaotic drive and response systems can be written as follows, respectively:

For the drive system (

If

In order to realize HPS for the fractional-order chaotic system (

Define the HPS errors between the response system (

Let

Now, we assume that the errors vector

Rewrite the controller

Now, Theorem

Choose the following controller:

If all the eigenvalues of

According to the drive system (

According to (

Because all the eigenvalues of

Since

because all the eigenvalues of

According to

In order to use the stability theory of linear fractional-order systems [

For the complex fractional-order multiscroll chaotic systems [

In order to illustrate the effectiveness of the proposed hybrid projective synchronization scheme obtained in Section

The fractional-order Chen system [

Tavazoei and Haeri pointed out that fractional-order Chen system (

Chaotic attractor of fractional-order Chen chaotic system (

According to the HPS scheme presented in the previous section, the response system is described by

Now, the term of

So, we can choose

According to Theorem

For example, choose reversible matrix

So, the eigenvalues of

The HPS result between the drive system (

Min et al. reported a fractional-order hyperchaotic Lu system [

The chaotic attractor of fractional-order hyperchaotic Lu system for

Chaotic attractor of fractional-order hyperchaotic Lu system (

Taking system (

Now, we can choose

According to Theorem

For example, choose reversible matrix

So, the eigenvalues of

The HPS result between the drive system (

We proposed a new synchronization scheme to achieve hybrid projective synchronization for two identical fractional-order chaotic systems in this paper. The drive system and response system could be synchronized up to a vector function factor, and this synchronization scheme needs not to absorb all the nonlinear terms of response system. The synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. Numerical simulations are used to illustrate the effectiveness of the proposed synchronization method.

The authors are very grateful to the reviewers for their valuable comments and suggestions, which have led to the improved presentation of this paper. This work is supported by Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ110525.