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An adaptive hybrid function projective synchronization (AHFPS) scheme between different fractional-order chaotic systems with uncertain system parameter is addressed in this paper. In this proposed scheme, the drive and response system could be synchronized up to a vector function factor. This proposed scheme is different with the function projective synchronization (FPS) scheme, in which the drive and response system could be synchronized up to a scaling function factor. The adaptive controller and the parameter update law are gained. Two examples are presented to demonstrate the effectiveness of the proposed scheme.

In nonlinear science, chaos synchronization is a hot topic, which has attracted much attention from scientists and engineers in the last few years [

At present, the FPS mentioned so far involved mainly the integer-order chaotic systems, and the parameters are exactly known in advance. But in many practical situations, many fractional-order systems yet exhibit chaotic behavior. The parameters of these fractional-order systems in social science and biological science cannot be known entirely. To the best of our knowledge, there are few results about the FPS for fractional-order chaotic systems with uncertain system parameter and there are few results about the FPS for a vector function factor. Motivated by the above discussion, an adaptive hybrid function projective synchronization (AHFPS) scheme between different fractional-order chaotic systems with uncertain system parameters is investigated in this paper. The drive and response system could be synchronized up to a vector function factor in this proposed scheme. This technique is applied to achieve the AHFPS between different fractional-order Lorenz systems with one uncertain system parameter, and the AHFPS between the fractional-order Lorenz system with one uncertain system parameter and the fractional-order Chen system. The numerical simulations demonstrate the validity and feasibility of the proposed method.

The organization of this paper is as follows. In Section

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 described as follows, respectively:

If parameter

For the drive system (

If

Based on the idea of tracking control, in order to achieve the goal of

In the next, we will discuss how to choose a controller

First, the “true” value of the “unknown” parameter

According to the controller (

In generally, we can get

Second, we define vector function

From (

Finally, let the parameter update law be

By (

According to the above,

If real matrix

Assume that

Multiplying the above equation left by

By a similar argument, we also can obtain

According to (

So, we can obtain

According to the stability theory of fractional-order systems [

Therefore,

This

In this section, to illustrate the effectiveness of the proposed synchronization scheme, the AHFPS between different fractional-order Lorenz systems with one uncertain system parameter and the AHFPS between the fractional-order Lorenz system with one uncertain system parameter and the fractional-order Chen system are considered and the numerical simulations are performed.

First, we introduce the numerical solution of fractional differential equations in [

The error of this approximation is described as follows:

The famous Lorenz system [

Chaotic attractors of the fractional-order Lorenz system (

If fractional-order

According to the above, we can get the controlled response system (

According to the above, we can obtain

Now, the parameter update laws and real matrix

Therefore,

Choosing real symmetric positive semidefinite matrix

So, the AHFPS between fractional-order Lorenz system (

Time evolution of the AHFPS error.

Chen and Ueta introduced another chaotic system, called Chen chaotic system, which is similar but not topologically equivalent to the Lorenz system. Chen chaotic system [

Chaotic attractors of the fractional-order Chen system (

Now, let the fractional-order Chen system (

If we choose real symmetric positive definite matrix

So, the AHFPS between the fractional-order Chen system (

Time evolution of the AHFPS error.

Chaotic attractors of the fractional-order Lorenz system (

According to the numerical results in Figures

In this paper, an adaptive hybrid function projective synchronization (AHFPS) scheme between different fractional-order chaotic systems with uncertain system parameter is addressed. The drive and response system could be synchronized up to a vector function factor in this proposed scheme. This is different with the function projective synchronization (FPS) scheme, in which the drive and response system could be synchronized up to a scaling function factor. Based on the stability theory of fractional-order system, an adaptive controller and the parameter update law are obtained. The AHFPS between different fractional-order Lorenz chaotic system with uncertain system parameter and the AHFPS between the fractional-order Lorenz chaotic systems with uncertain system parameter and the fractional-order Chen chaotic system are discussed. The numerical simulations demonstrate the validity and feasibility of the proposed scheme.

The paper is supported jointly by Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ110525, National Natural Science Foundation of China under Grant 61004042, and Natural Science Foundation Project of CQ CSTC 2009BB2417.