#### Abstract

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.

#### 1. Introduction

In nonlinear science, chaos synchronization is a hot topic, which has attracted much attention from scientists and engineers in the last few years [1–13]. Projective synchronization (PS) first reported by Mainieri and Rehacek [14] has been extensively investigated in recent years because it can obtain faster communication. Projective synchronization (PS) is characterized that the drive and response system could be synchronized up to a scaling factor. This proportional feature can be used to extend binary digital to *M*-nary digital communication [14] for getting faster communication. Recently, a new type of projective synchronization method [15–17], called function projective synchronization (FPS), is put forward. The drive and response system could be synchronized up to a scaling function factor in function projective synchronization (FPS). FPS could be used to get more security in application to secure communications, because the unpredictability of the scaling function in FPS can additionally enhance the security of communication.

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 2, the definition of the AHFPS is given and the AHFPS scheme between different fractional-order chaotic systems with uncertain system parameter is presented. In Section 3, two examples are used to verify the effectiveness of the proposed scheme. The conclusion is finally drawn in Section 4.

#### 2. The Fractional Derivatives and AHFPS Scheme

The Caputo definition of the fractional derivative, which is sometimes called smooth fractional derivative, is described as where denotes the Caputo definition of the fractional derivative, is the smallest integer larger than , is the -order derivative in the usual sense, is the gamma function.

The fractional-order chaotic drive and response systems can be described as follows, respectively: where and are fractional orders satisfying , , and may be different with . and are the state vectors of the drive system (2.2) and response system (2.3), respectively. is the system parameter. may be different continuous nonlinear vector functions.

If parameter in system (2.3) is unknown, a parameter update law is designed, and a controller is added to the original system (2.3), we obtain the controlled response system (2.4) with parameter update laws (2.5) where is unknown parameter. Controller is an real matrix to be designed, and is a real scalar function to be designed.

*Definition 2.1. *For the drive system (2.2) and controlled response system (2.4) with parameter update laws (2.5), it is said to be adaptive hybrid function projective synchronization (AHFPS) if there exist a controller and a real scalar function such that
where is the Euclidean norm and is the “true” value of the “unknown” parameter . is an real matrix, and matrix elements are continuous bounded functions. and are called the AHFPS error.

*Remark 2.2. *If and is a constant, then the AHFPS problem will be reduced to adaptive projective synchronization (APS), where is an identity matrix. If and are constant, then the adaptive modified projective synchronization (AMPS) will appear. And if , and is a constant matrix, then the adaptive hybrid projective synchronization (AHPS) will appear. If and are continuous bounded functions, then the adaptive function projective synchronization (AFPS) will appear, that is, the AFPS is also the special case of the proposed scheme.

*Remark 2.3. *Based on the idea of tracking control, in order to achieve the goal of , we can let be a reference signal. Then, AHFPS between fractional-order chaotic system (2.2) and fractional-order chaotic system (2.4) belongs to the problem of tracking control, that is, the output signal in system (2.4) follows the reference signal ultimately.

In the next, we will discuss how to choose a controller and a parameter update laws.

First, the “true” value of the “unknown” parameter is chosen as , and we define a compensation controller for response system (2.4), and choose controller as where is an vector function which will be designed later.

According to the controller (2.8) and the compensation controller (2.7), the response system (2.4) can be depicted as Using , one has

In generally, we can get where is an real matrix and is an real matrix.

Second, we define vector function as where is an real matrix to be designed.

From (2.10), (2.11), and (2.12), we have

Finally, let the parameter update law be where is an real matrix to be designed later. Because the Caputo derivative of a constant is zero, (2.14) can be rewritten as According to (2.13) and (2.15), we have where is an real matrix.

By (2.16), we know that the AHFPS between fractional-order system (2.2) with controlled response fractional-order system (2.4) and the uncertain parameter could be identified, and transformed into the following problem: choose suitable real matrix and real matrix such that system (2.16) is asymptotically convergent to zero.

*Remark 2.4. *According to the above, may be an arbitrary given reference signal, therefore the AHFPS between fractional-order chaotic systems and integer-order chaotic systems belongs to this class of problems if we choose the reference signal as being the output of one of the integer-order chaotic systems.

Theorem 2.5. *If real matrix and in system (2.16) are selected such that
**
where is a real symmetric positive definite matrix, is a real symmetric positive semidefinite matrix, and stands for conjugate transpose of a matrix, then
*

*Proof. *Assume that is one of the eigenvalues of matrix and the corresponding nonzero eigenvector is , that is,

Multiplying the above equation left by , we get

By a similar argument, we also can obtain

According to (2.20) and (2.21), we can obtain
Since and , are real symmetric positive definite matrix and real symmetric positive semidefinite matrix, respectively, then

So, we can obtain

According to the stability theory of fractional-order systems [18], the equilibrium point in (2.16) is asymptotically stable.

Therefore,
The proof is completed.

This ** theorem** indicates that system (2.16) can asymptotically converge to zero. It implies that the AHFPS between drive system (2.2) and controlled response system (2.4) with uncertain parameter will be obtained.

#### 3. Applications

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 [19]. All the numerical simulation of fractional-order system in this paper is based on [19]. Consider the following fractional-order system: with initial condition . Now, set , (). The previous system can be discretized as follows: where and, for ,

The error of this approximation is described as follows:

##### 3.1. The AHFPS between the Fractional-Order Lorenz Chaotic System with Different Fractional Order

The famous Lorenz system [20], the first chaotic attractor model in a 3D autonomous system, is described as follows: where system parameters . I. Grigorenko and E. Grigorenko [20] pointed out that fractional-order Lorenz system (3.6) exhibits chaotic behavior for fractional order . The chaotic attractor for is shown in Figure 1.

If fractional-order in fractional-order Lorenz system is 0.995, we can rewrite the fractional-order Lorenz system as where . Now, let the fractional-order Lorenz system (3.7) be drive system and parameter unknown in fractional-order Lorenz system (3.6). The fractional-order system (3.6) with uncertain parameters is described by

According to the above, we can get the controlled response system (3.9) with uncertain parameter , and parameter update laws (3.10):

According to the above, we can obtain

Now, the parameter update laws and real matrix are chosen as

Therefore, Choosing real symmetric positive definite matrix , we can get

Choosing real symmetric positive semidefinite matrix , we can obtain

So, the AHFPS between fractional-order Lorenz system (3.7) and controlled response system (3.9) with uncertain parameters can be achieved. For example, choose The corresponding numerical result is shown in Figure 2, in which the initial conditions are , , , and and the “true” value of the “unknown” parameter is chosen as , respectively.

**(a)**

**(b)**

**(c)**

**(d)**

##### 3.2. The AHFPS between the Fractional-Order Lorenz Chaotic System and the Fractional Order Chen Chaotic System

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 [21] is given by where . Its corresponding fractional-order system is described as follows, Tavazoei and Haeri [22] pointed out that fractional-order Chen system exhibits chaotic behavior for fractional order . When , the chaotic attractor is shown in Figure 3.

Now, let the fractional-order Chen system (3.18) be drive system and fractional-order Lorenz system (3.6) with unknown parameter as response system. According to the above, we can get the controlled response system with uncertain parameter (3.9), and parameter update laws (3.10). Similar to the above, we can obtain The parameter update laws and real matrix are chosen as

If we choose real symmetric positive definite matrix and real symmetric positive semidefinite matrix , we can get

So, the AHFPS between the fractional-order Chen system (3.18) and controlled response system (3.9) with uncertain parameters can be achieved. For example, choose The corresponding numerical result is shown in Figure 4, in which the initial conditions are , , and , and the “true” value of the “unknown” parameter is chosen as , respectively. The chaotic attractor of the fractional-order Lorenz system (3.6) with for is shown in Figure 5.

**(a)**

**(b)**

**(c)**

**(d)**

According to the numerical results in Figures 2 and 4, we can obtain that the errors are indeed close to zero. This means that the adaptive hybrid function projective synchronization (AHFPS) between different chaotic systems can be achieved finally.

#### 4. Conclusion

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.

#### Acknowledgments

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.