A Novel Hybrid Function Projective Synchronization between Different Fractional-Order Chaotic Systems

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.


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 fractionalorder 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.

The Fractional Derivatives and AHFPS Scheme
The Caputo definition of the fractional derivative, which is sometimes called smooth fractional derivative, is described as where D q denotes the Caputo definition of the fractional derivative, m is the smallest integer larger than q, f m t is the m-order derivative in the usual sense, Γ • is the gamma function.
The fractional-order chaotic drive and response systems can be described as follows, respectively: where q d and q r are fractional orders satisfying 0 < q d < 1, 0 < q r < 1, and q r may be different with q d .x ∈ R n and z ∈ R n are the state vectors of the drive system 2.2 and response system 2.3 , respectively.β is the system parameter.D, R : R n → R n may be different continuous nonlinear vector functions.
If parameter β in system 2.3 is unknown, a parameter update law is designed, and a controller ψ is added to the original system 2.3 , we obtain the controlled response system 2.4 with parameter update laws 2.5 where β is unknown parameter.Controller ψ is an n × 1 real matrix to be designed, and p x, z, β 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 ψ and a real scalar function p x, z, β such that where • is the Euclidean norm and β 0 is the "true" value of the "unknown" parameter β.K x is an n × n real matrix, and matrix elements k ij x i, j 1, 2, . . ., n are continuous bounded functions.e i z i − n j 1 k ij x j and e β β − β 0 are called the AHFPS error.
Remark 2.2.If K x kI and k ∈ R is a constant, then the AHFPS problem will be reduced to adaptive projective synchronization APS , where I is an n × n identity matrix.If K x diag k 1 , k 2 , . . ., k n and k i ∈ R are constant, then the adaptive modified projective synchronization AMPS will appear.And if K x K, and K is a constant matrix, then the adaptive hybrid projective synchronization AHPS will appear.If K x diag k 1 x , k 2 x , . . ., k n x and k i x i 1, 2, . . .n 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 lim t → ∞ e lim t → ∞ z − K x x 0, we can let K x x 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 z in system 2.4 follows the reference signal K x x ultimately.
In the next, we will discuss how to choose a controller ψ and a parameter update laws.First, the "true" value of the "unknown" parameter β is chosen as β 0 , and we define a compensation controller ψ 1 x ∈ R n for response system 2.4 , and choose controller ψ as where ψ 2 is an n × 1 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 e y − K x x, one has

2.10
In generally, we can get where ξ 2 x, z, β 0 is an n × n 1 real matrix to be designed.From 2.10 , 2.11 , and 2.12 , we have

2.13
Finally, let the parameter update law be where τ is an 1 × n 1 real matrix to be designed later.Because the Caputo derivative of a constant is zero, 2.14 can be rewritten as

2.15
According to 2.13 and 2.15 , we have where 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 β could be identified, and transformed into the following problem: choose suitable n × n 1 real matrix ξ 2 x, z, β 0 and 1 × n 1 real matrix τ such that system 2.16 is asymptotically convergent to zero.
Remark 2.4.According to the above, K x x 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 K x x as being the output of one of the integer-order chaotic systems.
Theorem 2.5.If real matrix ξ 2 x, z, β 0 and τ in system 2.16 are selected such that where P is a real symmetric positive definite matrix, Q is a real symmetric positive semidefinite matrix, and H stands for conjugate transpose of a matrix, then Proof.Assume that λ is one of the eigenvalues of matrix ξ 1 x,z,β 0 ξ 2 x,z,β 0 τ and the corresponding nonzero eigenvector is Ψ, that is,

2.19
Multiplying the above equation left by Ψ H P , we get By a similar argument, we also can obtain

2.21
According to 2.20 and 2.21 , we can obtain λ λ

2.23
So, we can obtain

2.24
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 β will be obtained.

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 x 0 , y 0 .Now, set h T/N, t n nh n 0, 1, 2 . . ., N .The previous system can be discretized as follows:

3.4
The error of this approximation is described as follows: 3.5

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 σ, β, γ 10, 28, 8/3 .I. Grigorenko and E. Grigorenko 20 pointed out that fractional-order Lorenz system 3.6 exhibits chaotic behavior for fractional order q r ≥ 0.993.The chaotic attractor for q r 0.998 is shown in Figure 1.

3.6
If fractional-order q in fractional-order Lorenz system is 0.995, we can rewrite the fractional-order Lorenz system as

3.7
where q d 0.995.Now, let the fractional-order Lorenz system 3.7 be drive system and parameter β unknown in fractional-order Lorenz system 3.6 .The fractional-order system 3.6 with uncertain parameters β is described by

3.8
According to the above, we can get the controlled response system 3.9 with uncertain parameter β, and parameter update laws 3.10 :

3.10
According to the above, we can obtain Now, the parameter update laws and real matrix ξ 2 x, z, β 0 are chosen as

3.14
Choosing real symmetric positive semidefinite matrix Q diag 20, 2, 16/3 , 0 , we can obtain So, the AHFPS between fractional-order Lorenz system 3.7 and controlled response system 3.9 with uncertain parameters β can be achieved.For example, choose The corresponding numerical result is shown in Figure 2, in which the initial conditions are x 0 0, 1, 2 T , z 0 5.5, 0, 4 T , β 0 31, and ep e β and the "true" value of the "unknown" parameter is chosen as β 0 28, respectively.

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 a, b, c 35, 3, 28 .Its corresponding fractional-order system is described as follows,

3.18
Tavazoei and Haeri 22 pointed out that fractional-order Chen system exhibits chaotic behavior for fractional order q d ≥ 0.83.When q d 0.95, 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 β as response system.According to the above,  we can get the controlled response system with uncertain parameter β 3.9 , and parameter update laws 3.10 .Similar to the above, we can obtain

3.19
The parameter update laws and real matrix ξ 2 x, z, β 0 are chosen as If we choose real symmetric positive definite matrix P diag 1, 1, 1, 1 and real symmetric positive semidefinite matrix Q diag 20, 2, 16/3, 0 , we can get So, the AHFPS between the fractional-order Chen system 3.18 and controlled response system 3.9 with uncertain parameters β can be achieved.For example, choose The corresponding numerical result is shown in Figure 4, in which the initial conditions are x 0 0, 1, 2 , z 0 5.5, 1, 2 , and β 0 43, and the "true" value of the "unknown" parameter is chosen as β 0 40, respectively.The chaotic attractor of the fractional-order Lorenz system 3.6 with σ, α, β 10, 40, 8/3 for q r 0.998 is shown in Figure 5.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.

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 fractionalorder 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.

3 Figure 1 :
Figure 1:Chaotic attractors of the fractional-order Lorenz system 3.6 for q r 0.998.

Figure 2 :
Figure 2: Time evolution of the AHFPS error.

3 Figure 3 :
Figure 3: Chaotic attractors of the fractional-order Chen system 3.18 for q d 0.95.

Figure 4 :
Figure 4: Time evolution of the AHFPS error.