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

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 as𝐷^𝑞𝑓(𝑡)=1Γ𝑚−𝑞^𝑡₀𝑓^(𝑚)(𝜏)(𝑡−𝜏)^{𝑞+1−𝑚}𝑑𝜏,𝑚−1<𝑞<𝑚, (2.1) 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:𝐷^𝑞𝑑(𝑥),𝐷𝑥=𝐷_𝑞𝑟𝑧=𝑅𝑧,𝛽, (2.2)(2.3) 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)𝐷^𝑞𝑟𝛽𝐷𝑧=𝑅𝑧,+𝜓,_𝑞𝑟𝛽=𝑝𝑥,𝑧,𝛽, (2.4)(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 lim_𝑡→+∞‖𝑒‖=lim_𝑡→+∞‖𝑧−𝐾(𝑥)𝑥‖=0,lim_𝑡→+∞𝑒_𝛽=lim_𝑡→+∞𝛽−𝛽₀=0, (2.6) where ‖⋅‖ is the Euclidean norm and 𝛽₀ 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=1𝑘_𝑖𝑗𝑥_𝑗 and 𝑒_𝛽=𝛽−𝛽₀ 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(𝑘₁,𝑘₂,…,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(𝑘₁(𝑥),𝑘₂(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 𝛽₀, and we define a compensation controller 𝜓₁(𝑥)∈𝑅^𝑛 for response system (2.4), 𝜓₁(𝑥)=𝐷^𝑞𝑟(𝐾(𝑥)𝑥)−𝑅𝐾(𝑥)𝑥,𝛽₀, (2.7) and choose controller 𝜓 as 𝜓=𝜓₁(𝑥)+𝜓₂, (2.8) where 𝜓₂ 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 𝐷^𝑞𝑟𝑧=𝑅𝑧,𝛽+𝐷^𝑞𝑟(𝐾(𝑥)𝑥)−𝑅𝐾(𝑥)𝑥,𝛽₀+𝜓₂. (2.9) Using 𝑒=𝑦−𝐾(𝑥)𝑥, one has 𝐷^𝑞𝑟𝑒=𝑅𝑧,𝛽−𝑅𝐾(𝑥)𝑥,𝛽₀+𝜓₂. (2.10)

In generally, we can get 𝑅𝑧,𝛽−𝑅𝐾(𝑥)𝑥,𝛽₀=𝜉₁𝑥,𝑧,𝛽₀𝑒𝑒_𝛽. (2.11) where 𝜉₁(𝑥,𝑧,𝛽₀) is an 𝑛×(𝑛+1) real matrix and ^𝑒_𝑒𝛽=(𝑒₁,𝑒₂,…,𝑒_𝑛,𝑒_𝛽)^𝑇 is an (𝑛+1)×1 real matrix.

Second, we define vector function 𝜓₂ as 𝜓₂=𝜉₂𝑥,𝑧,𝛽₀𝑒𝑒_𝛽, (2.12) where 𝜉₂(𝑥,𝑧,𝛽₀) is an 𝑛×(𝑛+1) real matrix to be designed.

From (2.10), (2.11), and (2.12), we have 𝐷^𝑞𝑟𝑒=𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝑒𝑒_𝛽. (2.13)

Finally, let the parameter update law be 𝐷^𝑞𝑟𝛽𝑒𝑒𝛽=𝑝𝑥,𝑧,=𝜏_𝛽, (2.14) 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 𝐷^𝑞𝑟𝛽−𝛽₀=𝐷^𝑞𝑟𝑒_𝛽𝑒𝑒=𝜏_𝛽. (2.15) According to (2.13) and (2.15), we have 𝐷^𝑞𝑟𝑒𝐷^𝑞𝑟𝑒_𝛽=𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏𝑒𝑒_𝛽, (2.16) where 𝜉₁(𝑥,𝑧,𝛽₀)+𝜉₂(𝑥,𝑧,𝛽₀)𝜏 is an (𝑛+1)×(𝑛+1) 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 𝛽 could be identified, and transformed into the following problem: choose suitable 𝑛×(𝑛+1) real matrix 𝜉₂(𝑥,𝑧,𝛽₀) 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 𝜉₂(𝑥,𝑧,𝛽₀) and 𝜏 in system (2.16) are selected such that 𝑃𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏+𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏^𝐻𝑃=−𝑄, (2.17) 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 lim_𝑡→+∞‖𝑒‖=lim_𝑡→+∞‖𝑧−𝐾(𝑥)𝑥‖=0,lim_𝑡→+∞𝑒_𝛽=lim_𝑡→+∞𝛽−𝛽₀=0. (2.18)

Proof. Assume that 𝜆 is one of the eigenvalues of matrix𝜉₁(𝑥,𝑧,𝛽₀)+𝜉₂(𝑥,𝑧,𝛽₀)𝜏 and the corresponding nonzero eigenvector is Ψ, that is, 𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏Ψ=𝜆Ψ. (2.19)
Multiplying the above equation left by Ψ^𝐻𝑃, we getΨ^𝐻𝑃𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏Ψ=Ψ^𝐻𝑃(𝜆Ψ). (2.20)
By a similar argument, we also can obtainΨ^𝐻𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏^𝐻𝑃Ψ=𝜆Ψ^𝐻𝑃Ψ. (2.21)
According to (2.20) and (2.21), we can obtain𝜆+Ψ𝜆=^𝐻𝑃^{𝜉1𝑥,𝑧,𝛽0+𝜉2𝑥,𝑧,𝛽0}_𝜏+^{𝜉1𝑥,𝑧,𝛽0+𝜉2𝑥,𝑧,𝛽0}_𝜏^𝐻𝑃ΨΨ^𝐻𝑃Ψ. (2.22) Since 𝑃^{𝜉1(𝑥,𝑧,𝛽0)+𝜉2(𝑥,𝑧,𝛽0)}_𝜏+^{𝜉1(𝑥,𝑧,𝛽0)+𝜉2(𝑥,𝑧,𝛽0)}_𝜏^𝐻𝑃=−𝑄 and 𝑃, 𝑄 are real symmetric positive definite matrix and real symmetric positive semidefinite matrix, respectively, then Ψ^𝐻𝑄Ψ≥0,Ψ^𝐻𝑃Ψ>0,𝜆+Ψ𝜆=−^𝐻𝑄ΨΨ^𝐻𝑃Ψ≤0. (2.23)
So, we can obtain≥𝜋arg𝜆2>𝑞_𝑟𝜋2. (2.24)
According to the stability theory of fractional-order systems [18], the equilibrium point in (2.16) is asymptotically stable.
Therefore, lim_𝑡→+∞‖𝑒‖=lim_𝑡→+∞‖𝑧−𝐾(𝑥)𝑥‖=0,lim_𝑡→+∞𝑒_𝛽=lim_𝑡→+∞𝛽−𝛽₀=0. (2.25) 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.

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:𝑑^𝑞1𝑥𝑑𝑡^𝑞1,𝑑=𝑓𝑥,𝑦_𝑞2𝑦𝑑𝑡^𝑞2=𝑔𝑥,𝑦,0<𝑞₁,𝑞₂<1, (3.1) with initial condition (𝑥₀,𝑦₀). Now, set ℎ=𝑇/𝑁, 𝑡_𝑛=𝑛ℎ (𝑛=0,1,2…,𝑁). The previous system can be discretized as follows:𝑥_𝑛+1=𝑥₀+ℎ^𝑞1Γ𝑞₁𝑓𝑥+2^𝑝𝑛+1,𝑦^𝑝_𝑛+1+^𝑛_𝑗=0𝛼_{1,𝑗,𝑛+1}𝑓𝑥_𝑗,𝑦_𝑗,𝑦_𝑛+1=𝑦₀+ℎ^𝑞2Γ𝑞₂𝑔𝑥+2^𝑝𝑛+1,𝑦^𝑝_𝑛+1+^𝑛_𝑗=0𝛼_{2,𝑗,𝑛+1}𝑔𝑥_𝑗,𝑦_𝑗, (3.2) where𝑥^𝑝_𝑛+1=𝑥₀+1Γ𝑞₁^𝑛_𝑗=0𝑏_{1,𝑗,𝑛+1}𝑓𝑥_𝑗,𝑦_𝑗,𝑦^𝑝_𝑛+1=𝑦₀+1Γ𝑞₂^𝑛_𝑗=0𝑏_{2,𝑗,𝑛+1}𝑔𝑥_𝑗,𝑦_𝑗, (3.3) and, for 𝑖=1,2,𝛼_{𝑖,𝑗,𝑛+1}=𝑛^𝑞𝑖+1−𝑛−𝑞_𝑖(𝑛+1)^𝑞𝑖,𝑗=0,𝑛−𝑗+2^𝑞𝑖+1+𝑛−𝑗^𝑞𝑖+1−2𝑛−𝑗+1^𝑞𝑖+1𝑏,1≤𝑗≤𝑛,1,𝑗=𝑛+1,_{𝑖,𝑗,𝑛+1}=ℎ^𝑞𝑖𝑞_𝑖𝑛−𝑗+1^𝑞𝑖−𝑛−𝑗^𝑞𝑖,0≤𝑗≤𝑛. (3.4)

The error of this approximation is described as follows: 𝑥𝑡_𝑛−𝑥_𝑛=𝑜ℎ^𝑝1,𝑝₁=min2,1+𝑞₁,𝑦𝑡_𝑛−𝑦_𝑛=𝑜ℎ^𝑝2,𝑝₂=min2,1+𝑞₂. (3.5)

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:𝐷^𝑞𝑟𝑧₁=𝜎𝑧₂−𝑧₁,𝐷^𝑞𝑟𝑧₂=𝛽𝑧₁−𝑧₁𝑧₃−𝑧₂,𝐷^𝑞𝑟𝑧₂=𝑧₁𝑧₂−𝛾𝑧₃. (3.6) 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.

Figure 1 

Chaotic attractors of the fractional-order Lorenz system (3.6) for 𝑞𝑟=0.998.

If fractional-order 𝑞 in fractional-order Lorenz system is 0.995, we can rewrite the fractional-order Lorenz system as𝐷^𝑞𝑑𝑥₁=𝜎𝑥₂−𝑥₁,𝐷^𝑞𝑑𝑥₂=𝛽𝑥₁−𝑥₁𝑥₃−𝑥₂,𝐷^𝑞𝑑𝑥₂=𝑥₁𝑥₂−𝛾𝑥₃, (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 by𝐷^𝑞𝑟𝑧₁=10𝑧₂−𝑧₁,𝐷^𝑞𝑟𝑧₂=𝛽𝑧₁−𝑧₁𝑧₃−𝑧₂,𝐷^𝑞𝑟𝑧₃=𝑧₁𝑧₂−8𝑧₃3. (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): 𝐷^𝑞𝑟𝑧₁𝐷^𝑞𝑟𝑧₂𝐷^𝑞𝑟𝑧₃=10𝑧₂−𝑧₁𝛽𝑧₁−𝑧₁𝑧₃−𝑧₂𝑧₁𝑧₂−8𝑧₃3𝐷+𝜓,_𝑞𝑟𝛽𝑒𝑒𝛽=𝑝𝑥,𝑧,=𝜏_𝛽. (3.9)(3.10)

According to the above, we can obtain𝜉₁𝑥,𝑧,𝛽₀=𝛽−101000₀−𝑧₃−1−³_𝑗=1𝑘_1𝑗(𝑥)𝑥_𝑗𝑧₁𝑧₂³_𝑗=1𝑘_1𝑗(𝑥)𝑥_𝑗−8/30. (3.11)

Now, the parameter update laws and real matrix 𝜉₂(𝑥,𝑧,𝛽₀) are chosen as𝐷^𝑞𝑟𝛽𝑒𝑒𝛽=𝑝𝑥,𝑧,=𝜏_𝛽=−𝑧₁𝑒₂,𝜉₂𝑥,𝑧,𝛽₀=0𝑧₃−𝛽₀−10−𝑧₂000000000. (3.12)

Therefore,𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏=−10𝑧₃−𝛽₀−𝑧₂0𝛽₀−𝑧₃−1−³_𝑗=1𝑘_1𝑗(𝑥)𝑥_𝑗𝑧₁𝑧₂³_𝑗=1𝑘_1𝑗(𝑥)𝑥_𝑗−8300−𝑧₁00. (3.13) Choosing real symmetric positive definite matrix 𝑃=diag(1,1,1,1), we can get𝑃𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏+𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏^𝐻𝑃=diag−20,−2,−163,0. (3.14)

Choosing real symmetric positive semidefinite matrix 𝑄=diag(20,2,16/3,0), we can obtain𝑃𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏+𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏^𝐻𝑃=−𝑄. (3.15)

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 𝐾(𝑥)=𝑥₁+𝑥₃0.5121+0.1𝑥₁𝑥₂−1−0.5−21+𝑥₂. (3.16) 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 𝛽₀=28, respectively.

(a)

(b)

(c)

(d)

Figure 2 

Time evolution of the AHFPS error.

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 by𝑑𝑥₁𝑥𝑑𝑡=𝑎₂−𝑥₁,𝑑𝑥₂=()𝑥𝑑𝑡𝑐−𝑎₁−𝑥₁𝑥₃+𝑐𝑥₂,𝑑𝑥₃𝑑𝑡=𝑥₁𝑥₂−𝑏𝑥₃, (3.17) where (𝑎,𝑏,𝑐)=(35,3,28). Its corresponding fractional-order system is described as follows,𝐷^𝑞𝑑𝑥₁=𝑎𝑥₂−𝑥₁,𝐷^𝑞𝑑𝑥₂=(𝑐−𝑎)𝑥₁−𝑥₁𝑥₃+𝑐𝑥₂,𝐷^𝑞𝑑𝑥₃=𝑥₁𝑥₂−𝑏𝑥₃, (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.

Figure 3 

Chaotic attractors of the fractional-order Chen system (3.18) for  𝑞𝑑=0.95.

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 obtain𝜉₁𝑥,𝑧,𝛽₀=𝛽−101000₀−𝑧₃−1−³_𝑗=1𝑘_1𝑗(𝑥)𝑥_𝑗𝑧₁𝑧₂³_𝑗=1𝑘_1𝑗(𝑥)𝑥_𝑗−830. (3.19) The parameter update laws and real matrix 𝜉₂(𝑥,𝑧,𝛽₀) are chosen as𝐷^𝑞𝑟𝛽=−𝑧₁𝑒₂,𝜉₂𝑥,𝑧,𝛽₀=0𝑧₃−𝛽₀−10−𝑧₂000000000. (3.20)

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 get𝑃𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏+𝜉₁𝑥,𝑧,𝛽₀+𝜉₂𝑥,𝑧,𝛽₀𝜏^𝐻𝑃=−𝑄. (3.21)

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 𝐾(𝑥)=1+0.1𝑥₁𝑥₂0.512𝑥₁+𝑥₃−1−0.5−2𝑥₂+𝑥₁. (3.22) 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 𝛽₀=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.

(a)

(b)

(c)

(d)

Figure 4 

Time evolution of the AHFPS error.

Figure 5 

Chaotic attractors of the fractional-order Lorenz system (3.6) with (𝜎,𝛼,𝛽)=(10,40,8/3) for 𝑞𝑟=0.998.

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.