Hybrid Projective Synchronization for Two Identical Fractional-Order Chaotic Systems

Abstract

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.

1. Introduction

Most recently, many authors begin to investigate the chaotic dynamics and synchronization for fractional-order dynamical systems [1–6]. Chaos synchronization of the fractional-order systems is just beginning to attract some attention due to its potential applications in secure communications and control processing [7–12]. Several types of chaos synchronization are well known, which include complete synchronization (CS), antisynchronization (AS), phase synchronization, generalized synchronization (GS), projective synchronization (PS), and modified projective synchronization (MPS). Among all patterns of synchronization, the most noticeable one may be projective synchronization (PS), which was first studied by Mainieri and Rehacek [13]. Projective synchronization (PS) has been extensively considered because it can obtain faster communication. The drive and response system could be synchronized up to a scaling factor in projective synchronization. In application to secure communications, this proportional feature can be used to extend binary digital to M-nary digital communication for getting faster communication [14, 15].

However, most of projective synchronizations for the fractional-order systems have concentrated on studying the same scaling factor [16–19], and some projective synchronization schemes [16, 17] are suitable for a class of fractional-order systems or for some special fractional-order systems [19], and all the nonlinear terms of response system was absorbed in some previous works. Moreover, in order to increase the degree of secrecy for secure communications, the same scaling factor in PS can be replaced by vector function factor. Motivated by the above discussions, we propose a hybrid projective synchronization (HPS) scheme for two identical fractional-order chaotic systems in this paper. Hybrid projective synchronization (HPS) is a more general definition of projective synchronization, in which the drive system and response system could be synchronized up to a vector function factor. HPS is different from the PS. Furthermore, HPS could be used to get more secure communication than PS in application to secure communications, because it is obvious that the unpredictability of the vector function factor in HPS is more than that of the same scaling factor in PS. The main contribution of this paper is as follows: the HPS scheme in this paper is suitable for a large number of fractional-order systems (not for special fractional-order system), and this HPS scheme needs not to absorb all the nonlinear terms of response system. This is different from some previous works [16, 17, 19–21].

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 2, the definition of HPS is given, and a HPS scheme for two identical fractional-order chaotic systems is presented. In Section 3, two groups of examples are used to verify the effectiveness of the proposed scheme. The conclusion is finally drawn in Section 4.

2. Hybrid Projective Synchronization Scheme

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 𝑑^𝑞𝑓(𝑡)𝑑𝑡^𝑞=1Γ𝑚−𝑞^𝑡₀𝑓^(𝑚)(𝜏)(𝑡−𝜏)^{𝑞+1−𝑚}𝑑𝜏,𝑚−1<𝑞<𝑚, (2.1) where 0<𝑞≤1 is fractional order and 𝑑^𝑞/𝑑𝑡^𝑞 denote the Caputo definition of the fractional derivative. 𝑚 is the smallest integer larger than 𝑞, and 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 written as follows, respectively: 𝑑^𝑞𝑥𝑑𝑡^𝑞(𝑥),𝑑=𝑓_𝑞𝑦𝑑𝑡^𝑞=𝑔𝑦+Ω𝑥,𝑦, (2.2)(2.3) where 𝑥∈𝑅^𝑛, 𝑦∈𝑅^𝑛 are state vectors of the drive system (2.2) and the response system (2.3) and 𝑓,𝑔∶𝑅^𝑛→𝑅^𝑛 are continuous vector functions, respectively. Ω(𝑥,𝑦) is a vector controller.

Definition 2.1. For the drive system (2.2) and the response system (2.3), it is said to be hybrid projective synchronization (HPS) if there exists an 𝑛×𝑛 reversible matrix 𝐴 such that lim_𝑡→∞𝐴𝑦−𝑥=0, (2.4) where ‖⋅‖ is the Euclidean norm.

Remark 2.2. If 𝐴=𝐼, and 𝐼 is a unit matrix, then this synchronization is called complete synchronization (CS); if 𝐴=−𝐼, then this synchronization is called antisynchronization (AS); if 𝐴=𝑎𝐼, and 𝑎≠±1 is a nonzero real constant, then this synchronization is called projective synchronization (PS); if 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛), and 𝑎₁,𝑎₂,…,𝑎_𝑛 are not the same nonzero constant, then this synchronization is called modified projective synchronization (MPS). Therefore, CS, AS, PS, and MPS are the special cases of the hybrid projective synchronization scheme (HPS) in this paper.

In order to realize HPS for the fractional-order chaotic system (2.2), we take the fractional-order chaotic system (2.2) as drive system and construct a response system as follows: 𝑑^𝑞𝑦𝑑𝑡^𝑞=𝐴⁻¹𝑓𝐴𝑦+Ω𝑥,𝑦, (2.5) where 𝐴⁻¹ is the reverse matrix of the reversible matrix 𝐴, 𝑦∈𝑅^𝑛 are state vector of the response system (2.5), and Ω(𝑥,𝑦) is a controller which will be designed.

Define the HPS errors between the response system (2.5) and the drive system (2.2) as 𝑒=𝐴𝑦−𝑥, (2.6) where𝑒=𝑒₁,𝑒₂,…,𝑒_𝑛^𝐓,𝑒_𝑖=^𝑛_𝑗=1𝑎_𝑖𝑗𝑦_𝑗−𝑥_𝑖𝑖,𝑗=1,2,…𝑛. (2.7)

Let𝑓𝐴𝑦−𝑓(𝑥)=𝐹(𝑥,𝑒). (2.8)

Now, we assume that the errors vector 𝑒 can be divided into 𝑒_𝛼=(𝑒_𝑠1,…,𝑒_𝑠𝑚)^𝐓 and 𝑒_𝛽=(𝑒_𝑠𝑚+1,…,𝑒_𝑠𝑛)^𝐓, such that 𝐹(𝑥,𝑒) has the form of 𝐹()=𝐵𝑥,𝑒^𝛼𝑒_𝛼+ℎ₁𝑥,𝑒_𝛼,𝑒_𝛽𝐵_𝛽𝑒_𝛽+ℎ₂₁𝑥,𝑒_𝛼,𝑒_𝛽+ℎ₂₂𝑥,𝑒_𝛼,𝑒_𝛽, (2.9) where ℎ₁(𝑥,𝑒_𝛼,𝑒_𝛽)∈𝑅^𝑚, ℎ₂₁(𝑥,𝑒_𝛼,𝑒_𝛽)∈𝑅^𝑛−𝑚, ℎ₂₂(𝑥,𝑒_𝛼,𝑒_𝛽)∈𝑅^𝑛−𝑚 and lim_𝑒𝛼→0ℎ₂₁(𝑥,𝑒_𝛼,𝑒_𝛽)=0, respectively. 𝐵_𝛼∈𝑅^𝑚×𝑚 and 𝐵_𝛽∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)} are real constant matrix.

Rewrite the controller Ω(𝑥,𝑦) as follows: Ω()=𝜇𝑥,𝑦=𝜇𝑥,𝑒^𝛼()𝜇𝑥,𝑒_𝛽(𝑥,𝑒), (2.10) where 𝜇_𝛼(𝑥,𝑒)∈𝑅^𝑚 and 𝜇_𝛽(𝑥,𝑒)∈𝑅^𝑛−𝑚, respectively.

Now, Theorem 2.3 is given based on the previously mentioned conditions in order to achieve the HPS between the drive system (2.2) and the response system (2.5).

Theorem 2.3. Choose the following controller: Ω()=𝜇𝑥,𝑦=𝜇𝑥,𝑒^𝛼()𝜇𝑥,𝑒_𝛽()=Λ𝑥,𝑒^𝛼𝑒_𝛼−ℎ₁𝑥,𝑒_𝛼,𝑒_𝛽Λ_𝛽𝑒_𝛽−ℎ₂₂𝑥,𝑒_𝛼,𝑒_𝛽, (2.11) where Λ_𝛼∈𝑅^𝑚×𝑚 and Λ_𝛽∈𝑅^{(𝑛−𝑚)×(𝑛−𝑚)} are suitable constant matrix, respectively.

If all the eigenvalues of 𝐵_𝛼+Λ_𝛼 satisfy |arg𝜆_𝑖|>0.5𝜋𝑞 (𝑖=1,2,…,𝑚) and all the eigenvalues of 𝐵_𝛽+Λ_𝛽 satisfy |arg𝜆_𝑗|>0.5𝜋𝑞  (𝑗=1,2,…,𝑛−𝑚), then hybrid projective synchronization between the drive system (2.2) and the response system (2.5) can be achieved.

Proof. According to the drive system (2.2) and the response system (2.5), the error dynamic system of hybrid projective synchronization can be obtained as follows: 𝑑^𝑞𝑒𝑑𝑡^𝑞=𝐴𝑑^𝑞𝑦𝑑𝑡^𝑞−𝑑^𝑞𝑥𝑑𝑡^𝑞=𝑓𝐴𝑦−𝑓(𝑥)+Ω𝑥,𝑦=𝐹(𝑥,𝑒)+𝜇(𝑥,𝑒). (2.12)
According to (2.9) and (2.10), the error dynamic system (2.12) can be rewritten as 𝑑^𝑞𝑒_𝛼𝑑𝑡^𝑞=𝐵_𝛼𝑒_𝛼+ℎ₁𝑥,𝑒_𝛼,𝑒_𝛽+𝜇_𝛼(),𝑑𝑥,𝑒_𝑞𝑒_𝛽𝑑𝑡^𝑞=𝐵_𝛽𝑒_𝛽+ℎ₂₁𝑥,𝑒_𝛼,𝑒_𝛽+ℎ₂₂𝑥,𝑒_𝛼,𝑒_𝛽+𝜇_𝛽(𝑥,𝑒), (2.13) because 𝜇_𝛼()𝜇𝑥,𝑒_𝛽()=Λ𝑥,𝑒^𝛼𝑒_𝛼−ℎ₁𝑥,𝑒_𝛼,𝑒_𝛽Λ_𝛽𝑒_𝛽−ℎ₂₂𝑥,𝑒_𝛼,𝑒_𝛽. (2.14) So, 𝑑^𝑞𝑒_𝛼𝑑𝑡^𝑞=𝐵_𝛼+Λ_𝛼𝑒_𝛼,𝑑^𝑞𝑒_𝛽𝑑𝑡^𝑞=𝐵_𝛽+Λ_𝛽𝑒_𝛽+ℎ₂₁𝑥,𝑒_𝛼,𝑒_𝛽. (2.15)
Because all the eigenvalues of 𝐵_𝛼+Λ_𝛼 satisfy |arg𝜆_𝑖|>0.5𝜋𝑞 (𝑖=1,2,…,𝑚), according to the stability theory of linear fractional-order systems [22], the equilibrium point 𝑒_𝑖=0(𝑖=1,2,…,𝑚) in the first equation of system (2.15) is asymptotically stable, which indicates lim_𝑡→+∞𝑒_𝛼=0.
Since lim_𝑡→+∞𝑒_𝛼=0 and lim_𝑒𝛼→0ℎ₂₁(𝑥,𝑒_𝛼,𝑒_𝛽)=0, therefore when time 𝑡→+∞, the second equation of system (2.15) can be changed as 𝑑^𝑞𝑒_𝛽𝑑𝑡^𝑞=𝐵_𝛽+Λ_𝛽𝑒_𝛽, (2.16)
because all the eigenvalues of 𝐵_𝛽+Λ_𝛽 satisfy |arg𝜆_𝑗|>0.5𝜋𝑞 (𝑗=1,2,…,𝑛−𝑚). According to the stability theory of linear fractional-order systems [22], the equilibrium point 𝑒_𝑖=0(𝑖=1,2,…,𝑛−𝑚) of system (2.16) is asymptotically stable, which indicates lim_𝑡→+∞𝑒_𝛽=0.
According to lim_𝑡→+∞𝑒_𝛼=0 and lim_𝑡→+∞𝑒_𝛽=0, the hybrid projective synchronization between the drive system (2.2) and the response system (2.5) can be achieved. This finishes the proof.

Remark 2.4. In order to use the stability theory of linear fractional-order systems [22], the controller Ω(𝑥,𝑦) or 𝜇(𝑥,𝑒) are chosen as Λ_𝛼𝑒_𝛼−ℎ₁(𝑥,𝑒_𝛼,𝑒_𝛽)Λ_𝛽𝑒_𝛽−ℎ₂₂(𝑥,𝑒_𝛼,𝑒_𝛽). Moreover, the nonlinear term ℎ₂₁(𝑥,𝑒_𝛼,𝑒_𝛽)∈𝑅^𝑛−𝑚 in the error dynamic system (2.13) or response system (2.5) is preserved; this is different from some previous works [17, 19–21] which need to absorb all the nonlinear terms of response system or error dynamic system.

Remark 2.5. For the complex fractional-order multiscroll chaotic systems [23–25] and the complex dynamical network or the small-world dynamical networks [26–28], the hybrid projective synchronization would be much more complex. Further work on this issue is an ongoing research topic in our group.

3. Applications

In order to illustrate the effectiveness of the proposed hybrid projective synchronization scheme obtained in Section 2, two examples are considered in this section, which are HPS for the fractional-order Chen system and HPS for the fractional-order hyperchaotic Lu system.

3.1. HPS for the Fractional-Order Chen System

The fractional-order Chen system [23] is described as follows: 𝑑^𝑞𝑥₁𝑑𝑡^𝑞=35𝑥₂−𝑥₁,𝑑^𝑞𝑥₂𝑑𝑡^𝑞=−7𝑥₁+28𝑥₂−𝑥₁𝑥₃,𝑑^𝑞𝑥₃𝑑𝑡^𝑞=𝑥₁𝑥₂−3𝑥₃. (3.1)

Tavazoei and Haeri pointed out that fractional-order Chen system (3.1) exhibits chaotic behavior for 𝑞≥0.83 [29]. The chaotic attractor of fractional-order Chen system for 𝑞=0.9, is depicted in Figure 1.

(a)

(b)

Figure 1 

Chaotic attractor of fractional-order Chen chaotic system (3.1) for 𝑞=0.9.

According to the HPS scheme presented in the previous section, the response system is described by 𝑑^𝑞𝑦₁𝑑𝑡^𝑞𝑑^𝑞𝑦₂𝑑𝑡^𝑞𝑑^𝑞𝑦₃𝑑𝑡^𝑞=𝐴⁻¹35∑³_𝑗=1𝑎_2𝑗𝑦_𝑗−∑³_𝑗=1𝑎_1𝑗𝑦_𝑗−7×∑³_𝑗=1𝑎_1𝑗𝑦_𝑗+28×∑³_𝑗=1𝑎_2𝑗𝑦_𝑗−∑³_𝑗=1𝑎_1𝑗𝑦_𝑗×∑³_𝑗=1𝑎_3𝑗𝑦_𝑗3𝑗∑₌₁𝑎_1𝑗𝑦_𝑗×∑³_𝑗=1𝑎_2𝑗𝑦_𝑗−3×∑³_𝑗=1𝑎_3𝑗𝑦_𝑗+Ω𝑥,𝑦, (3.2) where 𝐴=^{𝑎11𝑎12𝑎13𝑎21𝑎22𝑎23}_{𝑎31𝑎32𝑎33} is a reversible matrix and 𝐴⁻¹ is its reverse matrix.

Now, the term of 𝑓(𝐴𝑦)−𝑓(𝑥)=𝐹(𝑥,𝑒) is yielded firstly: 𝑓𝐴𝑦−𝑓(𝑥)=𝐹(𝑥,𝑒)=35𝑒₂−35𝑒₁−7𝑒₁+28𝑒₂−𝑥₃𝑒₁−𝑥₁𝑒₃−𝑒₁𝑒₃𝑥₂𝑒₁+𝑥₁𝑒₂+𝑒₁𝑒₂−3𝑒₃. (3.3)

So, we can choose 𝑒_𝛼=𝑒₁ and 𝑒_𝛽=(𝑒₂,𝑒₃)^𝐓. Therefore, ℎ₁(𝑥,𝑒_𝛼,𝑒_𝛽)=−35𝑒₂, ℎ₂₁(𝑥,𝑒_𝛼,𝑒_𝛽)=^{−7𝑒1−𝑥3𝑒1−𝑒1𝑒3}_{𝑥2𝑒1+𝑒1𝑒2}, ℎ₂₂(𝑥,𝑒_𝛼,𝑒_𝛽)=^{−𝑥1𝑒3}_𝑥1𝑒2, 𝐵_𝛼=−35, and 𝐵_𝛽=²⁸⁰₀₋₃, respectively. Obviously, lim_𝑒𝛼→0ℎ₂₁(𝑥,𝑒_𝛼,𝑒_𝛽)=0.

According to Theorem 2.3 in Section 2, the controller Ω(𝑥,𝑦) can be chosen as follows: Ω=𝜇𝑥,𝑦^𝛼()𝜇𝑥,𝑒_𝛽()=Λ𝑥,𝑒^𝛼𝑒_𝛼−ℎ₁𝑥,𝑒_𝛼,𝑒_𝛽Λ_𝛽𝑒_𝛽−ℎ₂₂𝑥,𝑒_𝛼,𝑒_𝛽, (3.4) where 𝜇_𝛼(𝑥,𝑒)=Λ_𝛼𝑒₁−35𝑒₂, 𝜇_𝛽(𝑥,𝑒)=Λ_𝛽^𝑒2_𝑒3−^{−𝑥1𝑒3}_𝑥1𝑒2, Λ_𝛼∈𝑅^1×1, and Λ_𝛽∈𝑅^2×2, respectively. Therefore, choose suitable matrix Λ_𝛼∈𝑅^1×1 and Λ_𝛽∈𝑅^2×2. If the eigenvalues of 𝐵_𝛼+Λ_𝛼 satisfy |arg𝜆|>0.5𝜋𝑞 and if all the eigenvalues of 𝐵_𝛽+Λ_𝛽 satisfy |arg𝜆_𝑗|>0.5𝜋𝑞 (𝑗=1,2), then hybrid projective synchronization between the drive system (3.1) and its response system (3.2) can be achieved.

For example, choose reversible matrix 𝐴=−220−11101−0.5,Λ_𝛼=30,Λ_𝛽=−2410−50. (3.5)

So, the eigenvalues of 𝐵_𝛼+Λ_𝛼 are −5, and all the eigenvalues of 𝐵_𝛽+Λ_𝛽 are 0.5±6.1441𝑗 and |arg𝜆(𝐵_𝛽+Λ_𝛽)|=0.9483×𝜋/2>0.5𝜋𝑞 (𝑞=0.9), respectively. Therefore, the hybrid projective synchronization between drive system (3.1) and its response system (3.2) can be achieved. The corresponding numerical result is shown in Figure 2, in which the initial conditions are (𝑥₁₀, 𝑥₂₀, 𝑥₃₀) = (3, 4, 5) for the drive system (3.1), and (𝑦₁₀, 𝑦₂₀, 𝑦₃₀) = (8, 10, 4) for the response system (3.2), respectively, and 𝜀=(∑³_𝑖=1𝑒²_𝑖)^1/2.

Figure 2 

The HPS result between the drive system (3.1) and its response system (3.2) for 𝑞=0.9.

3.2. HPS for Fractional-Order Hyperchaotic Lǔ System

Min et al. reported a fractional-order hyperchaotic Lu system [30, 31] based on the hyperchaotic Lu system, which is described as follows:𝑑^𝑞𝑥₁𝑑𝑡^𝑞=36𝑥₂−𝑥₁+𝑥₄,𝑑^𝑞𝑥₂𝑑𝑡^𝑞=20𝑥₂−𝑥₁𝑥₃,𝑑^𝑞𝑥₃𝑑𝑡^𝑞=𝑥₁𝑥₂−3𝑥₃,𝑑^𝑞𝑥₄𝑑𝑡^𝑞=1.3𝑥₄+𝑥₁𝑥₃. (3.6)

The chaotic attractor of fractional-order hyperchaotic Lu system for 𝑞=0.95, is shown in Figure 3.

(a)

(b)

Figure 3 

Chaotic attractor of fractional-order hyperchaotic Lu system (3.6) for 𝑞=0.95.

Taking system (3.6) as the drive system, according to the HPS scheme presented in Section 2, the response system is described by 𝑑^𝑞𝑦₁𝑑𝑡^𝑞𝑑^𝑞𝑦₂𝑑𝑡^𝑞𝑑^𝑞𝑦₃𝑑𝑡^𝑞𝑑^𝑞𝑦₄𝑑𝑡^𝑞=𝐴⁻¹36×∑⁴_𝑗=1𝑎_2𝑗𝑦_𝑗−∑⁴_𝑗=1𝑎_1𝑗𝑦_𝑗+∑⁴_𝑗=1𝑎_4𝑗𝑦_𝑗20×∑⁴_𝑗=1𝑎_2𝑗𝑦_𝑗−∑⁴_𝑗=1𝑎_1𝑗𝑦_𝑗×∑⁴_𝑗=1𝑎_3𝑗𝑦_𝑗4𝑗∑₌₁𝑎_1𝑗𝑦_𝑗×∑⁴_𝑗=1𝑎_2𝑗𝑦_𝑗−3×∑⁴_𝑗=1𝑎_3𝑗𝑦_𝑗1.3×∑⁴_𝑗=1𝑎_4𝑗𝑦_𝑗+∑⁴_𝑗=1𝑎_1𝑗𝑦_𝑗×∑⁴_𝑗=1𝑎_3𝑗𝑦_𝑗+Ω𝑥,𝑦, (3.7) where 𝐴=^{𝑎11𝑎12𝑎13𝑎14𝑎21𝑎22𝑎23𝑎24}_{𝑎31𝑎32𝑎33𝑎34𝑎41𝑎42𝑎43𝑎44} is a reversible matrix and 𝐴⁻¹ is its reverse matrix. As the same as what is mentioned previously, we can obtain 𝑓𝐴𝑦−𝑓(𝑥)=𝐹(𝑥,𝑒)=36𝑒₂−36𝑒₁+𝑒₄20𝑒₂−𝑥₃𝑒₁−𝑥₁𝑒₃−𝑒₁𝑒₃𝑥₂𝑒₁+𝑥₁𝑒₂+𝑒₁𝑒₂−3𝑒₃1.3𝑒₄+𝑥₃𝑒₁+𝑥₁𝑒₃+𝑒₁𝑒₃. (3.8)

Now, we can choose 𝑒_𝛼=(𝑒₁,𝑒₂)^𝐓 and 𝑒_𝛽=(𝑒₃,𝑒₄)^𝐓. Therefore, ℎ₁(𝑥,𝑒_𝛼,𝑒_𝛽)=^𝑒4_{−𝑥3𝑒1−𝑥1𝑒3−𝑒1𝑒3}, ℎ₂₁(𝑥,𝑒_𝛼,𝑒_𝛽)=^{𝑥2𝑒1+𝑥1𝑒2+𝑒1𝑒2}_{𝑥3𝑒1+𝑒1𝑒3}, ℎ₂₂(𝑥,𝑒_𝛼,𝑒_𝛽)=⁰_𝑥1𝑒3, 𝐵_𝛼=⁻³⁶³⁶₀₂₀, and 𝐵_𝛽=⁻³⁰_01.3, respectively. Obviously, lim_𝑒𝛼→0ℎ₂₁(𝑥,𝑒_𝛼,𝑒_𝛽)=0.

According to Theorem 2.3 in Section 2, the controller Ω(𝑥,𝑦) can be chosen as follows: Ω=𝜇𝑥,𝑦^𝛼()𝜇𝑥,𝑒_𝛽()=Λ𝑥,𝑒^𝛼𝑒_𝛼−ℎ₁𝑥,𝑒_𝛼,𝑒_𝛽Λ_𝛽𝑒_𝛽−ℎ₂₂𝑥,𝑒_𝛼,𝑒_𝛽, (3.9) where 𝜇_𝛼(𝑥,𝑒)=Λ_𝛼^𝑒1_𝑒2−^𝑒4_{−𝑥3𝑒1−𝑥1𝑒3−𝑒1𝑒3}, 𝜇_𝛽(𝑥,𝑒)=Λ_𝛽^𝑒2_𝑒3−⁰_𝑥1𝑒3, Λ_𝛼∈𝑅^2×2 and Λ_𝛽∈𝑅^2×2, respectively. Therefore, choose suitable matrix Λ_𝛼∈𝑅^2×2, and Λ_𝛽∈𝑅^2×2. If the eigenvalues of 𝐵_𝛼+Λ_𝛼 satisfy |arg𝜆_𝑖|>0.5𝜋𝑞(𝑖=1,2) and if all the eigenvalues of 𝐵_𝛽+Λ_𝛽 satisfy |arg𝜆_𝑗|>0.5𝜋𝑞 (𝑗=1,2), then the hybrid projective synchronization between the drive system (3.6) and its response system (3.7) can be achieved.

For example, choose reversible matrix𝐴=0.50000−2200−111001−0.5,Λ_𝛼=000−30,Λ_𝛽=000−2.3. (3.10)

So, the eigenvalues of 𝐵_𝛼+Λ_𝛼 are −36 and −10, and the eigenvalues of 𝐵_𝛽+Λ_𝛽 are −3 and −1, respectively. Therefore, the hybrid projective synchronization between drive system (3.6) and its response system (3.7) can be achieved. The corresponding numerical result is shown in Figure 4, in which the initial conditions are (𝑥₁₀, 𝑥₂₀, 𝑥₃₀,𝑥₄₀) = (3,4,5,6) for the drive system (3.6) and (𝑦₁₀, 𝑦₂₀, 𝑦₃₀, 𝑦₄₀) = (16,7,12,4) for the response system (3.7), respectively, and 𝜀=(∑⁴_𝑖=1𝑒²_𝑖)^1/2.

Figure 4 

The HPS result between the drive system (3.6) and its response system (3.7) for 𝑞=0.95.

4. Conclusions

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.

Acknowledgments

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.