Hybrid Projective Synchronization for Two Identical Fractional-Order Chaotic Systems

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.


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.

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 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: 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 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 a 1 , a 2 , . . ., a n , and a 1 , a 2 , . . ., 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: 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. If all the eigenvalues of B α Λ α satisfy | arg λ i | > 0.5πq i 1, 2, . . ., m and all the eigenvalues of B β Λ β satisfy | arg λ 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: Ad q y dt q − d q x dt q f Ay − f x Ω x, 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 q e α dt q B α e α h 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 α e α −h 1 x,e α ,e β Λ β e β −h 22 x,e α ,e β .Moreover, the nonlinear term h 21 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.

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

HPS for the Fractional-Order Chen System
The fractional-order Chen system 23 is described as follows:

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.
According to the HPS scheme presented in the previous section, the response system is described by where A a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 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: x 2 e 1 x 1 e 2 e 1 e 2 − 3e 3 The HPS result between the drive system 3.1 and its response system 3.2 for q 0.9.x 1 e 2 , Λ α ∈ R 1 × 1 , and

So, we can choose
5πq and if all the eigenvalues of B β Λ β satisfy | arg λ 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 So, the eigenvalues of B α Λ α are −5, and all the eigenvalues of B β Λ β are 0.5 ± 6.1441 j and | arg λ 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 x 10 , x 20 , x 30 3, 4, 5 for the drive system 3.1 , and y 10 , y 20 , y 30 8, 10, 4 for the response system 3.2 , respectively, and ε

HPS for Fractional-Order Hyperchaotic L ǔ System
Min et al. reported a fractional-order hyperchaotic Lu system 30, 31 based on the hyperchaotic Lu system, which is described as follows:

3.6
The chaotic attractor of fractional-order hyperchaotic Lu system for q 0.95, is shown in Figure 3.
Taking system 3.6 as the drive system, according to the HPS scheme presented in Section 2, the response system is described by 2 and if all the eigenvalues of B β Λ β satisfy | arg λ 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 matrix 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 x 10 , x 20 , x 30 , x 40 3,4,5,6 for the drive system 3.6 and y 10 , y 20 , y 30 , y 40 16,7,12,4 for the response system 3.7 , respectively, and ε

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 The HPS result between the drive system 3.6 and its response system 3.7 for q 0.95.rigorous.Numerical simulations are used to illustrate the effectiveness of the proposed synchronization method.

Figure 2 :
Figure 2:The HPS result between the drive system 3.1 and its response system 3.2 for q 0.9.

Figure 4 :
Figure 4:The HPS result between the drive system 3.6 and its response system 3.7 for q 0.95.
Now, we assume that the errors vector e can be divided into e α e s 1 , . .., e s m ∈ R m , h 21 x, e α , e β ∈ R n−m , h 22 x, e α , e β ∈ R n−m and lim e α → 0 h 21 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: 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 .
11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44is a reversible matrix and A −1 is its reverse matrix.As the same as what is mentioned previously, we can obtain − x 3 e 1 − x 1 e 3 − e 1 e 3 x 2 e 1 x 1 e 2 e 1 e 2 − 3e 3 1.3e 4 x 3 e 1 x 1 e 3 e 1 e 3 −x 3 e 1 −x 1 e 3 −e 1 e 3 , h 21 x, e α , e β x 2 e 1 x 1 e 2 e 1 e 2x 3 e 1 e 1 e 3 , respectively.Obviously, lim e α → 0 h 21 x, e α , e β 0. According to Theorem 2.3 in Section 2, the controller Ω x, y can be chosen as follows: a e 4 −x 3 e 1 −x 1 e 3 −e 1 e 3 , μ β x, e Λ β e 2