Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 768587, 11 pages

http://dx.doi.org/10.1155/2012/768587

## Hybrid Projective Synchronization for Two Identical Fractional-Order Chaotic Systems

^{1}Research Center for System Theory and Applications, Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{2}Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received 23 May 2012; Accepted 27 June 2012

Academic Editor: Vimal Singh

Copyright © 2012 Ping Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 where is fractional order and denote the Caputo definition of the fractional derivative. is the smallest integer larger than , and is the -order derivative in the usual sense. is the gamma function.

The fractional-order chaotic drive and response systems can be written as follows, respectively: where , are state vectors of the drive system (2.2) and the response system (2.3) and are continuous vector functions, respectively. 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 reversible matrix such that
where is the Euclidean norm.

*Remark 2.2. *If , and is a unit matrix, then this synchronization is called complete synchronization (CS); if , then this synchronization is called antisynchronization (AS); if , and is a nonzero real constant, then this synchronization is called projective synchronization (PS); if , and 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 is the reverse matrix of the reversible matrix , are state vector of the response system (2.5), and is a controller which will be designed.

Define the HPS errors between the response system (2.5) and the drive system (2.2) as where

Let

Now, we assume that the errors vector can be divided into and , such that has the form of where , , and , respectively. and are real constant matrix.

Rewrite the controller as follows: where and , 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:
**
where and are suitable constant matrix, respectively.*

If all the eigenvalues of satisfy () and all the eigenvalues of satisfy (), 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:

According to (2.9) and (2.10), the error dynamic system (2.12) can be rewritten as
because
So,

Because all the eigenvalues of satisfy (), according to the stability theory of linear fractional-order systems [22], the equilibrium point in the first equation of system (2.15) is asymptotically stable, which indicates .

Since and , therefore when time , the second equation of system (2.15) can be changed as

because all the eigenvalues of satisfy (). According to the stability theory of linear fractional-order systems [22], the equilibrium point of system (2.16) is asymptotically stable, which indicates .

According to and , 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 or are chosen as . Moreover, the nonlinear term 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:

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

According to the HPS scheme presented in the previous section, the response system is described by where is a reversible matrix and is its reverse matrix.

Now, the term of is yielded firstly:

So, we can choose and . Therefore, , , , , and , respectively. Obviously, .

According to Theorem 2.3 in Section 2, the controller can be chosen as follows: where , , , and , respectively. Therefore, choose suitable matrix and . If the eigenvalues of satisfy and if all the eigenvalues of satisfy (), 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 are , and all the eigenvalues of are and (), 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 (, , ) = (3, 4, 5) for the drive system (3.1), and (, , ) = (8, 10, 4) for the response system (3.2), respectively, and .

##### 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:

The chaotic attractor of fractional-order hyperchaotic Lu system for , 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 where is a reversible matrix and is its reverse matrix. As the same as what is mentioned previously, we can obtain

Now, we can choose and . Therefore, , , , , and , respectively. Obviously, .

According to Theorem 2.3 in Section 2, the controller can be chosen as follows: where , , and , respectively. Therefore, choose suitable matrix , and . If the eigenvalues of satisfy and if all the eigenvalues of satisfy (), 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 are −36 and −10, and the eigenvalues of 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 (, , ) = (3,4,5,6) for the drive system (3.6) and (, , , ) = (16,7,12,4) for the response system (3.7), respectively, and .

#### 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.

#### References

- I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,”
*Physical Review Letters*, vol. 91, no. 3, Article ID 034101, 4 pages, 2003. View at Google Scholar · View at Scopus - R. X. Zhang and S. P. Yang, “Adaptive synchronization of fractional-order chaotic systems via a single driving variable,”
*Nonlinear Dynamics*, vol. 66, no. 4, pp. 831–837, 2011. View at Publisher · View at Google Scholar - D. Cafagna and G. Grassi, “Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems,”
*Nonlinear Dynamics*, vol. 68, no. 1-2, pp. 117–128, 2012. View at Google Scholar - Z. M. Odibat, “A note on phase synchronization in coupled chaotic fractional order systems,”
*Nonlinear Analysis. Real World Applications*, vol. 13, no. 2, pp. 779–789, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. T. He and M. k. Luo, “Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control,”
*Applied Mathematics and Mechanics*, vol. 33, no. 5, pp. 567–582, 2012. View at Google Scholar - Z. M. Odibat, N. Corson, M. A. Aziz-Alaoui, and C. Bertelle, “Synchronization of chaotic fractional-order systems via linear control,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 20, no. 1, pp. 81–97, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. K. Agrawal, M. Srivastava, and S. Das, “Synchronization of fractional order chaotic systems using active control method,”
*Chaos, Solitons and Fractals*, vol. 45, no. 6, pp. 737–752, 2012. View at Publisher · View at Google Scholar - R. Zhang and S. Yang, “Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller,”
*Nonlinear Dynamics*, vol. 68, no. 1-2, pp. 45–51, 2012. View at Publisher · View at Google Scholar - C.-M. Chang and H.-K. Chen, “Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen-Lee systems,”
*Nonlinear Dynamics*, vol. 62, no. 4, pp. 851–858, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. J. Wu and H. Wang, “A new chaotic system with fractional order and its projective synchronization,”
*Nonlinear Dynamics*, vol. 61, no. 3, pp. 407–417, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Pan, W. E. Zhou, L. Zhou, and K. Sun, “Chaos synchronization between two different fractional-order hyperchaotic systems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 6, pp. 2628–2640, 2011. View at Google Scholar - S. S. Delshad, M. M. Asheghan, and M. H. Beheshti, “Synchronization of N-coupled incommensurate fractional-order chaotic systems with ring connection,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 9, pp. 3815–3824, 2011. View at Publisher · View at Google Scholar - R. Mainieri and J. Rehacek, “Projective synchronization in three-dimensional chaotic systems,”
*Physical Review Letters*, vol. 82, no. 15, pp. 3042–3045, 1999. View at Google Scholar · View at Scopus - C. Y. Chee and D. Xu, “Secure digital communication using controlled projective synchronisation of chaos,”
*Chaos, Solitons and Fractals*, vol. 23, no. 3, pp. 1063–1070, 2005. View at Publisher · View at Google Scholar · View at Scopus - K.-Z. Li, M.-C. Zhao, and X.-C. Fu, “Projective synchronization of driving-response systems and its application to secure communication,”
*IEEE Transactions on Circuits and Systems. I*, vol. 56, no. 10, pp. 2280–2291, 2009. View at Publisher · View at Google Scholar - G. Peng and Y. Jiang, “Generalized projective synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal,”
*Physics Letters A*, vol. 372, no. 22, pp. 3963–3970, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. R. Chen, C. X. Liu, and Y. X. Li, “Nonlinear observer based full-state projective synchronization for a class of fractional-order chaotic systems,”
*Acta Physica Sinica*, vol. 57, no. 3, pp. 1453–1457, 2008. View at Google Scholar - G. Peng, Y. Jiang, and F. Chen, “Generalized projective synchronization of fractional order chaotic systems,”
*Physica A*, vol. 387, no. 14, pp. 3738–3746, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Shao, “Controlling general projective synchronization of fractional order Rossler systems,”
*Chaos, Solitons and Fractals*, vol. 39, no. 4, pp. 1572–1577, 2009. View at Publisher · View at Google Scholar · View at Scopus - X. Y. Wang and Y. He, “Projective synchronization of fractional order chaotic system based on linear separation,”
*Physics Letters, Section A*, vol. 372, no. 4, pp. 435–441, 2008. View at Publisher · View at Google Scholar · View at Scopus - X. Y. Wang and Y. J. He, “Projective synchronization of a fractional order unified system,”
*Acta Physica Sinica*, vol. 57, no. 3, pp. 1485–1492, 2008. View at Google Scholar - D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in
*Proceedings of the IMACS-IEEE Multiconference on Computational Engineering in Systems Applications*, pp. 963–968, Lille, France, 1996. - J. Lü and G. Chen, “Generating multiscroll chaotic attractors: theories, methods and applications,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 16, no. 4, pp. 775–858, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Lü, S. Yu, H. Leung, and G. Chen, “Experimental verification of multidirectional multiscroll chaotic attractors,”
*IEEE Transactions on Circuits and Systems I*, vol. 53, no. 1, pp. 149–165, 2006. View at Publisher · View at Google Scholar - J. Lü, F. Han, X. Yu, and G. Chen, “Generating 3-D multi-scroll chaotic attractors: a hysteresis series switching method,”
*Automatica*, vol. 40, no. 10, pp. 1677–1687, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Lü and G. Chen, “A time-varying complex dynamical network model and its controlled synchronization criteria,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 6, pp. 841–846, 2005. View at Publisher · View at Google Scholar - J. Zhou, J.-A. Lu, and J. Lü, “Adaptive synchronization of an uncertain complex dynamical network,”
*IEEE Transactions on Automatic Control*, vol. 51, no. 4, pp. 652–656, 2006. View at Publisher · View at Google Scholar - J. Lü, X. Yu, G. Chen, and D. Cheng, “Characterizing the synchronizability of small-world dynamical networks,”
*IEEE Transactions on Circuits and Systems. I*, vol. 51, no. 4, pp. 787–796, 2004. View at Publisher · View at Google Scholar - M. S. Tavazoei and M. Haeri, “A necessary condition for double scroll attractor existence in fractional-order systems,”
*Physics Letters, Section A*, vol. 367, no. 1-2, pp. 102–113, 2007. View at Publisher · View at Google Scholar - A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,”
*Physica A*, vol. 364, pp. 103–110, 2006. View at Publisher · View at Google Scholar - F.-H. Min, Y. Yu, and C.-J. Ge, “Circuit implementation and tracking control of the fractional-order hyper-chaotic Lǔ system,”
*Acta Physica Sinica*, vol. 58, no. 3, pp. 1456–1461, 2009. View at Google Scholar