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Discrete Dynamics in Nature and Society

Volume 2011 (2011), Article ID 430724, 9 pages

http://dx.doi.org/10.1155/2011/430724

## Synchronization of Coupled Nonidentical Fractional-Order Hyperchaotic Systems

School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China

Received 24 June 2011; Revised 12 August 2011; Accepted 26 August 2011

Academic Editor: Wei-Der Chang

Copyright © 2011 Zhouchao Wei. 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

Synchronization of coupled nonidentical fractional-order hyperchaotic systems is addressed by the active sliding mode method. By designing an active sliding mode controller and choosing proper control parameters, the master and slave systems are synchronized. Furthermore, synchronizing fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system is performed to show the effectiveness of the proposed controller.

#### 1. Introduction

Fractional calculus has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz and L’Hôspital in 1695. Although it is a mathematical topic with more than 300 years old history, the applications of fractional calculus to physics and engineering are just a recent focus of interest [1, 2]. Nowadays, by utilizing fractional calculus technique, many investigations were devoted to the chaotic and hyperchaotic behaviors of fractional-order systems, such as fractional-order Chua circuit [3], fractional-order Lorenz system [4], fractional-order Rössler system [5], fractional-order Chen system [6], and fractional-order conjugate Lorenz system [7].

Over the last two decades, synchronization of chaotic systems has become more and more interesting to researchers in different fields. Since the synchronization of fractional-order chaotic systems was firstly investigated in [8], it has recently attracted increasing attention due to its potential applications in secure communication and control processing [9–13]. Moreover, many theoretical analysis and numerical simulation results about the synchronization of fractional-order chaotic systems are obtained [14–19]. Such synchronization may be safer than those of the classical chaotic systems in secure communications. This can be seen from two aspects: (i) the order of fractional derivatives can be regarded as a parameter and (ii) the fractional derivatives are nonlocal thus more complicated than the regular derivatives.

This paper focuses on synchronization of coupled fractional-order hyperchaotic nonidentical systems. The active sliding mode synchronization method is chosen to achieve this goal. The active sliding mode synchronization technique is a discontinuous control strategy, which relies on two stages of design. The first stage is to select an appropriate active controller to facilitate the design of the sequent sliding mode controller. The second stage is to design a sliding mode controller to achieve the synchronization. This process is verified when active sliding mode synchronization method is used to synchronize fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system.

This paper is organized as follows. In Section 2, basic definitions in fractional calculus are briefly presented. In Section 3, design of active sliding mode controller is proposed to synchronize coupled nonidentical fractional-order hyperchaotic systems. The application of the proposed method on fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system is numerically investigated in Section 4. Finally, conclusions in Section 5 close the paper.

#### 2. Basic Definitions

There are some definitions for fractional derivatives [20]. Three most commonly used definitions are Grünwald-Letnikov, Riemann-Liouville, and Caputo definitions.

*Definition 2.1. *The fractional derivative of Grünwald-Letnikov definition is given by

*Definition 2.2. *Let , , the Riemann-Liouville fractional derivative of order of any function is defined as follows:

*Definition 2.3. *Let , , the Caputo fractional derivative of is defined by

Note that the main advantage of Caputo approach is that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for integer-order differential equations. Therefore, in the rest of this paper, the notation indicates the Caputo fractional derivative.

#### 3. Active Sliding Mode Controller Design and Analysis

Consider a fractional-order hyperchaotic system of order , described by where denotes the system’s 4-dimensional state vector, represents the linear part of the system, and is the nonlinear part of the system. The system (3.1) represents the master system. The controller is added into the slave system, given by where , and imply the same roles as , , and in the master system.

To synchronize the master system (3.1) and the slave system (3.2), the synchronization errors dynamics is designed as follows: where and . The aim is to design the controller such that

##### 3.1. Designing the Active Controller

According to the active control design procedure [21–23], the nonlinear part of the error system is eliminated by the following choice of the input vector: Then the error system (3.3) is rewritten as On the other hand, based on a sliding mode control law, is designed as where denotes a constant vector and is the control input that satisfies where is a switching surface which prescribes the desired dynamics. Therefore, the error system becomes

##### 3.2. Designing the Sliding Surface

The sliding surface can be defined by in which is a constant vector. The equivalent control is found by the fact that is a necessary condition for the state trajectory to stay on the switching surface . Hence, when in sliding mode, the controlled system satisfies the following conditions: From (3.9)–(3.11), it follows Hence, The equivalent control is a solution of (3.13): which is realizable whenever assumes nonzero value.

Replacing in (3.9) by in (3.14), the error dynamics on the sliding surface are determined by the following relation:

##### 3.3. Designing the Sliding Mode Controller

Based on the constant plus proportional rate reaching law [24–27], the reaching law is chosen as where denotes the sign function. and are determined such that the sliding condition is satisfied and the sliding mode motion occurs.

Based on(3.9) and (3.10), it follows Therefore, the control input is determined as

##### 3.4. Stability Analysis

First, two stability theorems on fractional order systems are introduced.

Theorem 3.1 (see [28]). *The following autonomous system:
**
with , , and , is asymptotically stable if and only if is satisfied for all eigenvalues of matrix . Also, this system is stable if and only if is satisfied for all eigenvalues of matrix with those critical eigenvalues satisfying having geometric multiplicity of one. The geometric multiplicity of an eigenvalue of the matrix is the dimension of the subspace of vectors for which .*

Theorem 3.2 (see [28]). *Consider a system given by the following linear state space form with inner dimension **
with , , and . Also, assume that the triplet is minimal. System (3.20) is bounded-input bounded-output stable if and only if . When system (3.20) is externally stable, each component of its impulse response behaves like at infinity.*

According to Theorem 3.2, the sliding surface is bounded since the dynamic of the sliding surface is linear with bounded input ( for and for ).

Substituting (3.18) into (3.9), the error system reads As a linear fractional-order system with bounded input ( for and for ), the error system is stable if In this case, the error system is asymptotically stable when . The error signals will not converge to zero if . Parameter can be used to enhance the robustness of the controller in the presence of noise and mismatches.

Furthermore, it is easy to check that the eigenvalues of matrix [] are . This means that the eigenvalues of matrix [] are . Then the rank of matrix satisfies and one of eigenvalues of matrix [] is always 0. Thus,

Therefore, one of the eigenvalues of matrix [] is always . It can be shown that the three other eigenvalues are independent from and determined by the other control parameters ( and ). These three eigenvalues must satisfy condition (3.22).

*Remark 3.3. *Since we consider the fractional-order hyperchaotic systems (3.1) and (3.2) in the general form, the approach in this section is generic and can also be used for other fractional-order chaotic or hyperchaotic systems.

*Remark 3.4. *Since the chain rule is not valid in fractional-order systems, the expression (3.12) is significantly different from the integer-order one. In addition, the classical Lyapunov stability approach is difficult in checking the asymptotical stability of the error system. To overcome this shortcoming, we utilize the obtained stability results on fractional order systems to deduce the condition (3.22). Therefore, our aim is to find the controller such that the condition (3.22) is satisfied.

#### 4. Numerical Results

The hyperchaotic Lorenz system [29] was found by adding a nonlinear controller to the classical Lorenz system [30]. It has been shown that the fractional-order hyperchaotic Lorenz system can exhibit hyperchaotic behavior [31]. The fractional-order hyperchaotic Lorenz system reads

By adding a nonlinear controller to the Chen system [32], the authors obtained a hyperchaotic Chen system [33], which can demonstrate hyperchaotic behavior. The fractional-order hyperchaotic Chen system reads

In what follows, the active sliding mode synchronization method is applied to synchronize fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system. In this case, matrix and are given as

Assume that order of the master and slave systems is 2.94 () and system parameters are and . The controller parameters are chosen as , , and . This selection of parameters results in eigenvalues that are located in the stable region.

The numerical simulation has been carried out using MATLAB subroutines written based on the predictor-corrector scheme [34]. The time step size employed in the simulation is 0.001 , and the initial conditions of master and slave systems are and . The simulation results are given in Figure 1. As one can see, the designed controller is effective to synchronize fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system.

#### 5. Conclusions

In this paper, the active sliding mode method for synchronization of coupled nonidentical fractional-order hyperchaotic systems is addressed. By designing the active sliding mode controller and choosing proper control parameters (, , and ), the master and slave systems are synchronized. Furthermore, the application of the proposed method on fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system is investigated. Numerical results show the efficiency of the proposed controller to synchronize coupled nonidentical fractional-order hyperchaotic systems.

#### Acknowledgments

The author acknowledges the referees and the editor for carefully reading this paper and suggesting many helpful comments. This work was supported by the National Natural Science Foundation of China (no. 10871074) and the Special Fund for Basic Scientific Research of Central Colleges, South-Central University for Nationalities (no. CZQ11034).

#### References

- K. B. Oldham and J. Spanier,
*The Fractional Calculus*, Academic Pres, New York, NY, USA, 1974. View at Zentralblatt MATH - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific, River Edge, NJ, USA, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua's system,”
*IEEE Transactions on Circuits and Systems I*, vol. 42, no. 8, pp. 485–490, 1995. View at Publisher · View at Google Scholar · View at Scopus - 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 - C. G. Li and G. R. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,”
*Physica A*, vol. 341, no. 1–4, pp. 55–61, 2004. View at Publisher · View at Google Scholar - C. P. Li and G. R. Chen, “Chaos in the fractional order Chen system and its control,”
*Chaos, Solitons and Fractals*, vol. 22, no. 3, pp. 549–554, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Q. G. Yang and C. B. Zeng, “Chaos in fractional conjugate Lorenz system and its scaling attractors,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 12, pp. 4041–4051, 2010. View at Publisher · View at Google Scholar - C. G. Li, X. X. Liao, and J. B. Yu, “Synchronization of fractional order chaotic systems,”
*Physical Review E*, vol. 68, Article ID 067203, 3 pages, 2003. View at Google Scholar - T. S. Zhou and C. P. Li, “Synchronization in fractional-order differential systems,”
*Physica D*, vol. 212, no. 1-2, pp. 111–125, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. P. Li, W. H. Deng, and D. Xu, “Chaos synchronization of the Chua system with a fractional order,”
*Physica A*, vol. 360, no. 2, pp. 171–185, 2006. View at Publisher · View at Google Scholar - X. Y. Wang, Y. J. He, and M. J. Wang, “Chaos control of a fractional order modified coupled dynamos system,”
*Nonlinear Analysis*, vol. 71, no. 12, pp. 6126–6134, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Lei, J. Y. Yang, and S. Y. Xu, “Chaos synchronization for a class of nonlinear oscillators with fractional order,”
*Nonlinear Analysis*, vol. 72, no. 5, pp. 2326–2336, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Y. Chen, Y. X. Liu, X. Y. Ma, and R. F. Zhang, “Control of a class of fractional-order chaotic systems via sliding mode,”
*Nonlinear Dynamics*. In press. View at Publisher · View at Google Scholar - C. B. Zeng, Q. G. Yang, and J. W. Wang, “Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci,”
*Nonlinear Dynamics*, vol. 65, pp. 457–466, 2011. View at Publisher · View at Google Scholar · View at Scopus - X. Y. Wang and Y. J. 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 Zentralblatt MATH · View at Scopus - X. Y. Wang and M. J. Wang, “Dynamic analysis of the fractional-order Liu system and its synchronization,”
*Chaos*, vol. 17, no. 3, Article ID 033106, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. J. Niu, X. Y. Wang, F. Z. Nian, and M. J. Wang, “Dynamic analysis of a new chaotic system with fractional order and its generalized projective synchronization,”
*Chinese Physics B*, vol. 19, no. 12, Article ID 120507, 2010. View at Publisher · View at Google Scholar - M. J. Wang and X. Y. Wang, “Generalized synchronization of fractional order chaotic systems,”
*International Journal of Modern Physics B*, vol. 25, no. 9, pp. 1283–1292, 2011. View at Publisher · View at Google Scholar - X. Y. Wang and Y. J. He, “Chaotic synchronization of fractional-order modified coupled dynamos system,”
*International Journal of Modern Physics B*, vol. 23, no. 31, pp. 5769–5777, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999. - E. W. Bai and K. E. Lonngren, “Synchronization of two Lorenz systems using active control,”
*Chaos, Solitons and Fractals*, vol. 8, no. 1, pp. 51–58, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. N. Agiza and M. T. Yassen, “Synchronization of Rössler and Chen chaotic dynamical systems using active control,”
*Physics Letters A*, vol. 278, no. 4, pp. 191–197, 2001. View at Publisher · View at Google Scholar - M. Ch. Ho and Y. Ch. Hung, “Synchronization of two different systems by using generalized active control,”
*Physics Letters A*, vol. 301, no. 5-6, pp. 424–428, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. T. Yau, “Design of adaptive sliding mode controller for chaos synchronization with uncertainties,”
*Chaos, Solitons and Fractals*, vol. 22, no. 2, pp. 341–347, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Haeri and A. A. Emadzadeh, “Synchronizing different chaotic systems using active sliding mode control,”
*Chaos, Solitons and Fractals*, vol. 31, no. 1, pp. 119–129, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. S. Tavazoei and M. Haeri, “Determination of active sliding mode controller parameters in synchronizing different chaotic systems,”
*Chaos, Solitons and Fractals*, vol. 32, no. 2, pp. 583–591, 2007. View at Publisher · View at Google Scholar · View at Scopus - M. S. Tavazoei and M. Haeri, “Synchronization of chaotic fractional-order systems via active sliding mode controller,”
*Physica A*, vol. 387, no. 1, pp. 57–70, 2008. View at Publisher · View at Google Scholar · View at Scopus - D. Matignon, “Stability result on fractional differential equations with applications to control processing,” in
*Proceedings of the IMACS International Conference on Systems, Man, and Cybernetics*, pp. 963–968, Lille, France, 1996. - X. Y. Wang and M. J. Wang, “A hyperchaos generated from Lorenz system,”
*Physica A. Statistical Mechanics and its Applications*, vol. 387, no. 14, pp. 3751–3758, 2008. View at Publisher · View at Google Scholar - E. N. Lorenz, “Deterministic nonperodic flow,”
*Journal of the Atmospheric Sciences*, vol. 20, pp. 130–141, 1963. View at Google Scholar - X. Y. Wang and J. M. Song, “Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 8, pp. 3351–3357, 2009. View at Publisher · View at Google Scholar - G. R. Chen and T. Ueta, “Yet another chaotic attractor,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 9, no. 7, pp. 1465–1466, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. X. Li, W. K. S. Tang, and G. R. Chen, “Generating hyperchaos via state feedback control,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 15, no. 10, pp. 3367–3375, 2005. View at Publisher · View at Google Scholar · View at Scopus - K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,”
*Nonlinear Dynamics*, vol. 29, no. 1–4, pp. 3–22, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH