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Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 430724, 9 pages
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.
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.
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 , fractional-order Lorenz system , fractional-order Rössler system , fractional-order Chen system , and fractional-order conjugate Lorenz system .
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 , 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 . 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.
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.
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.
3.4. Stability Analysis
First, two stability theorems on fractional order systems are introduced.
Theorem 3.1 (see ). 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 ). 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  was found by adding a nonlinear controller to the classical Lorenz system . It has been shown that the fractional-order hyperchaotic Lorenz system can exhibit hyperchaotic behavior . The fractional-order hyperchaotic Lorenz system reads
By adding a nonlinear controller to the Chen system , the authors obtained a hyperchaotic Chen system , 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 . 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.
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.
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).
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