Function Projective Synchronization of Fractional-Order Hyperchaotic System Based on Open-Plus-Closed-Looping

Commun. Theor. Phys. 55 (2011) 617 621 Vol. 55, No. 4, April 15, 2011 Function Projective Synchronization of Fractional-Order Hyperchaotic System Based on Open-Plus-Closed-Looping WANG Xing-Yuan ( ), LIU Rong ( Â), and ZHANG Na ( ) School of Electronic & Information Engineering, Dalian University of Technology, Dalian 116024, China (Received October 26, 2010; revised manuscript received December 10, 2010) Abstract The purpose of this paper is to analyze the dynamic behavior of fractional-order four-order hyperchaotic Lü system, and use the Open-Plus-Closed-Looping (OPCL) coupling method to construct the system s corresponding response system, and then implement function projective synchronization (FPS) of fractional-order drive-response system with system parameters perturbation or not. Finally, the numerical simulations verify the effectiveness and robustness of this scheme. PACS numbers: 05.45.Xt, 05.45.Gg, 05.45.Pq Key words: hyperchaotic, FPS, OPCL, fractional-order 1 Introduction In 1990, Pecora and Carroll presented the conception of chaotic synchronization [1] and introduced a method to synchronize two identical chaotic systems with different initial conditions. Since then, the chaos synchronization becomes a great interest topic. At the same time, chaos synchronization has also received extensive attention in the field of secure communication. Its potential applications have been always the object of researchers. So far, a variety of synchronization methods have been mentioned. [2 6] Synchronization programs can be summarized into the following categories: complete synchronization, anti-synchronization, phase synchronization, lag synchronization, generalized synchronization, and impulse synchronization and so on. In 1999, Mainieri and related colleges [7] found that the state output phase of the coupled drive-response system had been locked and the amplitude of the corresponding state showed an evolution phenomena according to a proportional relationship when they coupled some of the linear chaotic systems. According to that, projective synchronization came out. In 2002, Xu and related colleges [8] added a control to the response system, which achieved the projective synchronization of some nonlinear systems, that is to say the projective synchronization systems is no longer limited to the linear ones. Since then, projective synchronization methods are widely used in chaotic cryptography and other fields. However, projective synchronization mentioned above is just evoluted according to a particular ratio. But in some actual problems, it is necessary for the drive-response systems to evolute in accordance with the proportional function of time. Recently Chen and related colleges [9] proposed an FPS method. The method replaced the constant scaling factor α to a time proportion function α(t) firstly. Secondly access output chaotic signals whose ratio is different from the original drive system via choosing different α(t). The FPS increase the difficulties of interception of outputting the chaotic signals, which make the chaotic secure communication more secure. In recent years, people pay more and more attention to the fractional-order chaotic systems. [10 16] People found that many systems such as fractional-order Chua system, [10] fractional-order Chen system, [11] fractionalorder Rössler system [12] and fractional-order hyperchaotic Rössler system [13] showed chaotic state when the value of the order is very low. Now, the fractional-order chaotic systems theory has been widely used in the related field such as oscillation, [17] turbulence, [18] control, [19] and many other fields. Integral-order is a special case of fractional. Fractional-order chaotic systems can describe the actual physical system more accurately. Therefore, the fractional-order chaotic systems have important theoretical value and application prospect. Recently, some scholars proposed a number of FPS methods [20 27] about integral-order chaotic systems, but there are less researched to fractional-order chaotic systems. In this paper, we study the FPS of the fractional-order hyperchaotic systems when the systems have parameters perturbation. Firstly, select the fractional hyperchaotic system as the drive system, and then construct the corresponding response system using the OPCL coupling method. [28 30] Finally, the FPS of the drive-response systems can be implemented. Take the parameters perturbation fractional- Supported by National Natural Science Foundation of China under Grant Nos. 60573172, 60973152, and Doctoral Program Foundation of Institution of Higher Education of China under Grant No. 20070141014, and the Natural Science Foundation of Liaoning Province of China under Grant No. 20082165 E-mail: wangxy@dlut.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

618 Communications in Theoretical Physics Vol. 55 order hyperchaotic Lü system as an example to simulate and the numerical simulations prove the effectiveness and robust of this scheme. 2 Method Introductions 2.1 Fractional-Order Differential System Currently, fractional-order differential system has not been formed a unified definition. In this paper, we will introduce the most common equation, which is the definition called Riemann Liouville (RL), the mathematical expression is: D α x(t) = J m α x (m) (t) (α > 0), here, m is the smallest integer, which is bigger or equal than α, x (m) represents the m derivative in the common sense. J β (β > 0) is β derivative Reimann Liouville integral operator, here we have J β x(t) = 1 Γ(β) t 0 (t τ) β 1 x(τ)dτ, Γ is the Gamma Function. D α is α derivative Caputo differential operator. [31] 2.2 OPCL Function As to the fractional-order chaotic systems, we will briefly describe how to construct the corresponding response system through the OPCL coupling method. Assume that the fractional-order chaotic system in the drive part is as follows d q x dt q = f(x) + f(x), x Rn, (1) here, f(x) is the perturbation part of parameters. q (0, 1) is the order of fractional-order system. If the system parameters have not been disturbed in the theory, we set zero to the value of f(x). Through the procedure of coupling, we can construct the fractionalorder responsive system as d q y dt q = f(y) + D(y, g), y Rn. (2) The coupling function is: ( D(y, g) = ġ f(g) + H f(g) ) (y g), (3) here, f(g)/ is the Jacobian matrix of the dynamic system. H (n n) is an arbitrary Hurwitz matrix constant, whose eigenvalues are negative. Through the equation and the deviation of g in g = α(t)x we can get ġ = α(t)ẋ + α(t)x, which α(t) is any scaling function, and it should meet continuously differentiable and limited boundary in [0, ], when α(t) is a constant of 1 or 1, the system is the complete synchronization or antisynchronization. Define the error dynamic system equation as e = y g, if we have lim y g = 0. t Then, we say that the drive-response systems achieve the FPS. Written f(y) as the form of Taylor series f(y) = f(g) + f(g) (x g). (4) Through the derivation of the error dynamic equation can get the following equation ė = ẏ ġ = f(y) + D(y, g) ġ. (5) Put Eqs. (3) and (4) into Eq. (5), we can obtain the result ė = H(x g) = He. (6) H is Hurwitz matrix and its eigenvalues are negative. So at time t we can get e 0 In this way, we can realize the FPS of the drive-response systems. 3 System Descriptions In this paper, we use Lü system as an example. The models of Lü system are at least two types, such as the third-order system model in [32], [33] and the fourth-order system model in [34]. Because the fourth-order chaotic system is better than the third-order system in the research field, in this paper we use the fourth-order chaotic system model ẋ 1 = a(x 2 x 1 ), ẋ 2 = bx 1 kx 1 x 3 + x 4, ẋ 3 = cx 3 + hx 2 1, ẋ 4 = dx 1, (7) here, a, b, c, d, h, and k are all the system s control parameters. When the parameters a = 10, b = 40, c = 2.5, d = 10.6, k = 1, and h = 4, the system (7) is in the situation of hyperchaotic. [34] The systems part of threedimensional and two-dimensional hyperchaotic attractors projective are shown in Fig. 1. We set fractional-order hyperchaotic Lü system model with parameters perturbation based on Eq. (7) as: d q x 1 dt q = (a + a)(x 2 x 1 ), d q x 2 dt q = (b + b)x 1 (k + k)x 1 x 3 + x 4, d q x 3 dt q = (c + c)x 3 + (h + h)x 2 1, d q x 4 dt q = (d + d)x 1. (8) Here, the values of the system control parameters a, b, c, d, h, and k are the same as Eq. (7). a, b, c, d, h, and k are the perturbation parts in the parameters. q is the system order. Assume that the system parameters perturbation is zero, when the system order q is different, the attractors projective of x 2 -x 3 plane are as shown in Fig. 2. Especially when q = 0.605 650 76, the fractional-order hyperchaotic Lü system s attractor is shown in Fig. 2(a), which is a limited cycle, it illustrates the system is in cycle state. And the equation (8) is no solution when q < 0.605 650 76. When q is increased from 0.605 650 76, the fractional-order hyperchaotic Lü system will be from cycle state to chaos state.

No. 4 Communications in Theoretical Physics 619 Fig. 1 Attractors projective of hyperchaotic Lü system: (a) x 4-x 1-x 2; (b) x 4-x 2-x 3; (c) x 2-x 3. Fig. 2 Attractors projective of x 2-x 3 plane of fractional-order Lü system with different system order q: (a) q = 0.605 650 76; (b) q = 0.75; (c) q = 0.799 999; (d) q = 0.9; (e) q = 0.99. Simplicity, we just do the research on the order of q = 0.99 in this paper, and its attractor projective of x 2 -x 3 plane is shown in Fig. 2(e). Figure 3 is the corresponding Lyapunov exponential spectrum. From Fig. 3 we can obviously see: the fractional-order hyperchaotic Lü system has two positive exponentials, so the system is in chaos state right now. Define Eq. (8) as the drive system and construct the corresponding response system by OPCL coupling method. The Jacobian matrix of system (8) is a a 0 0 f(x) x = b kx 3 0 kx 1 1 2hx 1 0 c 0. (9) d 0 0 0 Define the Hurwitz matrix H as the unit negative matrix I, then H f(g) = a 1 a 0 0 kαx 3 b 1 kαx 1 1 2hαx 1 0 c 1 0 d 0 0 1. (10) Therefore, the response system (after coupled structure) is as follows d q y 1 dt q = a(y 2 y 1 ) + (a 1)e 1 ae 2 + αx 1 + a(x 2 x 1 ), d q y 2 dt q = by 1 ky 1 y 3 + y 4 + (kαx 3 b)e 1 e 2 + kαx 1 e 3 e 4 + kx 1 x 3 α(α 1) + αx 2 + bx 1 kx 1 x 3,

620 Communications in Theoretical Physics Vol. 55 d q y 3 dt q = cy 3 + hy 2 1 2hαx 1 e 1 + (c 1)e 3 + αhx 2 1 (1 α) + αx 3 cx 3 + hx 2 1, d q y 4 dt q = dy 1 + de 1 e 4 + αx 4 dx 1. (11) Fig. 3 Lyapunov exponential spectrum when q = 0.99. 4 Numerical Simulations 4.1 No Parameters Perturbation When system parameters are not disturbed, it is to say the disturbance parameters a, b, c, d, h, and k are all zero. Select (x 1, x 2, x 3, x 4 ) = ( 1, 3, 0.2, 0.5) and (y 1, y 2, y 3, y 4 ) = (0.5, 1, 2, 3) as the initial values of the drive system (8) and response system (11) respectively. Time step t is 0.01 seconds. The proportion function α(t) = 20 + cos(0.2πt + 25). The numerical simulation results are shown in Fig. 4. From Fig. 4, we can obviously see that the drive system (8) and response system (11) have achieved FPS. 4.2 With Parameters Perturbation When system parameters are disturbed and the disturbance parameters are a = b = k = 0, c = 3, d = 5, and h = 5. Select the initial values of drive system (8) as (x 1, x 2, x 3, x 4 ) = ( 0.3, 0.7, 0.2, 0.5), and the response system (11) as (y 1, y 2, y 3, y 4 ) = (0.5, 0.8, 0.2, 0.6). Time step t is 0.01 seconds. Fig. 4 The simulation results of FPS of drive system (8) and response system (11) when there is no parameters perturbation: (a) Time series of state variable x 1 and y 1; (b) Time series of state variable x 3 and y 3; (c) FPS errors. Fig. 5 The simulation results of FPS of drive system (8) and response system (11) with parameters perturbation: (a) Time series of state variable x 1 and y 1; (b) Time series of state variable x 3 and y 3; (c) FPS errors. The proportion function α(t) = 25 + 2 cos(0.2πt + 20). And then the numerical simulation results are shown in Fig. 5. From Fig. 5 we can see that the drive system (8) and response system (11) have achieved FPS.

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