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International Journal of Advances in Science and Technology, Vol. 2, No.4, 2011 Global Chaos Synchronization of Shimizu-Morioka and Liu-Chen Chaotic Systems by Active Nonlinear Control Dr. V. Sundarapandian1 1 R & D Centre, Vel Tech Dr. RR & Dr. SR Technical University, Avadi-600 062, Chennai, INDIA sundarvtu@gmail.com Abstract This paper investigates the global chaos synchronization of identical Shimizu-Morioka systems (1980), identical Liu-Chen systems (2004) and non-identical Shimizu-Morioka and Liu-Chen systems. In this paper, active nonlinear control has been deployed to synchronize the identical and non-identical Shimizu-Morioka and Liu-Chen chaotic systems and our stability results have been established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active nonlinear control method is effective and convenient to synchronize Shimizu-Morioka and Liu-Chen chaotic systems. Numerical simulations are shown to demonstrate the effectiveness of the synchronization schemes derived in this paper for Shimizu-Morioka and Liu- Chen chaotic systems. Keywords: chaos synchronization, active control, Shimizu-Morioka system, Liu-Chen system. 1. Introduction Chaotic systems are dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly known as the butterfly effect [1]. Chaos synchronization problem was first described by Fujisaka and Yemada [2] in 1983. This problem did not receive great attention until Pecora and Carroll [3-4] published their results on chaos synchronization in early 1990s. From then on, chaos synchronization has been extensively and intensively studied in the last three decades [3-22]. Chaos theory has been explored in a variety of fields including physical systems [5], chemical systems [6] and ecological systems [7], secure communications [8-10] etc. Synchronization of chaotic systems is a phenomenon that may occur when two or more chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator. Because of the butterfly effect which causes the exponential divergence of the trajectories of two identical chaotic systems started with nearly the same initial conditions, synchronizing two chaotic systems is seemingly a very challenging problem. In most of the chaos synchronization approaches, the master-slave or drive-response formalism is used. If a particular chaotic system is called the master or drive system and another chaotic system is called the slave or response system, then the idea of the synchronization is to use the output of the master system to control the slave system so that the output of the slave system tracks the output of the master system asymptotically. Since the seminal work by Pecora and Carroll [3-4], a variety of impressive approaches have been proposed for the synchronization for the chaotic systems such as PC method [3-4], the sampled-data feedback synchronization method [10-11], OGY method [12], time-delay feedback approach [13], backstepping design method [14], adaptive design method [15-19], sliding mode control method [20], Lyapunov stability theory method [21], hyperchaos [22], etc. In this paper, we derive new results for the global chaos synchronization for identical and different Shimizu-Morioka and Liu-Chen chaotic systems using active nonlinear control. Explicitly, using active control and Lyapunov stability theory, we achieve global chaos synchronization for identical Shimizu- Morioka, identical Liu-Chen systems and non-identical Shimizu-Morioka and Liu-Chen systems. April Issue Page 11 of 95 ISSN 2229 5216 International Journal of Advances in Science and Technology This paper has been organized as follows. In Section 2, we give the problem statement and our methodology. In Section 3, we discuss the chaos synchronization of two identical Shimizu-Morioka chaotic systems ([23], 1980). In Section 4, we discuss the chaos synchronization of two identical Liu- Chen chaotic systems ([24], 2004). In Section 5, we discuss the chaos synchronization of Shimizu- Morioka and Liu-Chen chaotic systems. In Section 6, we summarize the main results of this paper. 2. Problem Statement and Our Methodology Consider the chaotic system described by the dynamics x Ax f ( x) (1) where x R is the state of the system, A is the n n matrix of the system parameters and n f : Rn Rn is the nonlinear part of the system. We consider the system (1) as the master or drive system. As the slave or response system, we consider the following chaotic system described by the dynamics y By g ( y) u (2) where y R is the state of the system, B is the n n matrix of the system parameters, n g : Rn Rn is the nonlinear part of the system and u Rn is the controller of the slave system. If A B and f g , then x and y are the states of two identical chaotic systems. If A B and f g , then x and y are the states of two different chaotic systems. In the nonlinear feedback control approach, we design a feedback controller u , which synchronizes the states of the master system (1) and the slave system (2) for all initial conditions x(0), z (0) R . n If we define the synchronization error as e y x, (3) then the synchronization error dynamics is obtained as e By Ax g ( y) f ( x) u (4) Thus, the global synchronization problem is essentially to find a feedback controller u so as to stabilize the error dynamics (4) for all initial conditions e(0) R . n Hence, we find a feedback controller u so that lim e(t ) 0 for all e(0) Rn . (5) t We take as a candidate Lyapunov function V (e) eT Pe, where P is a positive definite matrix. Note that V : Rn R is a positive definite function by construction. We assume that the parameters of the master and slave system are known and that the states of both systems (1) and (2) are measurable. If we find a feedback controller u so that V (e) eT Qe, where Q is a positive definite matrix, then V : Rn R is a negative definite function. Thus, by Lyapunov stability theory [25], the error dynamics (4) is globally exponentially stable and hence the condition (5) will be satisfied. Hence, the states of the master system (1) and the slave system (2) will be globally and exponentially synchronized. April Issue Page 12 of 95 ISSN 2229 5216 International Journal of Advances in Science and Technology 3. Synchronization of Identical Shimizu-Morioka Systems 3.1 Theoretical Results In this section, we apply the nonlinear control technique for the synchronization of two identical Shimizu-Morioka systems ([23], 1980). The Shimizu-Morioka system is a well-known three- dimensional chaotic system. Thus, the master system is described by the Shimizu-Morioka dynamics x1 x2 x2 x1 x2 x1 x3 (6) x3 x3 x12 where x1 , x2 , x3 are the states of the system and 0, 0 are constant parameters of the system. The slave system is described by the controlled Shimizu-Morioka dynamics y1 y2 u1 y2 y1 y2 y1 y3 u2 (7) y3 y3 y12 u3 where y1 , y2 , y3 are the states of the system and u1 , u2 , u3 are the active nonlinear controllers to be designed. The Shimizu-Morioka system (6) is chaotic when 0.75 and 0.45. Figure 1 depicts the chaotic state portrait of the Shimizu-Morioka system (6). Figure 1. State Orbits of the Shimizu-Morioka System The synchronization error e is defined by ei yi xi , (i 1, 2,3) (8) April Issue Page 13 of 95 ISSN 2229 5216 International Journal of Advances in Science and Technology The error dynamics is obtained as e1 e2 u1 e2 e1 e2 y1 y3 x1 x3 u2 (9) e3 e3 y12 x12 u3 We choose the nonlinear controller as u1 e1 e2 u2 e1 y1 y3 x1 x3 (10) u3 y12 x12 Substituting (10) into (9), we obtain the linear system e1 e1 , e2 e2 , e3 e3 (11) We consider the quadratic Lyapunov function defined by V (e) e1 e2 e3 , 1 2 2 2 2 (12) which is a positive definite function on R . 3 Differentiating (12) along the trajectories of (11), we get V (e) e12 e2 e3 , 2 2 (13) which is a negative definite function on R . 3 Thus, the error dynamics (11) is globally exponentially stable. Hence, we arrive at the following result. Theorem 1. The identical Shimizu-Morioka systems (6) and (7) are exponentially and globally synchronized for all initial conditions by the active nonlinear controller defined by (10). 3.2 Numerical Results 6 For simulations, the fourth-order Runge-Kutta method with time-step h 10 is used to solve the differential equations (6) and (7) with the active nonlinear controller (10). Figure 2. Synchronization of the Identical Shimizu-Morioka Systems April Issue Page 14 of 95 ISSN 2229 5216 International Journal of Advances in Science and Technology The parameters of the Shimizu-Morioka chaotic systems are chosen as 0.75 and 0.45. The initial conditions of the master and slave systems are chosen as x(0) (20,36,14) and y(0) (43,10, 26). Figure 2 shows the synchronization of the master system (6) and slave system (7). 4. Synchronization of Identical Liu-Chen Systems 4.1 Theoretical Results In this section, we apply the nonlinear control technique for the synchronization of two identical Liu-Chen systems ([24], 2004). The Liu-Chen system is a well-known three-dimensional chaotic system, known as a four-scroll chaotic attractor. Thus, the master system is described by the Liu-Chen dynamics x1 ax1 x2 x3 x2 bx2 x1 x3 (14) x3 cx3 x1 x2 where x1 , x2 , x3 are the states of the system and a, b, c are constant, positive parameters of the system such that b c a. The slave system is described by the controlled Liu-Chen dynamics y1 ay1 y2 y3 u1 y2 by2 y1 y3 u2 (15) y3 cy3 y1 y2 u3 where y1 , y2 , y3 are the states of the system and u1 , u2 , u3 are the active nonlinear controllers to be designed. The Liu-Chen system (14) is chaotic when a 0.4, b 12 and c 5 Figure 3. State Orbits of the Liu-Chen Four-Scroll System April Issue Page 15 of 95 ISSN 2229 5216 International Journal of Advances in Science and Technology Figure 3 depicts the state orbits of the Liu-Chen four-scroll system (14). The synchronization error e is defined by e1 y1 x1 e2 y2 x2 (16) e3 y3 x3 The error dynamics is obtained as e1 ae1 y2 y3 x2 x3 u1 e2 be2 y1 y3 x1 x3 u2 (17) e3 ce3 y1 y2 x1 x2 u3 We choose the nonlinear controller as u1 (a 1)e1 y2 y3 x2 x3 u2 y1 y3 x1 x3 (18) u3 y1 y2 x1 x2 Substituting (18) into (17), we obtain the linear system e1 e1 e2 be2 (19) e3 ce3 We consider the quadratic Lyapunov function defined by V (e) e1 e2 e3 , 1 2 2 2 2 (20) which is a positive definite function on R . 3 Differentiating (20) along the trajectories of (19), we get V (e) e12 be2 ce3 , 2 2 (21) which is a negative definite function on R . 3 Thus, the error dynamics (19) is globally exponentially stable. Hence, we arrive at the following result. Theorem 2. The identical Liu-Chen systems (14) and (15) are exponentially and globally synchronized for all initial conditions by the active nonlinear controller defined by (18). 4.2 Numerical Results 6 For simulations, the fourth-order Runge-Kutta method with time-step h 10 is used to solve the differential equations (14) and (15) with the active nonlinear controller (18). The parameters of the Liu-Chen chaotic systems are chosen as a 0.4, b 12 and c 5 The initial conditions of the master system are chosen as x(0) (16, 22,18) The initial conditions of the slave system are chosen as y(0) (30,12, 24) Figure 4 shows the synchronization of the master system (14) and slave system (15). April Issue Page 16 of 95 ISSN 2229 5216 International Journal of Advances in Science and Technology Figure 4. Synchronization of the Identical Liu-Chen Systems 5. Synchronization of Shimizu-Morioka and Liu-Chen Systems 5.1 Theoretical Results In this section, we apply the active control method for the synchronization of non-identical chaotic systems, viz. Shimizu-Morioka system ([22], 1980) and Liu-Chen system ([23], 2004). We take the Shimizu-Morioka system as the master system and Liu-Chen system as the slave system for the global chaos synchronization. Thus, the master system is described by the Shimizu-Morioka dynamics x1 x2 x2 x1 x2 x1 x3 (22) x3 x3 x12 where x1 , x2 , x3 are the states of the system and , are constant, positive parameters of the system. The slave system is described by the controlled Liu-Chen dynamics y1 ay1 y2 y3 u1 y2 by2 y1 y3 u2 (23) y3 cy3 y1 y2 u3 where y1 , y2 , y3 are the states of the system and u1 , u2 , u3 are the active nonlinear controllers to be designed. April Issue Page 17 of 95 ISSN 2229 5216 International Journal of Advances in Science and Technology The synchronization error e is defined by e1 y1 x1 e2 y2 x2 (24) e3 y3 x3 The error dynamics is obtained as e1 ae1 ax1 x2 y2 y3 u1 e2 be2 x1 ( b) x2 y1 y3 x1 x3 u2 (25) e3 ce3 (c ) x3 x y1 y2 u3 2 1 We choose the nonlinear controller as u1 (a 1)e1 ax1 x2 y2 y3 u2 x1 ( b) x2 y1 y3 x1 x3 (26) u3 x (c ) x3 y1 y2 2 1 Substituting (26) into (25), we obtain the linear system e1 e1 e2 be2 (27) e3 ce3 We consider the quadratic Lyapunov function defined by V (e) e1 e2 e3 , 1 2 2 2 2 (28) which is a positive definite function on R . 3 Differentiating (28) along the trajectories of (27), we get V (e) e12 be2 ce3 , 2 2 (29) which is a negative definite function on R . 3 Thus, the error dynamics (27) is globally exponentially stable. Hence, we arrive at the following result. Theorem 3. The Shimizu-Morioka system (22) and the Liu-Chen system (23) are exponentially and globally synchronized for all initial conditions with the nonlinear controller defined by (26). 5.2 Numerical Results 6 For simulations, the fourth-order Runge-Kutta method with time-step h 10 is used to solve the differential equations (22) and (23) with the active nonlinear controller (26). The parameters of the Shimizu-Morioka and Liu-Chen chaotic systems are chosen as 0.75, 0.45, a 0.4, b 12 and c 5 The initial conditions of the master system are chosen as x(0) (33, 21,16) The initial conditions of the slave system are chosen as y(0) (14, 26,11) Figure 5 shows the synchronization of the master system (22) and slave system (23). April Issue Page 18 of 95 ISSN 2229 5216 International Journal of Advances in Science and Technology Figure 5. Synchronization of the Shimizu-Morioka and Liu-Chen Systems 6. Conclusions In this paper, we have used nonlinear control method and Lyapunov stability theory to achieve global chaos synchronization for the following three cases of chaotic systems: (A) Identical Shimizu-Morioka chaotic systems (1980) (B) Identical Liu-Chen chaotic systems (2004) (C) Non-Identical Shimizu-Morioka and Liu-Chen systems. Since the Lyapunov exponents are not needed for these calculations, the nonlinear control method is very effective and convenient to achieve global chaos synchronization for the three cases of chaotic systems studied in this paper. Numerical simulations have been shown to illustrate the effectiveness of the synchronization schemes derived in this paper. 7. References [1] K.T. Alligood, T. Aauer and J.A. Yorka, Chaos: An Introduction to Dynamical Systems, Springer, New York, 1997. [2] H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,” Progress of Theoretical Physics, vol. 69, pp. 32-47, 1983. [3] L.M. 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Yau, “Design of adaptive sliding mode controller for chaos synchronization with uncertainties,” Chaos, Solitons and Fractals, vol. 22, pp. 341-347, 2004 [21] X. Zhang and H. Zhu, “Anti-synchronization of two different hyperchaotic systems with active and adaptive control,” International Journal of Nonlinear Science, vol. 6, pp. 216-223, 2008. [22] R. Vicente, J. Dauden, P. Colet and R. Toral, “Analysis and characterization of the hyperchaos generated by semiconductor laser object,” IEEE J. Quantum Electronics, vol. 41, pp. 541-548, 2005. [23] T. Shimizu and N. Morioka, “On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model,” Phys. Lett. A, vol. 76, pp. 201-204, 1980. [24] W. Liu and G. Chen, “Can a three-dimensional smooth autonomous quadratic chaotic system generate a single four-scroll attractor?” Internat. J. Bifurcat. Chaos, vol. 14, pp. 1395-1403, 2004.. [25] W. Hahn, The Stability of Motion, Springer, Berlin, 1967. Authors Profile Dr. V. Sundarapandian received his D.Sc. degree from the Department of Electrical and Systems Engineering, Washington University, Saint Louis, USA in 1996. He is currently working as Professor (Systems & Control Engineering) at the Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University, Chennai. He has published two text-books (Numerical Linear Algebra, Probability, Statistics and Queueing Theory) with Prentice Hall of India, New Delhi. He has published over 100 refereed international journal papers and presented over 140 papers in National and International Conferences. April Issue Page 20 of 95 ISSN 2229 5216