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Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters
Applied Mathematics and Computation 215 (2009) 557–561
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters Congxu Zhu School of Information Science and Engineering, Central South University, Changsha 410083, China
a r t i c l e
i n f o
Keywords: Hyperchaotic system Adaptive control Chaos synchronization Lyapunov stability theorem
a b s t r a c t This paper investigates adaptive synchronization between two novel different hyperchaotic systems with partly uncertain parameters. Based on the Lyapunov stability theorem and the adaptive control theory, synchronization between these two hyperchaotic systems is achieved by proposing a new adaptive controller and a parameter estimation update law. Numerical simulations are presented to demonstrate the analytical results. 2009 Elsevier Inc. All rights reserved.
1. Introduction Since Rössler [1] first introduced the hyperchaotic dynamical system in 1979, many hyperchaotic systems have been proposed and studied in the last few decades, such as hyperchaotic Lorenz system [2], hyperchaotic Chen system [3], hyperchaotic Lü system [4], just to name a few. Hyperchaotic systems possess more complex dynamical behaviors than chaotic systems such as having more than one positive Lyapunov exponent, therefore, they have broader potential applications, particularly in secure communications. Chaos synchronization plays an important role for understanding the cooperative behavior in coupled chaotic oscillators [5]. Since Pecora and Carroll [6] introduced a method to synchronize two identical chaotic systems with different initial conditions, chaos synchronization has attracted a great deal of attention from various fields during the last two decades. A variety of approaches have been proposed for the synchronization of chaotic and hyperchaotic systems such as linear and nonlinear feedback synchronization methods [7,8], adaptive synchronization methods [9,10], backstepping design methods [11,12], and sliding mode control methods [13], etc. However, to our best knowledge, most of the methods mentioned above and many other existing synchronization methods mainly concern the synchronization of two identical chaotic or hyperchaotic systems, the methods of synchronization of two different chaotic or hyperchaotic systems are far from being straightforward because of their different structures and parameter mismatch. Moreover, most of the methods synchronize only two systems with exactly knowing of their structure and parameters. But in practical situations, some or all of the systems’ parameters cannot be exactly known in priori. As a result, more and more applications of chaos synchronization in secure communication have made it much more important to synchronize two different hyperchaotic systems with uncertain parameters in recent years. In this regard, some works on synchronization of two different hyperchaotic systems with uncertain parameters have been performed [14–16]. Recently, Chen et al. [17] proposed a novel hyperchaotic system by introducing state feedback control and constant multipliers to the two quadratic terms in the system reported in [18]. This novel hyperchaotic system takes the following form:
E-mail address: [email protected] 0096-3003/$ - see front matter 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.05.026
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C. Zhu / Applied Mathematics and Computation 215 (2009) 557–561
8 x_ ¼ aðy xÞ þ eyz; > > > < y_ ¼ cx dxz þ y þ w; > z_ ¼ xy bz; > > : _ ¼ gy; w
ð1Þ
where x; y; z and w are state variables, and a; b; c; d; e and g are constant parameters. System (1) has only one unstable equilibrium O(0, 0, 0, 0) and has bigger positive Lyapunov exponents than those already known hyperchaotic systems. It can generate complex dynamics within wide parameter ranges, including periodic orbit, quasi-periodic orbit, chaos and hyperchaos. In particular, when a = 35; b = 4.9; c = 25; d = 5; e = 35 and varying g from 10 to 126, or a = 35; c = 25; d = 5; e = 35; g = 100 and ranging b between 3.8 and 11, system (1) exhibits hyperchaos. This hyperchaotic attractor is given in Fig. 1. Very recently, Wang et al. [19] generated another new hyperchaotic system from Lorenz system. The new hyperchaotic system is described by
8 x_ ¼ hðy xÞ þ w; > > > < y_ ¼ px y xz; > z_ ¼ xy kz; > > : _ ¼ yz þ rw; w
ð2Þ
where h, k, p and r are constant parameters. When h = 10, k = 8/3, p = 28 and r 2 (1.52, 0.06), system (2) has two positive Lyapunov exponents k1 = 0. 3381 and k2 = 0. 1586. Thus, system (2) shows hyperchaotic behavior. This hyperchaotic attractor is given in Fig. 2. To our knowledge, synchronizing hyperchaotic systems (1) and (2) with uncertain parameters has not been reported. This paper presents the different structure synchronization between these two novel hyperchaotic systems aforementioned with partly uncertain parameters. On the basis of the Lyapunov stability theory, we design a new adaptive synchronization controller with a novel parameter update law. With this adaptive controller, one can synchronize the two hyperchaotic systems effectively and identify the system parameters accurately. The rest of the paper is organized as follows. Section 2 presents hyperchaos synchronization between hyperchaotic system (1) and (2) via adaptive control. Section 3 provides a numerical example to demonstrate the effectiveness of the proposed method. Section 4 concludes the paper.
Fig. 1. Views of the hyperchaotic attractor of the system (1).
Fig. 2. Views of the hyperchaotic attractor of the system (2).
C. Zhu / Applied Mathematics and Computation 215 (2009) 557–561
559
2. Adaptive synchronization In order to observe synchronization behavior between the two hyperchaotic systems, we assume that the hyperchaotic system (1) is the drive system whose variables are denoted by subscript 1 and the parameters b and g cannot be exactly known in priori. The drive systems is described by the following equation:
8 x_ 1 ¼ aðy1 x1 Þ þ ey1 z1 ; > > > < y_ 1 ¼ cx1 dx1 z1 þ y1 þ w1 ; > > z_ 1 ¼ x1 y1 bz1 ; > : _ 1 ¼ gy1 : w
ð3Þ
The hyperchaotic system (2) is the response system whose variables are denoted by subscript 2 and the parameter r cannot be exactly known in priori. The response systems is described by the following equation:
8 x_ 2 ¼ hðy2 x2 Þ þ w2 þ u1 ; > > > < y_ 2 ¼ px2 y2 x2 z2 þ u2 ; > z > _ 2 ¼ x2 y2 kz2 þ u3 ; > : _ 2 ¼ y2 z2 þ rw2 þ u4 ; w
ð4Þ
where u1, u2, u3, u4 are four control functions to be designed. Without loss of generality, we only consider the parameters b, g and r as uncertain for the simple reason that they can generate hyperchaotic behaviors in systems (3) and (4) when varying in wide parameter ranges. Hence, it is more corresponded to engineering practice to consider the parameters b, g and r as uncertain while did not consider the others. In order to determine the control functions to realize synchronization between systems (3) and (4), we subtract (3) from (4) and get
8 e_ 1 > > > < e_ 2 > _3 e > > : e_ 4
¼ hðy2 x2 Þ aðy1 x1 Þ ey1 z1 þ w2 þ u1 ; ¼ px2 cx1 x2 z2 þ dx1 z1 y2 y1 w1 þ u2 ; ¼ x2 y2 kz2 x1 y1 þ bz1 þ u3 ;
ð5Þ
¼ y2 z2 þ rw2 þ gy1 þ u4 ;
where e1 = x2 x1, e2 = y2 y1, e3 = z2 z1, e4 = w2 w1. The goal of the control is to find an effective controller U = [u1, u2, u3, u4]T with a parameter estimation update law such that the response system (4) asymptotically synchronizes the drive system (3), i.e., limt?1jje(t)jj = 0, where e(t) = [e1, e2, e3, e4]T.
Fig. 3. Dynamics of the variables e1, e2, e3 and e4 for error system (8) with time t.
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For two systems (3) and (4) without controls (ui = 0, i = 1, 2, 3, 4), if the initial condition (x1(0), y1(0), z1(0), w1(0)) – (x2(0), y2(0), z2(0), w2(0)), then the trajectories of two systems will quickly separate from each other and become irrelevant. However, when controls are applied, the two systems will approach synchronization for any initial conditions through appropriate control functions. The following theorem shows that hyperchaotic systems (3) and (4) can be synchronized effectively by the following proposed adaptive controller. Theorem. Hyperchaotic system (3) and (4) can be synchronized globally and asymptotically for any different initial condition with the following adaptive controller:
8 u1 > > > u 3 > > : u4
¼ ðh 1Þðx2 y2 Þ þ ða 1Þðy1 x1 Þ þ ey1 z1 w2 ; ¼ ðc pÞx1 þ 2y1 ðd 1Þx1 z1 þ e1 z1 þ w1 ðp þ 1Þe1 ; ¼ kz2 e1 y1 b1 z1 þ e2 e4 e3 ;
ð6Þ
¼ y1 z1 þ e2 z1 þ e3 y1 r 1 w2 g 1 y1 e4 ;
and the following parameter estimation adaptive update law:
8 _ > < b1 ¼ z1 e3 ; g_ 1 ¼ y1 e4 ; > : r_ 1 ¼ ðw1 þ e4 Þe4 ;
ð7Þ
where b1, g1, r1 are the parameter estimations of b, g, r, respectively. Proof. Substituting (6) into (5) leads to the following error system:
8 e_ 1 > > > < e_ 2 > e_ 3 > > : e_ 4
¼ e1 þ e2 ; ¼ e1 e2 x1 e3 e1 e3 ; ~ 1 þ x1 e2 þ e1 e2 þ e2 e4 e3 ; ¼ bz
ð8Þ
¼ ~r ðw1 þ e4 Þ þ g~ y1 e2 e3 e4 ;
~ ¼ b b1 , g~ ¼ g g , ~r ¼ r r1 . Consider the following Lyapunov function: where b 1
VðtÞ ¼
1 2 ~2 þ g~2 þ ~r 2 : e1 þ e22 þ e23 þ e24 þ b 2
The time derivative of V is as follows:
Fig. 4. Estimated values of three parameters with parameter update law (7).
ð9Þ
C. Zhu / Applied Mathematics and Computation 215 (2009) 557–561
~b ~_ þ g~g~_ þ ~r~r_ ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3 þ e4 e_ 4 þ bð ~ b_ 1 Þ þ g~ðg_ 1 Þ þ ~r ðr_ 1 Þ: _ VðtÞ ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3 þ e4 e_ 4 þ b
561
ð10Þ
Inserting (7) and (8) into (10) yields the following result:
~ 1 þ x1 e2 þ e1 e2 þ e2 e4 e3 _ VðtÞ ¼ e1 ðe1 þ e2 Þ þ e2 ðe1 e2 x1 e3 e1 e3 Þ þ e3 bz ~ 1 e3 Þ g~ðy e4 Þ ~rðw1 e4 þ e2 Þ þ e4 ðw1~r þ e4~r þ g~y1 e2 e3 e4 Þ bðz 1 4 ¼ e21 e22 e23 e24 < 0:
ð11Þ
Since the Lyapunov function (9) is positive definite and its derivative is negative definite in the neighborhood of the zero solution of the error system (5). According to the Lyapunov stability theorem, the error dynamical system (5) can converge to the origin asymptotically, i.e., limt?1jje(t)jj = 0, "a, b, c, d, e, g, h, k, p, r 2 R, where e(t) = [e1, e2, e3, e4]T. Therefore, the synchronization between two different novel hyperchaotic systems is achieved with the adaptive controller (6) and the parameter estimation adaptive update law (7). This completes the proof. h
3. Numerical simulation In this section, Numerical simulation is given to show the effectiveness and feasibility of the proposed synchronization method. In the numerical simulation, the Runge–Kutta–Fehlberg algorithm with adaptive varied time step is used to solve the augment system consisting of (3), (7) and (8). The parameters are chosen to be a = 35, b = 5, c = 25, d = 5, e = 35, g = 10, h = 10, k = 8/3, p = 28 and r = 1 in the simulation so that system (3) and (4) exhibit hyperchaotic behaviors if no controls are applied. The initial values of the drive and response systems are (x1(0), y1(0), z1(0), w1(0)) = (10, 2, 3, 10) and (x2(0), y2(0), z2(0), w2(0)) = (14.5, 4.5, 6.5, 8.5), respectively; hence the initial errors of system (8) are (e1(0), e2(0), e3(0), e4(0)) = (4.5, 2.5, 3.5, 1.5). Moreover, the initial values of the estimated parameters are chosen as b1 = 3.8, g1 = 20 and r1 = 0.5. The simulation results are illustrated in Figs. 3 and 4. Fig. 3 shows that e1, e2, e3 and e4 tend to zero, which demonstrates that the response system (4) synchronizes the drive system (3). Fig. 4 shows that the estimated values b1, g1 and r1 of the uncertain parameters converge to b = 5, g = 10, and r = 1, respectively, as t ? 1. 4. Conclusion This paper addresses the problem of adaptive synchronization of two different new hyperchaotic systems with some uncertain parameters. On the basis of the Lyapunov stability theory and the adaptive control theory, a new adaptive synchronization control law and a novel parameter estimation update law are proposed to achieve synchronization between the two novel different hyperchaotic systems with uncertain parameters. Numerical simulations are given to demonstrate the effectiveness of the proposed synchronization scheme and verify the theoretical results. Acknowledgements This work was supported by the Chinese Provincial Natural Science Foundation of Hunan province (No: 06JJ5098). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
O.E. Rösler, An equation for hyperchaos, Phys. Lett. A 71 (1979) 155–157. Y. Li, W.K.S. Tang, G. Chen, Hyperchaos evolved from the generalized Lorenz equation, Int. J. Circuit Theory Appl. 33 (2005) 235–251. Y. Li, W.K.S. Tang, G. Chen, Generating hyperchaos via state feedback control, Int. J. Bifur. Chaos 15 (2005) 3367–3375. A. Chen, J. Lu, J. Lü, S. Yu, Generating hyperchaotic Lü attractor via state feedback control, Physica A: Stat. Mech. Appl. 364 (2006) 103–110. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001. L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. Z. Yan, Controlling hyperchaos in the new hyperchaotic Chen system, Appl. Math. Comput. 168 (2005) 1239–1250. F. Wang, C. Liu, A new criterion for chaos and hyperchaos synchronization using linear feedback control, Phys. Lett. A 360 (2006) 274–278. Q. Jia, Adaptive control and synchronization of a new hyperchaotic system with unknown parameters, Phys. Lett. A 362 (2007) 424–429. X. Zhou, Y. Wu, Y. Li, H. Xue, Adaptive control and synchronization of a novel hyperchaotic system with uncertain parameters, Appl. Math. Comput. 203 (2008) 80–85. H. Zhang, X. Ma, M. Li, J. Zou, Controlling and tracking hyperchaotic Rösler system via active backstepping design, Chaos Soliton Fract. 26 (2005) 353– 361. G.H. Li, S.P. Zhou, K. Yang, Generalized projective synchronization between two different chaotic systems using active backstepping control, Phys. Lett. A 355 (2006) 326–330. M.-J. Jang, C.-L. Chen, C.-K. Chen, Sliding mode control of hyperchaos in Rösler systems, Chaos Soliton Fract. 14 (2002) 1465–1476. M.T. Yassen, Synchronization hyperchaos of hyperchaotic systems, Chaos Soliton Fract. 37 (2008) 465–475. J. Huang, Adaptive synchronization between different hyperchaotic systems with fully uncertain parameters, Phys. Lett. A 372 (2008) 4799–4804. U.E. Vincent, Synchronization of identical and non-identical 4-D chaotic systems using active control, Chaos Soliton Fract. 37 (2008) 1065–1075. Z. Chen, Y. Yang, G. Qi, Z. Yuan, A novel hyperchaos system only with one equilibrium, Phys. Lett. A 360 (2007) 696–701. G. Qi, G. Chen, S. Du, Z. Chen, Z. Yuan, Analysis of a new chaotic system, Physica A: Stat. Mech. Appl. 352 (2005) 295–308. X. Wang, M. Wang, A hyperchaos generated from Lorenz system, Physica A: Stat. Mech. Appl. 387 (2008) 3751–3758.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters Congxu Zhu School of Information Science and Engineering, Central South University, Changsha 410083, China
a r t i c l e
i n f o
Keywords: Hyperchaotic system Adaptive control Chaos synchronization Lyapunov stability theorem
a b s t r a c t This paper investigates adaptive synchronization between two novel different hyperchaotic systems with partly uncertain parameters. Based on the Lyapunov stability theorem and the adaptive control theory, synchronization between these two hyperchaotic systems is achieved by proposing a new adaptive controller and a parameter estimation update law. Numerical simulations are presented to demonstrate the analytical results. 2009 Elsevier Inc. All rights reserved.
1. Introduction Since Rössler [1] first introduced the hyperchaotic dynamical system in 1979, many hyperchaotic systems have been proposed and studied in the last few decades, such as hyperchaotic Lorenz system [2], hyperchaotic Chen system [3], hyperchaotic Lü system [4], just to name a few. Hyperchaotic systems possess more complex dynamical behaviors than chaotic systems such as having more than one positive Lyapunov exponent, therefore, they have broader potential applications, particularly in secure communications. Chaos synchronization plays an important role for understanding the cooperative behavior in coupled chaotic oscillators [5]. Since Pecora and Carroll [6] introduced a method to synchronize two identical chaotic systems with different initial conditions, chaos synchronization has attracted a great deal of attention from various fields during the last two decades. A variety of approaches have been proposed for the synchronization of chaotic and hyperchaotic systems such as linear and nonlinear feedback synchronization methods [7,8], adaptive synchronization methods [9,10], backstepping design methods [11,12], and sliding mode control methods [13], etc. However, to our best knowledge, most of the methods mentioned above and many other existing synchronization methods mainly concern the synchronization of two identical chaotic or hyperchaotic systems, the methods of synchronization of two different chaotic or hyperchaotic systems are far from being straightforward because of their different structures and parameter mismatch. Moreover, most of the methods synchronize only two systems with exactly knowing of their structure and parameters. But in practical situations, some or all of the systems’ parameters cannot be exactly known in priori. As a result, more and more applications of chaos synchronization in secure communication have made it much more important to synchronize two different hyperchaotic systems with uncertain parameters in recent years. In this regard, some works on synchronization of two different hyperchaotic systems with uncertain parameters have been performed [14–16]. Recently, Chen et al. [17] proposed a novel hyperchaotic system by introducing state feedback control and constant multipliers to the two quadratic terms in the system reported in [18]. This novel hyperchaotic system takes the following form:
E-mail address: [email protected] 0096-3003/$ - see front matter 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.05.026
558
C. Zhu / Applied Mathematics and Computation 215 (2009) 557–561
8 x_ ¼ aðy xÞ þ eyz; > > > < y_ ¼ cx dxz þ y þ w; > z_ ¼ xy bz; > > : _ ¼ gy; w
ð1Þ
where x; y; z and w are state variables, and a; b; c; d; e and g are constant parameters. System (1) has only one unstable equilibrium O(0, 0, 0, 0) and has bigger positive Lyapunov exponents than those already known hyperchaotic systems. It can generate complex dynamics within wide parameter ranges, including periodic orbit, quasi-periodic orbit, chaos and hyperchaos. In particular, when a = 35; b = 4.9; c = 25; d = 5; e = 35 and varying g from 10 to 126, or a = 35; c = 25; d = 5; e = 35; g = 100 and ranging b between 3.8 and 11, system (1) exhibits hyperchaos. This hyperchaotic attractor is given in Fig. 1. Very recently, Wang et al. [19] generated another new hyperchaotic system from Lorenz system. The new hyperchaotic system is described by
8 x_ ¼ hðy xÞ þ w; > > > < y_ ¼ px y xz; > z_ ¼ xy kz; > > : _ ¼ yz þ rw; w
ð2Þ
where h, k, p and r are constant parameters. When h = 10, k = 8/3, p = 28 and r 2 (1.52, 0.06), system (2) has two positive Lyapunov exponents k1 = 0. 3381 and k2 = 0. 1586. Thus, system (2) shows hyperchaotic behavior. This hyperchaotic attractor is given in Fig. 2. To our knowledge, synchronizing hyperchaotic systems (1) and (2) with uncertain parameters has not been reported. This paper presents the different structure synchronization between these two novel hyperchaotic systems aforementioned with partly uncertain parameters. On the basis of the Lyapunov stability theory, we design a new adaptive synchronization controller with a novel parameter update law. With this adaptive controller, one can synchronize the two hyperchaotic systems effectively and identify the system parameters accurately. The rest of the paper is organized as follows. Section 2 presents hyperchaos synchronization between hyperchaotic system (1) and (2) via adaptive control. Section 3 provides a numerical example to demonstrate the effectiveness of the proposed method. Section 4 concludes the paper.
Fig. 1. Views of the hyperchaotic attractor of the system (1).
Fig. 2. Views of the hyperchaotic attractor of the system (2).
C. Zhu / Applied Mathematics and Computation 215 (2009) 557–561
559
2. Adaptive synchronization In order to observe synchronization behavior between the two hyperchaotic systems, we assume that the hyperchaotic system (1) is the drive system whose variables are denoted by subscript 1 and the parameters b and g cannot be exactly known in priori. The drive systems is described by the following equation:
8 x_ 1 ¼ aðy1 x1 Þ þ ey1 z1 ; > > > < y_ 1 ¼ cx1 dx1 z1 þ y1 þ w1 ; > > z_ 1 ¼ x1 y1 bz1 ; > : _ 1 ¼ gy1 : w
ð3Þ
The hyperchaotic system (2) is the response system whose variables are denoted by subscript 2 and the parameter r cannot be exactly known in priori. The response systems is described by the following equation:
8 x_ 2 ¼ hðy2 x2 Þ þ w2 þ u1 ; > > > < y_ 2 ¼ px2 y2 x2 z2 þ u2 ; > z > _ 2 ¼ x2 y2 kz2 þ u3 ; > : _ 2 ¼ y2 z2 þ rw2 þ u4 ; w
ð4Þ
where u1, u2, u3, u4 are four control functions to be designed. Without loss of generality, we only consider the parameters b, g and r as uncertain for the simple reason that they can generate hyperchaotic behaviors in systems (3) and (4) when varying in wide parameter ranges. Hence, it is more corresponded to engineering practice to consider the parameters b, g and r as uncertain while did not consider the others. In order to determine the control functions to realize synchronization between systems (3) and (4), we subtract (3) from (4) and get
8 e_ 1 > > > < e_ 2 > _3 e > > : e_ 4
¼ hðy2 x2 Þ aðy1 x1 Þ ey1 z1 þ w2 þ u1 ; ¼ px2 cx1 x2 z2 þ dx1 z1 y2 y1 w1 þ u2 ; ¼ x2 y2 kz2 x1 y1 þ bz1 þ u3 ;
ð5Þ
¼ y2 z2 þ rw2 þ gy1 þ u4 ;
where e1 = x2 x1, e2 = y2 y1, e3 = z2 z1, e4 = w2 w1. The goal of the control is to find an effective controller U = [u1, u2, u3, u4]T with a parameter estimation update law such that the response system (4) asymptotically synchronizes the drive system (3), i.e., limt?1jje(t)jj = 0, where e(t) = [e1, e2, e3, e4]T.
Fig. 3. Dynamics of the variables e1, e2, e3 and e4 for error system (8) with time t.
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C. Zhu / Applied Mathematics and Computation 215 (2009) 557–561
For two systems (3) and (4) without controls (ui = 0, i = 1, 2, 3, 4), if the initial condition (x1(0), y1(0), z1(0), w1(0)) – (x2(0), y2(0), z2(0), w2(0)), then the trajectories of two systems will quickly separate from each other and become irrelevant. However, when controls are applied, the two systems will approach synchronization for any initial conditions through appropriate control functions. The following theorem shows that hyperchaotic systems (3) and (4) can be synchronized effectively by the following proposed adaptive controller. Theorem. Hyperchaotic system (3) and (4) can be synchronized globally and asymptotically for any different initial condition with the following adaptive controller:
8 u1 > > > u 3 > > : u4
¼ ðh 1Þðx2 y2 Þ þ ða 1Þðy1 x1 Þ þ ey1 z1 w2 ; ¼ ðc pÞx1 þ 2y1 ðd 1Þx1 z1 þ e1 z1 þ w1 ðp þ 1Þe1 ; ¼ kz2 e1 y1 b1 z1 þ e2 e4 e3 ;
ð6Þ
¼ y1 z1 þ e2 z1 þ e3 y1 r 1 w2 g 1 y1 e4 ;
and the following parameter estimation adaptive update law:
8 _ > < b1 ¼ z1 e3 ; g_ 1 ¼ y1 e4 ; > : r_ 1 ¼ ðw1 þ e4 Þe4 ;
ð7Þ
where b1, g1, r1 are the parameter estimations of b, g, r, respectively. Proof. Substituting (6) into (5) leads to the following error system:
8 e_ 1 > > > < e_ 2 > e_ 3 > > : e_ 4
¼ e1 þ e2 ; ¼ e1 e2 x1 e3 e1 e3 ; ~ 1 þ x1 e2 þ e1 e2 þ e2 e4 e3 ; ¼ bz
ð8Þ
¼ ~r ðw1 þ e4 Þ þ g~ y1 e2 e3 e4 ;
~ ¼ b b1 , g~ ¼ g g , ~r ¼ r r1 . Consider the following Lyapunov function: where b 1
VðtÞ ¼
1 2 ~2 þ g~2 þ ~r 2 : e1 þ e22 þ e23 þ e24 þ b 2
The time derivative of V is as follows:
Fig. 4. Estimated values of three parameters with parameter update law (7).
ð9Þ
C. Zhu / Applied Mathematics and Computation 215 (2009) 557–561
~b ~_ þ g~g~_ þ ~r~r_ ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3 þ e4 e_ 4 þ bð ~ b_ 1 Þ þ g~ðg_ 1 Þ þ ~r ðr_ 1 Þ: _ VðtÞ ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3 þ e4 e_ 4 þ b
561
ð10Þ
Inserting (7) and (8) into (10) yields the following result:
~ 1 þ x1 e2 þ e1 e2 þ e2 e4 e3 _ VðtÞ ¼ e1 ðe1 þ e2 Þ þ e2 ðe1 e2 x1 e3 e1 e3 Þ þ e3 bz ~ 1 e3 Þ g~ðy e4 Þ ~rðw1 e4 þ e2 Þ þ e4 ðw1~r þ e4~r þ g~y1 e2 e3 e4 Þ bðz 1 4 ¼ e21 e22 e23 e24 < 0:
ð11Þ
Since the Lyapunov function (9) is positive definite and its derivative is negative definite in the neighborhood of the zero solution of the error system (5). According to the Lyapunov stability theorem, the error dynamical system (5) can converge to the origin asymptotically, i.e., limt?1jje(t)jj = 0, "a, b, c, d, e, g, h, k, p, r 2 R, where e(t) = [e1, e2, e3, e4]T. Therefore, the synchronization between two different novel hyperchaotic systems is achieved with the adaptive controller (6) and the parameter estimation adaptive update law (7). This completes the proof. h
3. Numerical simulation In this section, Numerical simulation is given to show the effectiveness and feasibility of the proposed synchronization method. In the numerical simulation, the Runge–Kutta–Fehlberg algorithm with adaptive varied time step is used to solve the augment system consisting of (3), (7) and (8). The parameters are chosen to be a = 35, b = 5, c = 25, d = 5, e = 35, g = 10, h = 10, k = 8/3, p = 28 and r = 1 in the simulation so that system (3) and (4) exhibit hyperchaotic behaviors if no controls are applied. The initial values of the drive and response systems are (x1(0), y1(0), z1(0), w1(0)) = (10, 2, 3, 10) and (x2(0), y2(0), z2(0), w2(0)) = (14.5, 4.5, 6.5, 8.5), respectively; hence the initial errors of system (8) are (e1(0), e2(0), e3(0), e4(0)) = (4.5, 2.5, 3.5, 1.5). Moreover, the initial values of the estimated parameters are chosen as b1 = 3.8, g1 = 20 and r1 = 0.5. The simulation results are illustrated in Figs. 3 and 4. Fig. 3 shows that e1, e2, e3 and e4 tend to zero, which demonstrates that the response system (4) synchronizes the drive system (3). Fig. 4 shows that the estimated values b1, g1 and r1 of the uncertain parameters converge to b = 5, g = 10, and r = 1, respectively, as t ? 1. 4. Conclusion This paper addresses the problem of adaptive synchronization of two different new hyperchaotic systems with some uncertain parameters. On the basis of the Lyapunov stability theory and the adaptive control theory, a new adaptive synchronization control law and a novel parameter estimation update law are proposed to achieve synchronization between the two novel different hyperchaotic systems with uncertain parameters. Numerical simulations are given to demonstrate the effectiveness of the proposed synchronization scheme and verify the theoretical results. Acknowledgements This work was supported by the Chinese Provincial Natural Science Foundation of Hunan province (No: 06JJ5098). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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