www.ccsenet.org/cis Coputer and Inforation Science Vol., No. ; August 00 Finite Tie Synchronization between Two Different Chaotic Systes with Uncertain Paraeters Abstract Wanli Yang, Xiaodong Xia, Yucai Dong & Suwen Zheng Institute of Nonlinear Science, Acadey of Arored force Engineering, Beijing 0007, China This paper deals with the finite-tie chaos synchronization between two different chaotic systes with uncertain paraeters by using active control. Based on the finite-tie stability theory, a control law is proposed to realize finite-tie chaos synchronization for the uncertain systes Lorenz and Lü. The controller is siple and robust against the uncertainty in syste paraeters. Nuerical results are presented to show the effectiveness of the proposed control technique. Keywords: Finite-tie synchronization, Active control, Finite-tie stability theory, Uncertain paraeters. Introduction Synchronization of chaotic systes has been a hot topic since the pioneering work of Pecocra and Carroll (Pecora LM, Carroll TL. 990). It can be applied in various fields such as cheical reactors, power converters, biological systes, inforation processing, secure counication(colet P, Roy R. 994)(Sugawara T, Tachikawa M, Tsukaoto T, Shiisu T. 994)(Lu JA, Wu XQ, Lü JH. 00), etc. A wide variety of approaches have been eployed in the synchronization of chaotic systes which ost of the are designed to synchronize two identical chaotic systes(chen S, Lü J. 00)(Agiza HN, Yassen MT. 00)(Ho M-C, Hung Y-C. 00)(Huang L, Feng R, Wang M. 004)(Liao T-L. 998)(Liao T-L, Lin S-H. 999)(Yassen MT. 00). The synchronization of two different chaotic systes, however, is not straightforward. Difference in the structure of the systes akes the synchronization a challenging proble in this case(m.-ch. Ho, Y.-Ch. Hung, 00)(H. Zhang, W. Huang, Z. Wang, T. Chai, 006)( M.-T. Yassen, 005). Soe of the proposed approaches in the identical case are not applicable here or required different procedure to be designed. This proble becoes ore difficult when two chaotic systes have soe uncertain paraeters. In the literatures (Shahra Eteadi, Aria Alasty. 007) used active sliding ode control to synchronize two different chaotic systes with uncertain paraeters. However, the convergence of the synchronization procedure in (Shahra Eteadi, Aria Alasty. 007) is exponential with infinite settling tie. To attain fast convergence speed, any effective ethods have been introduced and finite-tie control is one of the. Finite-tie synchronization eans the optiality in convergence tie. Moreover, the finite-tie control techniques have deonstrated better robustness and disturbance rejection properties(bhat S, Bernstein D. 997)(Hua Wang, 008). In this paper, the goal is to force the two different chaotic systes with uncertain paraeters to be synchronized in finite tie. The ethod of active control is applied to control the chaos synchronization syste. Based on finite-tie stability theory, a controller is designed to achieve finite-tie synchronization. Siulation results show that the proposed controller synchronizes the Lorenz and Lü chaotic systes in finite tie.. Preliinary definitions and leas Finite-tie synchronization eans that the state of the slave syste can track the state of the aster syste after a finite-tie. The precise definition of finite-tie synchronization is given below. Definition. Consider the following two chaotic systes: Where x, x& = f( x) x& = hx (, x) s s x s are two n diensional state vectors. The subscripts and s stand for the aster and slave n n n n systes, respectively. f : R R and h: R R are vector-valued functions. If there exists a constant T > 0, such that li x x = 0, and x x = 0, if t T,then synchronization of the syste () is achieved in a finite-tie. t T s s () 74 ISSN 9-8989 E-ISSN 9-8997
www.ccsenet.org/cis Coputer and Inforation Science Vol., No. ; August 00 Lea. Suppose there exists a continuous function V : D R such that the following hold :. V() t is positive definite..there exists real nubers c > 0 and (0,) that α and an open neighborhood N D V& η ( x) cv ( x), x N \{0} () Then the origin is a finite-tie stable equilibriu. Lea. when ab, and c< are all positive nubers, the following inequality holds: of the origin such c c c ( a+ b) a + b. (). Systes description Lorenz syste is considered a paradig, since it captures any of the feathers of chaotic dynaics. The Lorenz syste is described by the following nonlinear equations: which has a chaotic attractor when 8 = a( y x) = cx xz y = xy bz a = 0, b=, c= 8. Chen syste is a typical chaos anti-control odel, which has a ore coplicated topological structure than Lorenz attractor. The nonlinear differential equations that describe the Chen syste are = α( y x) = ( γ α) x xz + γ y = xy β z which has a chaotic attractor when α = 5, β =, γ = 8. Lü syste is a typical transition syste, which connects the Lorenz and Chen attractors and represents the transition fro one to the other. The Lü syste is described by = ρ( y x) = xz + vy (6) = xy μz which has a chaotic attractor when ρ = 6, μ =. v = 0. In the next sections, we will study the chaos synchronization between the chaotic dynaical systes Lorenz and Lü with uncertain paraeters. 4. Finite-tie synchronization between Lorenz and Lü systes with paraeters uncertainty This subsection deals with finite-tie synchronization of uncertain Lorenz and Lü systes. It is valuable because practical systes are often disturbed by different factors. It is assued that both the aster syste and the slave syste hold uncertainties. Consider the following chaotic syste with uncertain paraeters: and = a( y x) = ( c+δ ) x x z y = xy ( b+δ) z (4) (5) (7) Published by Canadian Center of Science and Education 75
www.ccsenet.org/cis Coputer and Inforation Science Vol., No. ; August 00 = ρ( y x) + u( t) = xz + ( v+δ ) y + u( t) = xy ( μ +Δ 4) z + u( t) where Δ i, i =,,,4denote the bounded uncertain paraeters, i.e. Δi δi, We have introduced three control functionsu () t, u () t, u () t in (8). Our goal is to deterine the control functionsu () t, u () t, u () t. In order to estiate the control functions, we subtract (7) fro (8). We define the error syste as the differences between the Lorenz syste (7) and the controlled Lü syste (8). Let us define the state errors as e = x x, e = y y, e = z z (9) Use of the definition in (9), the error dynaics can be written as e& = ρ( e e) + ( ρ a)( y x) + u( t) e& = ve + vy xe ze ee cx+ y+δy Δ x+ u() t e& = μe+ ey+ ex+ ee ( μ b) z ( Δ4 Δ) z Δ e+ u( t) We define the active control functionsu () t, u () t, and u () t as follows Hence the error syste (0) becoes u = W() t ( ρ a)( y x) u = W () t vy + ze + cx y u = W() t ey+ ( μ b) z e& = ρ( e e) + W( t) e& = xe ee + ve +Δy Δ x+ W() t e& = μe+ ex+ ee ( Δ4 Δ) z Δ e+ W( t) Our ai is to design a controller that can achieve the finite-tie synchronization of uncertain Lorenz (7) and Lü systes (8). This proble can be converted to design a controller to attain finite-tie stable of the error syste (). The design procedure consists of two steps as follows. Step: Let W = ρe e sign( e), (0,),substituting this control input W into the first equation of () yields Choose a candidate Lyaupunov function = ρ The derivative of V along the trajectory of () is (8) (0) () () e& e e sign( e ) () V = e + + + + = = ρ = V& ee& e e e V Fro Lea, the syste () is finite-tie stable. That eans that is a T > 0 such that e = 0 provided that t T. Step : When t > T, e = 0. The last two equation of syste () becoe 76 ISSN 9-8989 E-ISSN 9-8997
www.ccsenet.org/cis Coputer and Inforation Science Vol., No. ; August 00 e& = xe + ve +Δy Δ x+ W() t e& = μe+ ex ( Δ4 Δ) z Δ e+ W( t) Select W t ve e sign e γ x sign e γ y sign e ( ) = ( ) ( ) ( ), γ δ, γ δ,and (4) γ W () t = e sign( e ) z sign( e ),where γ δ + δ4. Substitute W andw into the syste (4) and consider the following candidate Lyapunov function: V = ( e + e ) The derivative of V along the trajectory of (4) is V& e( xe y x e signe ( ) x signe ( ) y signe ( )) = +Δ Δ γ γ + e ( μe + e x ( Δ Δ ) z Δ e e sign( e ) γ z sign( e )) 4 + + ye xe e γxe γye ( μ ) e ( 4 ) ze γze e + + γ xe γye γze e e +Δye Δxe ( Δ4 Δ) ze =Δ Δ +Δ Δ Δ = ( γ +Δ sign( x e )) x e ( γ Δ sign( y e )) y e ( γ + ( Δ Δ ) sign( z e )) z e e 4 + + e + + + + e e V = Fro Lea, it follows that (4) is finite-tie stabilized. Thus, the uncertain slave syste (8) can synchronize the uncertain aster syste (7) in finite tie. 5. Siulations results In this siulation, the 4th order Runge Kutta algorith was used to solve the sets of differential equations related to the aster and slave systes. We select the paraeters of Lorenz syste as a = 0, b= 8/, c = 8 and the paraeters of Lü syste as ρ = 6, v = 0, μ =. The initial values of Lorenz syste and Lü syste are[ x(0) y(0) z (0)] = [5 6 9],[ x(0) y(0) z (0)] = [5 7 0]. The initial errors are e(0) = 0, e(0) =, e(0) =. The uncertain paraeters of Lorenz syste and Lü syste are adopted as Δ = 0.5sin t, Δ = 0.5cos t, Δ = 0., Δ 4 = cos t, γ = 0.8, γ = 0.5, γ =.The controller paraeter is selected as / to satisfy given condition. The siulation results are given in Fig. for the case that the Lorenz syste drives the Lü syste.fig. shows the errors responses of the uncertain Lorenz and Lü systes. As we expect, the slave syste synchronizes with the aster syste and the syste have strong robustness to the uncertainties. 6. Conclusion In this paper, an effective control ethod for synchronizing different chaotic systes with uncertain paraeters has been proposed using active control. Based on the finite-tie stability theory, the proposed controller enables stabilization of synchronization error dynaics to zeros in finite tie. Finite tie synchronization between the pairs of the Lorenz and Lü systes is achieved. Nuerical siulations are also given to validate the synchronization approach. References Agiza HN, Yassen MT. (00). Synchronization of Rossler and Chen chaotic dynaical systes using active control. Phys Lett A, 00;78:9-97. Bhat S, Bernstein D. (997). Finite-tie stability of hoogeneous systes. In: Proceedings of ACC, Published by Canadian Center of Science and Education 77
www.ccsenet.org/cis Coputer and Inforation Science Vol., No. ; August 00 Albuquerque, NM; 997. p. 5 54. Chen S, Lü J. (00). Paraeters identification and synchronization of chaotic systes based upon adaptive control. Phys Lett A00;99:5-58. Colet P, Roy R. (994). Digital counication with synchronized chaotic lasers. Opt Lett 994;9:056 058. H. Zhang, W. Huang, Z. Wang, T. Chai. (006). Adaptive synchronization between two different chaotic systes with unknown paraeters, Phys. Lett. A 50 (006) 6 66. H.-N. Agiza, M.-T. Yassen, (00). Synchronization of Rossler and Chen chaotic dynaical systes using active control, Phys. Lett. A 78(00) 9 97. Ho M-C, Hung Y-C. (00). Synchronization of two different systes by using generalized active control. Phys Lett A 00;0:44-48. Hua Wang. (008). Finite-tie chaos synchronization of unified chaotic syste with uncertain paraeters, counication in nonlinear science and nuerical siulation, 008. Huang L, Feng R, Wang M. (004). Synchronization of chaotic systes via nonlinear control. Phys Lett A 004;0:7-75. Liao T-L, Lin S-H. (999). Adaptive control and synchronization of Lorenz systes. J Franklin Inst 999;6:95-97. Liao T-L. (998). Adaptive synchronization of two Lorenz systes. Chaos, Solitons & Fractals 998;9:555-56. Lu JA, Wu XQ, Lü JH. (00). Synchronization of a unified syste and the application in secure counication. Phys Lett A 00;05:65 70. M.-Ch. Ho, Y.-Ch. Hung. (00). Synchronization of two different systes by using generalized active control, Phys. Lett. A 0 (00) 44 48. M.-T. Yassen. (005). Chaos synchronization between two different chaotic systes using active control, Chaos Soliton Fract. (005) 40. M.-T. Yassen. (007). Controlling, synchronization and tracking chaotic Liu syste using active backstepping design, Phys. Lett. A 60(007) 58 587. Pecora LM, Carroll TL. (990). Synchronization in chaotic systes. Phys Rev Lett, 990;64():8-84. Shahra Eteadi, Aria Alasty. (007). Synchronization of uncertain chaotic systes using active sliding ode control. Chaos, Solitons & Fractals 007;:0-9. Sugawara T, Tachikawa M, Tsukaoto T, Shiisu T. (994). Observation of synchronization in laser chaos. Phys Rev Lett 994;7:50 505. Yassen MT. (00). Adaptive control and synchronization of a odified Chua s circuit syste. Appl Math Coput, 00;5():-8. 78 ISSN 9-8989 E-ISSN 9-8997
www.cccsenet.org/cis Coputer aand Inforationn Science Figgure. Resultss of synchronizzing Lorenz annd Lü systes. Publishhed by Canadiann Center of Scien nce and Educatiion Vool., No. ; Auggust 00 Figure. Thee synchronizatiion errors at t t [0,0] of controlleed Lü 79