Hybrid Projective Dislocated Synchronization of Liu Chaotic System Based on Parameters Identification

Transcription

1 Modern Applied Science Vol. 6, No. ; February Hybrid Projective Dilocated Synchronization of Liu Chaotic Sytem Baed on Parameter Identification Yanfei Chen College of Science, Guilin Univerity of echnology Guilin 44, China el: chenyanfei@6.com Zhen Jia (Correponding author) College of Science, Guilin Univerity of echnology Guangxi Key Laboratory of Spatial Information and Geomatic, Guilin Univerity of echnology Guilin 44, China el: jjjzzz@6.com he reearch i financed by the National Natural Science Foundation of China (Grand No.664) and the Natural Science Foundation of Guangxi, China (No.GXNSFA847). Received: December, Accepted: December 9, Publihed: February, doi:.9/ma.v6np6 URL: Abtract hi letter dicue hybrid projective dilocated ynchronization of Liu chaotic ytem with five uncertain parameter. Baed on adaptive technique, the hybrid projective dilocated ynchronization of Liu chaotic ytem i achieved by deigning a novel nonlinear controller. Furthermore, the parameter identification i realized imultaneouly. A ufficient condition i given and proved theoretically by Lyapunov tability theory and LaSalle invariance principle. Finally, the numerical imulation are provided to how the effectivene and feaibility of the propoed method. Keyword: Hybrid projective dilocated ynchronization, Parameter identification, Liu chaotic ytem, Adaptive technique. Introduction Since the pioneering work on chao ynchronization by Pecora and Carroll in 99 (Pecora, 99), chao ynchronization ha attracted much attraction due to it potential application in many practical engineering field, uch a ecure communication, information proceing, chemical reaction, and o on. In the pat two decade, many type of ynchronization phenomena have been tudied, namely, complete ynchronization (Lu, ), generalized ynchronization (Jia, 8), phae ynchronization (Ho, ), lag ynchronization (Chen, 7), etc. Meanwhile, many cheme for chao ynchronization have been propoed, including linear and nonlinear feedback approach (Wang, 6; Jia, 7), adaptive technique (Jia, 7), coupled method (Chen, ), impulive control method (Luo, 8), among many other. Mainieri and Rehacek conidered a type of chao ynchronization, called projective ynchronization (Mainieri, 999), where the correponding tate vector of drive-repone ytem could be ynchronized up to a contant caling factor. In Ref. (Hu, 7), Hu et al. propoed a dilocated ynchronization method. In thi paper, we invetigate the hybrid projective dilocated ynchronization and parameter identification of Liu chaotic ytem. In thi cheme, every tate variable of drive ytem ynchronize other mimatched tate variable of repone ytem with different caling factor. Baed on the adaptive technique, a novel controller and parameter adaptive law are deigned uch that parameter identification i realized, and hybrid projective dilocated ynchronization of Liu chaotic ytem i achieved imultaneouly. hi work i organized a follow. he drive and repone ytem are decribed and hybrid projective 6 ISSN E-ISSN 9-8

2 Modern Applied Science Vol. 6, No. ; February dilocated ynchronization error are theoretically analyzed in ection. In ection, a general cheme for hybrid projective dilocated ynchronization of Liu chaotic ytem and parameter identification i proved. Section 4 preent ome numerical imulation to how the effectivene of the propoed cheme. Finally, concluion are hown.. Problem Formulation he Liu chaotic ytem (Liu, 4) a the drive ytem i given by x a x x, x bxkxx, x cx hx, having a chaotic attractor when a, b 4, c., h 4, k. Here the Lyapunov exponent of ytem () are found to be L.648, L, L 4.4. he phae portrait i hown in Figure. Conidering the drive ytem (), the repone ytem i controlled Liu chaotic ytem a following y a y y u, y by kyy u, () y cy hy u, where the ytem parameter,,,, U u, u, u i the controller which hould be deigned. herefore, the goal of parameter identification and hybrid projective dilocated ynchronization i to find an appropriate controller U u, u, u and parameter adaptive law of a, b, c, h, k, uch that the ynchronization error e y x, e y x, e y x a t () and the unknown parameter a b c h k of () are unknown, lim a a, limb b, lim c c, lim h h, limk k. (4) t t t t t where,, are the caling factor. Remark When or, the hybrid projective dilocated ynchronization will appear. When, projective dilocated ynchronization will appear. More in general, dilocated ynchronization and dilocated anti-ynchronization will appear when i and i, i,,, repectively. Remark Here are another four type of hybrid projective dilocated ynchronization error () e y x, (Ⅰ) e y x, e y x. e y x, e yx, e y x, (Ⅱ) e y x, ( Ⅲ) e y x, ( Ⅳ) e y x, e y x. e y x. e y x. For thee cae, the dicuion are imilar to the method given in thi paper.. Hybrid Projective Dilocated Synchronization of Liu Chaotic Sytem In thi ection, baed upon the nonlinear adaptive feedback control technique, a ytematic deign proce of parameter identification and hybrid projective dilocated ynchronization of Liu chaotic ytem under the ituation of repone ytem with unknown parameter i provided. According to the ytem () and (), the error dynamical ytem can be obtained a follow. e aea ayay kxx bxax u, e ce b byk kyy kyy by cy hx u, () e ae c cy h hy hy cay ax u. Obviouly, hybrid projective dilocated ynchronization of ytem () and () appear if the error dynamical ytem () ha an aymptotically table equilibrium point e e, e, e. e, where Publihed by Canadian Center of Science and Education 7

3 Modern Applied Science Vol. 6, No. ; February hu, we deign the controller and parameter adaptive law a the following theorem. heorem Auming that the Liu chaotic ytem () drive the controlled Liu chaotic ytem (), take and parameter adaptive law u ay kxx bx ax, u kyy by cy hx, u hy cay ax, a ye, b ye, c ye, h ye, k yye. (6) (7) Sytem () and () can realize hybrid projective dilocated ynchronization and the unknown parameter will be identified, i.e., Eq. () and (4) will be achieved. Proof Eq. () can be converted to the following form under the controller (6) e ae a a y, e ce b byk kyy, e ae c cy h hy. Conider the following Lyapunov function V e e e a a b b c c h h k k, Obviouly, V i a poitive definite function. aking it time derivative along with the trajectorie of Eq. (8) and (7) lead to V ee e e e e a a a b b b c c c k k k h h h ae ce ae e Pe, It i obviou that V if and only if e,,, i i, namely the et M e, a ab, bc, c, h h, k k i the larget invariant et contained in E V for Eq. (8). So according to the LaSalle invariance principle (Laalle, 96), tarting with arbitrary initial value of Eq. (8), the trajectory converge aymptotically to the et M, i.e., e, a a, b b, c c, h h and k k a t. hi indicate that the hybrid projective dilocated ynchronization of Liu chaotic ytem i achieved and the unknown parameter a, b, c, h, k can be uccefully identified by uing controller (6) and parameter adaptive law (7). Now the proof i completed. Remark aking our adaptive ynchronization method, we can not only achieve ynchronization but alo identify the ytem parameter. 4. Numerical Simulation In thi ection, ome numerical imulation about the hybrid projective dilocated ynchronization and parameter identification between ytem () and () are given to verify the effectivene and feaibility of the propoed technique. In the numerical imulation, all the differential equation are olved by uing the fourth-order Runge-Kutta method. We aume that the drive ignal are from the Liu chaotic ytem () with ytem parameter a, b 4, c,, h 4, k and the initial value x.,.4,8, the initial value of controlled Liu chaotic ytem () i y 9,, and the unknown parameter have zero initial condition. he three caling factor are choen a,,.8. he imulation reult are hown in Figure and. Figure (a)-(c) diplay the error tate repone of ytem () and (), Figure how the identification reult of unknown parameter a, b, c, h, k. (8) (9) 8 ISSN E-ISSN 9-8

4 Modern Applied Science Vol. 6, No. ; February. Concluion In thi letter, we introduce an adaptive hybrid projective dilocated ynchronization and parameter identification cheme for the Liu chaotic ytem with the repone ytem parameter unknown. With thi cheme we can achieve hybrid projective dilocated ynchronization and parameter identification imultaneouly. heoretical proof and numerical imulation demontrate the effectivene of the propoed cheme. It hould be pointed out that, although thi proce i focued on the Liu chaotic ytem, the ytematic deign proce could be ued for many other complex dynamical ytem with unknown parameter. Reference Chen, Y., Chen, X., & Chen, S. (7). Lag ynchronization of tructurally nonequivalent chaotic ytem with time delay. Nonlinear Analyi, 66, Chen, J., Lu, J., & Wu, X. (). Bidirectionally coupled ynchronization of the generalized Lorenz ytem. J Syt Sci Complex, 4, Ho, M. C., Hung, Y. C., & Chou, C. H. (). Phae and anti-phae ynchronization of two chaotic ytem by uing active control. Phy. Lett. A., 96, Hu, M., & Xu, Z. (7). Nonlinear feedback mimatch ynchronization of Lorenz chaotic yytem. Syt Eng Electr, 9(8), Jia, Z. (8). Linear generalized ynchronization of chaotic ytem with uncertain parameter. J Syt Eng Electr, 9(4), Jia, Z., Lu, J. A., & Deng, G. M. (7). Nonlinear tate feedback and adaptive ynchronization of hyperchaotic Lü ytem. Syt Eng Electr, 9(4), Laalle, J. P. (96). he extent of aymptotic tability. Proc. Natl. Acad. Sci. U.S.A., 46(), Liu, C., Liu,., Liu, L., & Liu, K. (4). A new chaotic attractor. Chao, Soliton & Fractal,, Luo, R. Z. (8). Impulive control and ynchronization of a new chaotic ytem. Phy.Lett. A, 7, Lu, J., & Cao, J. (). Adaptive complete ynchronization of two identical or different chaotic (hyperchaotic) ytem with fully unknown parameter. Chao, (4), Mainieri, R., & Rehacek, J. (999). Projective ynchronization in three-dimenional chaotic ytem. Phy. Rev. Lett., 8(), Pecora, L. M., & Carroll,. L. (99). Synchronization in chaotic ytem. Phy. Rev. Lett., 64(8), Wang, F., & Liu, C. (6). A new criterion for chao and hyperchao ynchronization uing linear feedback control. Phy.Lett.A, 6, x x x Figure. he phae portrait of Liu chaotic ytem () with parameter value a, b 4, c., h 4, k Publihed by Canadian Center of Science and Education 9

5 Modern Applied Science Vol. 6, No. ; February e e t/ (a) - t/ (b) e t/ (c) Figure. he ynchronization error evolution of ytem () and (): (a) e y x; (b) e y.8x ; (c) e y x ISSN E-ISSN 9-8

6 Modern Applied Science Vol. 6, No. ; February a t/ (a) b t/ (b) c h t/ (c) - t/ (d) k t/ (e) Figure. he parameter identification reult of repone ytem (): (a) a ; (b) b 4 ; (c) c. ; (d) h 4 ; (e) k Publihed by Canadian Center of Science and Education