Email this article Generalized Synchronization of Typical Fractional Order Chaos System
Abstract
Based on the fractional calculus predictor-corrector algorithm and control design theory, generalized synchronization of dynamics of a chaotic system is investigated by linear or nonlinear feedback control, making the non-linear chaotic synchronization error system become linear systems, then synchronous controller was designed based on the stability theory of fractional linear systems [10], and the generalized synchronization of fractional order chaotic systems was fulfilled in different structures and the different order. The auxiliary system method is simple and effective. Detailed numerical results verify the effectiveness of our proposed new scheme.
Keywords
References
[1] Lorenz. Deterministic non-periodic flow. Journal of the Atmospheric Sciences, 1963, 20(1):130-141.
[2] Li T Y, Yorke J A. Period three implies chaos. American Mathematical Monthly, 1975, 82:985-992.
http://dx.doi.org/10.2307/2318254
[3] Gang, Chen. On feedback control of chaotic nonlinear dynamic system [J]. Bifurcation and chaos. 1992. 2: 407-411,
[4] PengZhen Dong, Jie Liu, XinJie Li. Anticipating Synchronization of Integral-order and Fractional-order Chaotic Liu Systems [M]. Proceeding of CCDC 2009, Guilin, 2009: 407-411
[5] Jitao Chen, Qidi Wu, Impulsive Control for The Stabilization and Synchronization of Lur’s Systems [J]. Applied Mathematics and Mechanics, 2004, 25: 322-328
[6] H. Richter, K. J. Reinschke. Optimization of local control of chaos by an evolutionary algorithm[J], Physic D 2000 309-334
[7] D. Coles. Transition in circular Cotter flow [J]. J. Fluid Mech.vol.21 1965 385-425
http://dx.doi.org/10.1017/S0022112065000241
[8] Jinhu Lǚ, The Generalized Lorenz-Like System [M]. International Journal of Bifurcation and Chaos,2002, 12:1531-1534
http://dx.doi.org/10.1142/S0218127402005273
[9] H.D Abarbanel, Rulkov N.F. Generalized synchronization of chaos: The auxiliary system approach [J]. Phys. Rev. E, 2007,53:28-45
[10] Jie Liu, Lu Jun’an. Different type of synchronization phenomena in unidirectional coupled unified chaotic systems [J]. Int. J. of Innovative Computing, In for. And Cont. 2007. 3(3):697-708
[11] Xinjie Li, Jie Liu. Observer designing for generalized synchronization of fractional order hyper chaotic Lü System[C]. Proceeding of CCDC 2009, Guilin, 2009. 43-45
[12] Junguo.Lu, Nonlinear observer design to synchronize fractional-order chaotic systems via a scalar transmitted signal [J]. Physical A. 2007, 359: 107-118
[13] Qun-Jiao Zhang, Lu Jun-An, Jia Zhen. Global Exponential Projective Synchronization and Lag Synchronization of Hyper chaotic Lǚ system. Chinese Physical Society and IOP Publishing Ltd, 2009. 51(4):679-683
[14] Jie Liu, Xinjie Li, Synchronization of fractional hyper chaotic Lü system via unidirectional coupling method[C], Proceedings of the 7th World Congress on Intelligent Control and Automation, 2008: 4653-4658
[15] Lu J.G, Chen G., A note on the fractional order Chen system [J]. Chaos, Solitions and Fractals, 2006, 27 (3): 685-688.
http://dx.doi.org/10.1016/j.chaos.2005.04.037
[16] Diethelm K, Ford N J, Freed AD, A predictor corrector approach for the numerical solution of fractional differential equations [J]. Nonlinear Dynamics, 2002, 29 (1): 3-22.
http://dx.doi.org/10.1023/A:1016592219341
[17] Matignon D, Stability results of fractional differential equations with applications to control processing [M]. Lille, France: IMACS, IEEE-SMC, 1996:963-968.
[18] M. S. Tavazoei, M. Haeri. A necessary condition for double scroll attractor existence in fractional order systems [J]. Physics Letters A, 2007, 367:102-113.
http://dx.doi.org/10.1016/j.physleta.2007.05.081
[19] Ivo Petrá. Method for simulation of the fractional order chaotic systems [J]. Acta Montanistica Slovaca, 2007, 11(4): 273-277.
[20] W.M. Ahmad, J.C. Sprott, Chaos in fractional-order autonomous nonlinear systems [Z]. Chaos, Solutions & Fractals 16, 2008, 339-351.
http://dx.doi.org/10.1016/S0960-0779(02)00438-1
[21] Lvjin Hu, Jun-an Lu, Shi-Hua Chen. Chaotic Time Series Analysis and Its Applications [M], Wuhan: Wuhan University Press, 2008.
[22] Chen Guan-Rong, Lvjin Hu. Lorenz system family dynamics analysis, control and synchronization of [M], Beijing: Science Press, 2009.
[23] M. S. Tavazoei, M. Haeri, Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems [J]. IET Signal Processing, 1(4), 2007, 171-181.
[24] M. S. Tavazoei, M. Haeri. A necessary condition for doublescroll attractor existence in fractional-order systems [J].Physics Letters A, 367, 1-2(16), 2007.102-113.
[25] Jie Liu, Lu Jun’an, Different type of synchronization phenomena in unidirectional coupled unified chaotic systems [J], Int. J. of Innovative Computing, In for. And Cont. 2008, 3(3):697-708
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