Function projective synchronization for fractional-order chaotic systems. (English) Zbl 1209.34065

Zhou, Ping; Zhu, Wei

Nonlinear Anal., Real World Appl. 12, No. 2, 811-816 (2011).

Summary: This letter investigates the function projective synchronization between fractional-order chaotic systems. Based on the stability theory of fractional-order systems and tracking control, a controller for the synchronization of two fractional-order chaotic systems is designed. This technique is applied to achieve synchronization between fractional-order Lorenz systems of different orders, and achieve synchronization between the fractional-order Lorenz system and the fractional-order Chen system. Numerical simulations demonstrate the validity and feasibility of the proposed method.

MSC:

34D06	Synchronization of solutions to ordinary differential equations
34A08	Fractional ordinary differential equations
34H05	Control problems involving ordinary differential equations
34C28	Complex behavior and chaotic systems of ordinary differential equations

Keywords:

function projective synchronization; fractional-order chaotic systems; stability theory of fractional-order systems; tracking control

References:

[1]	Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821-824 (1990) · Zbl 0938.37019
[2]	Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990) · Zbl 0964.37501
[3]	Chen, G.; Dong, X., From Chaos to Order: Methodologies, Perspectives and Applications (1998), World Scientific: World Scientific Singapore
[4]	Carroll, T. L.; Pecora, L. M., Synchronizing chaotic circuits, IEEE Trans. Circuits Syst. I, 38, 453-456 (1991)
[5]	Kacarev, L.; Parlitz, U., General approach for chaotic synchronization with application to communication, Phys. Rev. Lett., 74, 5028-5031 (1996)
[6]	Du, H. Y.; Zeng, Q. S.; Wang, C. H.; Ling, M. X., Function projective synchronization in coupled chaotic systems, Nonlinear Anal. RWA, 11, 705-712 (2010) · Zbl 1181.37039
[7]	Sebastian, K. S.; Sabir, M., Adaptive modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system with uncertain parameters, Phys. Lett. A, 373, 3743-3748 (2009) · Zbl 1233.93060
[8]	Wang, X. Y.; Song, J. M., Adaptive full state hybrid projective synchronization in the unified chaotic system, Modern Phys. Lett. B, 23, 1913-1921 (2009) · Zbl 1168.37313
[9]	Luo, R. Z., Adaptive function project synchronization of Rössler hyperchaotic system with uncertain parameters, Phys. Lett. A, 372, 3667-3671 (2008) · Zbl 1220.37025
[10]	Mainieri, R.; Rehacek, J., Projective synchronization in the three-dimensional chaotic systems, Phys. Rev. Lett., 82, 3042-3045 (1999)
[11]	Chen, Y.; Li, X., Function projective synchronization between two identical chaotic systems, Internat. J. Modern Phys. C, 18, 5, 883-888 (2007) · Zbl 1139.37301
[12]	Chen, Y.; An, H. L.; Li, Z. B., The function cascade synchronization approach with uncertain parameters or not for hyperchaotic systems, Appl. Math. Comput., 197, 96-110 (2008) · Zbl 1136.93410
[13]	An, H. L.; Chen, Y., The function cascade synchronization method and applications, Commun. Nonlinear Sci. Numer. Simul., 13, 2246-2255 (2008) · Zbl 1221.93124
[14]	Matignon, D., Stability results of fractional differential equations with applications to control processing, (IMACS (1996), IEEE-SMC: IEEE-SMC Lille, France)
[15]	Lorenz, E. N., Deterministic nonperodic flow, J. Atmospheric Sci., 20, 130-141 (1963) · Zbl 1417.37129
[16]	Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 91, 3, 034101 (2003)
[17]	Chen, G.; Ueta, T., Yet anthor chaotic attractor, Internat. J. Bifur. Chaos, 9, 1465-1466 (1999) · Zbl 0962.37013
[18]	Mohammad, S. T.; Mohammad, H., A necessary condition for double scroll attractor existence in fractional-order systems, Phys. Lett. A, 367, 102-113 (2007) · Zbl 1209.37037