Anti-Synchronization of Tigan and Li Systems with Unknown Parameters via Adaptive Control
In this paper, the adaptive nonlinear control method has been deployed to derive new results for the anti-synchronization of identical Tigan systems (2008), identical Li systems (2009) and nonidentical Tigan and Li systems. In adaptive anti-synchronization of identical chaotic systems, the parameters of the master and slave systems are unknown and the feedback control law has been derived using the estimates of the system parameters. In adaptive anti-synchronization of non-identical chaotic systems, the parameters of the master system are known, but the parameters of the slave system are unknown and accordingly, the feedback control law has been derived using the estimates of the parameters of the slave system. Our adaptive synchronization results derived in this paper for the uncertain Tigan and Li systems are established using Lyapunov stability theory. Numerical simulations are shown to demonstrate the effectiveness of the adaptive anti-synchronization schemes for the uncertain chaotic systems addressed in this paper.
[1] Alligood, K.T., Sauer, T. & Yorke, J.A., Chaos: An Introduction to Dynamical Systems, Springer, New York (1987).
[2] Lorenz, E., Deterministic nonperiodic flow, J. Atmos. Sciences, 20, 130-141 (1963).
[3] Pecora, L.M. & Carroll, T.L., Synchronization in chaotic systems, Phys. Rev. Letters, 64, 821-824 (1990).
[4] Lakshmanan, M. & Murali, K., Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientific, Singapore (1996).
[5] Han, S.K., Kerrer, C. & Kuramoto, Y., Dephasing and bursting in coupled neural oscillators, Phys. Rev. Letters, 75, 3190-3193 (1995)
[6] Blasius, B., Huppert, A. & Stone, L., Complex dynamics and phase synchronization in spatially extended ecological system, Nature, 399, 354-359 (1999).
[7] Cuomo, K.M. & Oppenheim, A.V. Circuit implementation of synchronized chaos with applications to communications, Phys. Rev. Letters, 71, 65-68 (1993).
[8] Li, Z., Li, K., Wen, C. & Soh, Y.C., A new chaotic secure communication system, IEEE Trans. Comm, 51 (8), 1306-1312 (2003).
[9] Ott, E., Grebogi, C. & Yorke, J.A., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990).
[10] Bai, E.W. & Longren, K.E., Synchronization of two Lorenz systems using active control, Chaos, Solit. Fractals, 8, 51-58 (1997).
[11] Ho, M.C. & Hung, Y.C., Synchronization of two different chaotic systems using generalized active control, Phys. Lett. A, 301, 424-428 (2002).
[12] Huang, L., Feng, R. & Wang, M., Synchronization of chaotic systems via nonlinear control, Phys. Lett. A, 320, 271-275 (2005).
[13] Lei, Y., Xu, W., Shen, J. & Fang, T., Global synchronization of two parametrically excited systems using active control, Chaos Solit. Fract., 28, 428-436 (2006).
[14] Chen, H.K., Global chaos synchronization of new chaotic systems via nonlinear control, Chaos Solit. Fract., 23, 1245-1251 (2005).
[15] Vincent, U.E., Synchronization of identical and non-identical 4-D systems via active control, Chaos Solit. Fract., 31, 119-129 (2007).
[16] Sundarapandian, V. & Karthikeyan, R., Global chaos synchronization of hyperchaotic Liu and hyperchaotic Chen systems by active nonlinear control, CIIT Int. J. Digital Signal Processing, 3 (3), 134-139 (2011).
[17] Sundarapandian, V. & Karthikeyan, R., Global chaos synchronization of Chen and Cai systems by active nonlinear control, CIIT Int. J. Digital Signal Processing, 3 (3), 140-144 (2011).
[18] Lu, J., Wu, X., Han, X. & Lü, J., Adaptive feedback stabilization of a unified chaotic system, Phys. Lett. A, 329, 327-333 (2004).
[19] Chen, S.H. & Lü, J., Synchronization of an uncertian unified system via adaptive control, Chaos Solit. Fract., 14, 643-647 (2002).
[20] Aghababa, M.P. & Aghababa, H.P., Adaptive finite-time stabilization of uncertain nonautonomous chaotic electromechanical gyrostat systems with unknown parameters, Mech. Research Commun., 38, 500-505 (2011).
[21] Aghababa, M.P., A novel adaptive finite-time controller for synchronizing chaotic gyros withnonlinear inputs, Chinese Phys. B, 20, 090505 (2011).
[22] Aghababa, M.P. & Aghababa, H.P., Synchronization of nonlinear chaotic electromechanical gyrostat systems with undertainties, Nonlinear Dynamics, doi:10.1007/s11071-011-0181-5 (2011).
[23] Aghababa, M.P. & Heydari, A., Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input nonlinearities, Applied Math. Modelling, doi:10.1016/j.apm.2011.09.023 (2011).
[24] Park, J.H. & Kwon, O.M., A novel criterion for delayed feedback control of time-delay chaotic systems, Chaos Solit. Fract., 17, 709- 716 (2003).
[25] Yu, Y.G. & Zhang, S.C., Adaptive backstepping synchronization of uncertain chaotic systems, Chaos Solit. Fract., 27, 1369- 1375 (2006).
[26] Idowu, B.A., Vincent, U.E. & Njah, A.N., Generalized adaptive backstepping synchronization for non-identical parametrically excited systems, Nonlinear Analysis: Modelling and Control, 14 (2), 165- 176 (2009).
[27] Zhao, J. & Lü, J., Using sampled-data feedback control and linear feedback synchronization in a new hyperchaotic system, Chaos Solit. Fract., 35, 376-382 (2006).
[28] Konishi, K., Hirai, M. & Kokame, H., Sliding mode control for a class of chaotic systems, Phys. Lett. A, 245, 511-517 (1998).
[29] Haeri, M. & Emazadeh, A.A., Synchronization of different chaotic systems using active sliding mode control, Chaos Solit. Fract., 119-129 (2007).
[30] Pourmahamood, M., Khanmohammadi, S. & Alizadeh, G., Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller, Commun. Nonlinear Sci. Numerical Simulat., 16, 2853-2868 (2011).
[31] Aghababa, M.P. & Khanmohammadi, S. & Alizadeh, G., Finite-time synchronization of two different chaotic systems with unknownparameters via sliding mode technique, Appl. Math. Model, 35, 3080-3091 (2011).
[32] Tigan, G. & Opris, D., Analysis of a 3D chaotic system, Chaos Solit. Fract., 36, 1315- 1319 (2008).
[33] Li, X.F., Chlouverakis, K.E. & Xu, D.L., Nonlinear dynamics and circuit realization of a new chaotic flow: A variant of Lorenz, Chen and Lü, Nonlinear Analysis, 10, 2357-2368 (2009).
[34] Hahn, W., The Stability of Motion, Springer, New York (1967).