Exponential generalized synchronization of uncertain coupled chaotic systems by adaptive control. (English) Zbl 1221.93244
Summary: The exponential generalized synchronization for a class of coupled systems with uncertainties is defined. A novel and powerful method is proposed to investigate the generalized synchronization based on the adaptive control technique. According to the Lyapunov stability theory, rigorous proof is given for the exponential stability of error system. In comparison with previous schemes, the presented method shortens the synchronization time and is more applicable in practice. Besides, it is shown that the synchronization effect is robust against the uncertain factors. Some typical chaotic and hyper-chaotic systems are taken as examples to illustrate above approach. The corresponding numerical simulations are demonstrated to verify the effectiveness of proposed method.
MSC:
| 93D21 | Adaptive or robust stabilization |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37N35 | Dynamical systems in control |
References:
| [1] | Fujisaka, H.; Yamada, T., Stability theory of synchronized motion in coupled-oscillator systems, Prog Theor Phys, 69, 32-47 (1983) · Zbl 1171.70306 |
| [2] | Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys Rev Lett, 64, 821-824 (1990) · Zbl 0938.37019 |
| [3] | Pecora, L. M.; Carroll, T. L.; Johnson, G. A.; Mar, D. J.; Heagy, J. F., Fundamentals of synchronization in chaotic systems: concepts, and applications, Chaos, 7, 520-543 (1997) · Zbl 0933.37030 |
| [4] | Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization—a universal concept in nonlinear science (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0993.37002 |
| [5] | Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D. L.; Zhou, C. S., The synchronization of chaotic systems, Phys Rep, 366, 1-101 (2001) · Zbl 0995.37022 |
| [6] | Guan, S. G.; Li, K.; Lai, C. H., Chaotic synchronization through coupling strategies, Chaos, 16, 023107 (2006) · Zbl 1146.37320 |
| [7] | Sundar, S.; Minai, A. A., Synchronization of randomly multiplexed chaotic systems with application to communication, Phys Rev Lett, 85, 5456-5459 (2000) |
| [8] | Wang, S. H.; Kuang, J. Y.; Li, J. H.; Luo, Y. L.; Lu, H. P.; Hu, G., Chaos-based secure communications in a large community, Phys Rev E, 66, 065202 (2002) |
| [9] | Alvarez, G.; Hernández, L.; Muňoz, J.; Montoya, F.; Li, S. J., Security analysis of communication system based on the synchronization of different order chaotic systems, Phys Lett A, 345, 245-250 (2005) · Zbl 1345.94032 |
| [10] | Huang, D. B., Simple adaptive-feedback controller for identical chaos synchronization, Phys Rev E, 71, 037203 (2005) |
| [11] | Wang, D. X.; ZHong, Y. L.; Chen, S. H., Lag synchronizing chaotic system based on a single controller, Commun Nonlinear Sci Number Simul, 13, 637-644 (2008) · Zbl 1130.34322 |
| [12] | Wen, G. L.; Xu, D. L., Observer-based control for full-state projective synchronization of a general class of chaotic maps in any dimension, Phys Lett A, 333, 420-425 (2004) · Zbl 1123.37326 |
| [13] | Lin, W.; He, Y. B., Complete synchronization of the noise-perturbed Chua’s circuits, Chaos, 15, 023705 (2005) |
| [14] | Osipov, G. V.; Pikovsky, A. S.; Kurths, J., Phase synchronization of chaotic rotators, Phys Rev Lett, 88, 054102 (2002) |
| [15] | Huang, Z. X.; Ruan, J., Synchronization of chaotic systems by linear feedback controller, Commun Nonlinear Sci Number Simul, 3, 27-30 (1998) · Zbl 0923.34050 |
| [16] | Yu, H. J.; Liu, Y. Z., Chaotic synchronization based on stability criterion of linear systems, Phys Lett A, 314, 292-298 (2003) · Zbl 1026.37024 |
| [17] | Anteneodo, C.; Batista, A. M.; Viana, R. L., Chaos synchronization in long-range coupled map lattices, Phys Lett A, 326, 227-233 (2004) · Zbl 1138.37338 |
| [18] | Terry, J. R.; VanWiggeren, G. D., Chaotic communication using generalized synchronization, Chaos Solitons Fract, 12, 145-152 (2001) |
| [19] | Rulkov, N. F.; Sushchik, M. M.; Tsimring, L. S.; Abarbanel, H. D.I., Generalized synchronization of chaos in directionally coupled chaotic systems, Phys Rev E, 51, 980-994 (1995) |
| [20] | Kocarev, L.; Parlitz, U., Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems, Phys Rev Lett, 76, 1816-1819 (1996) |
| [21] | Abarbanel, H. D.I.; Rulkov, N. F.; Sushchik, M. M., Generalized synchronization of chaos: the auxiliary system approach, Phys Rev E, 53, 4528-4535 (1996) |
| [22] | Yang, X. S., On the existence of generalized synchronization in unidirectionally coupled systems, Appl Math Comput, 122, 71-79 (2000) · Zbl 1036.34042 |
| [23] | Yang, S. S.; Duan, C. K., Generalized synchronization in chaotic systems, Chaos Solitons Fract, 9, 1703-1707 (1998) · Zbl 0946.34040 |
| [24] | Lu, J. G.; Xi, Y. G., Linear generalized synchronization of continuous-time chaotic systems, Chaos Solitons Fract, 17, 825-831 (2003) · Zbl 1043.93518 |
| [25] | Wang, Y. W.; Guan, Z. H., Generalized synchronization of continuous chaotic system, Chaos Solitons Fract, 27, 97-101 (2006) · Zbl 1083.37515 |
| [26] | Femat, R.; Kocarev, L.; VanGerven, L.; Monsivais-Pérez, M. E., Towards generalized synchronization of strictly different chaotic systems, Phys Lett A, 342, 247-255 (2005) · Zbl 1222.37108 |
| [27] | Zhang, G.; Liu, Z. R.; Ma, Z. J., Generalized synchronization of different dimensional chaotic dynamical systems, Chaos Solitons Fract, 32, 773-779 (2007) · Zbl 1138.37316 |
| [28] | Zhou, J.; Chen, T. P.; Xiang, L., Robust synchronization of delayed neural networks based on adaptive control and parameters identification, Chaos Solitons Fract, 27, 905-913 (2006) · Zbl 1091.93032 |
| [29] | Hale, J.; Lunel, S. V., Introduction to functional differential equations (1993), Springer: Springer New York · Zbl 0787.34002 |
| [30] | Hassan, K. K., Nonlinear systems (2002), Prentice Hall: Prentice Hall New Jersey · Zbl 1003.34002 |
| [31] | Lu, W. L., Comment on “Adaptive-feedback control algorithm”, Phys Rev E, 75, 018201 (2007) |
| [32] | Lin, W.; Ma, H. F., Failure of parameter identification based on adaptive synchronization techniques, Phys Rev E, 75, 066212 (2007) |
| [33] | Yu, W. W.; Chen, G. R.; Cao, J. D.; Lü, J. H.; Parlitz, U., Parameter identification of dynamical system from time series, Phys Rev E, 75, 067201 (2007) |
| [34] | Chen, S. H.; Hu, J.; Wang, C. P.; Lü, J. H., Adaptive synchronization of uncertain Rossler hyperchaotic system based on parameter identification, Phys Lett A, 321, 50-55 (2004) · Zbl 1118.81326 |
| [35] | Liu, C. X.; Liu, T.; Liu, L.; Liu, K., A new chaotic attractor, Chaos Solitons Fract, 22, 1031-1038 (2004) · Zbl 1060.37027 |
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