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Adaptive feedback controller for projective synchronization. (English) Zbl 1144.93364
Summary: Due to the unpredictability of the scaling factor of projective synchronization in coupled partially linear systems, it is hard to know for sure the terminal state of the synchronized dynamics. In this paper, a simple adaptive linear feedback control method is proposed for controlling the scaling factor onto a desired value, based on the invariance principle of differential equations. Firstly, we prove the synchronizability of the proposed simple adaptive projective synchronization control method from the viewpoint of mathematics. Then, two numerical examples are presented to illustrate the applications of the derived results. Finally, we propose a communication scheme based on the adaptive projective synchronization of the Lorenz chaotic system. Numerical simulation shows its feasibility.

93D21Adaptive or robust stabilization
37N35Dynamical systems in control
93C40Adaptive control systems
94A62Authentication and secret sharing
93C15Control systems governed by ODE
Full Text: DOI
[1] Agiza, H. N.: Chaos synchronization of Lü dynamical system, Nonlinear anal. 58, 11-20 (2004) · Zbl 1057.34042 · doi:10.1016/j.na.2004.04.002
[2] Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D. L.; Zhou, C. S.: The synchronization of chaotic systems, Phys. report 366, 1-101 (2002) · Zbl 0995.37022 · doi:10.1016/S0370-1573(02)00137-0
[3] Cao, J.; Lu, J.: Adaptive synchronization of neural networks with or without time-varing delay, Chaos 16, 013133 (2006) · Zbl 1144.37331 · doi:10.1063/1.2178448
[4] Chen, A.; Lu, J.; Lü, J.; Yu, S.: Generating hyperchaotic Lü attractor via state feedback control, Phys. A 364, 103-110 (2006)
[5] F.X. Chen, W.D. Zhang, LMI criteria for robust chaos synchronization of a class of chaotic systems. Nonlinear Anal., in press. · Zbl 1131.34038 · doi:10.1016/j.na.2006.10.020
[6] Chen, G.; Dong, X.: From chaos to order: perspective, methodologies and applications, (1998) · Zbl 0908.93005
[7] Chen, M.; Zhou, D.: Synchronization in uncertain complex networks, Chaos 16, 013101 (2006) · Zbl 1144.37338 · doi:10.1063/1.2126581
[8] Y. Chen, X.X. Chen, S.S. Gu, Lag synchronization of structurally nonequivalent chaotic systems with time delays, Nonlinear Anal. 66 (2007) 1929 -- 1937. · Zbl 1123.34336 · doi:10.1016/j.na.2006.02.033
[9] Feng, C.; Zhang, Y.; Wang, Y.: Projective synchronization in time-delayed chaotic systems, Chinese phys. Lett. 23, 1418-1421 (2006)
[10] Hu, M. F.; Xu, Z. Y.; Zhang, R.; Hu, A. H.: Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyper-chaotic) systems, Phys. lett. A 361, 231-237 (2007) · Zbl 1170.93365 · doi:10.1016/j.physleta.2006.08.092
[11] Huang, D.: Stabilizing near-nonhyperbolic chaotic systems with applications, Phys. rev. Lett. 93, 214101 (2004)
[12] Huang, D.: Simple adaptive-feedback controller for identical chaos synchronization, Phys. rev. E 71, 037203 (2005)
[13] Huang, D.: Adaptive-feedback control algorithm, Phys. rev. E 73, 066204 (2006)
[14] Kilic, R.: Experimental study on impulsive synchronization between two modified chuas circuits, Nonlinear anal: real world appl. 7, No. 5, 1298-1303 (2006) · Zbl 1130.37359 · doi:10.1016/j.nonrwa.2005.12.004
[15] Li, C. G.: Projective synchronization in fractional order chaotic systems and its control, Prog. theor. Phys. 115, 661-666 (2006)
[16] Li, G. H.; Zhou, S. P.; Yang, K.: Generalized projective synchronization between two different chaotic systems using active backstepping control, Phys. lett. A 355, 326-330 (2006)
[17] Li, Z.; Xu, D.: Stability criterion for projective synchronization in three-dimensional chaotic systems, Phys. lett. A 282, 175-179 (2001) · Zbl 0983.37036 · doi:10.1016/S0375-9601(01)00185-2
[18] Liu, J.; Chen, S. H.; Lu, J. A.: Projective synchronization in a unified chaotic system and its control, Acta phys. Sin 52, 1595-1599 (2003)
[19] Liu, Y. W.; Ge, G. M.; Zhao, H.; Wang, Y. H.: Synchronization of hyperchaotic harmonics in time-delay systems and its application to secure communication, Phys. rev. E 62, 7898-7904 (2000)
[20] Lu, J. A.; Wu, X. Q.; Lü, J. H.: Synchronization of unified chaotic system and the application in secure communication, Phys. lett. A 305, 365-370 (2002) · Zbl 1005.37012 · doi:10.1016/S0375-9601(02)01497-4
[21] Mainieri, R.; Rehacek, J.: Projective synchronization in three-dimensional chaotic systems, Phys. rev. Lett. 82, 3042-3045 (1999)
[22] A.N. Milioua, I.P. Antoniadesa, S.G. Stavrinides, A.N. Anagnostopoulos, Secure communication by chaotic synchronization: Robustness under noisy conditions, Nonlinear Anal: Real World Appl. 8 (2007) 1003 -- 1012. · Zbl 1187.94007 · doi:10.1016/j.nonrwa.2006.05.004
[23] Perora, L. M.; Carroll, T. L.: Synchronization in chaotic systems, Phys. rev. Lett. 64, 821-825 (1990)
[24] Rosenblum, M. G.; Pikovsky, A. S.; Kurths, J.: Phase synchronization of chaotic oscillators, Phys. rev. Lett. 76, 1804-1807 (1996)
[25] Rulkov, N. F.; Sushchik, M. M.; Tsimring, L. S.: Generalized synchronization of chaos in directionally coupled chaotic systems, Phys. rev. E 51, 980-994 (1995)
[26] Taherion, S.; Lai, Y. C.: Observability of lag synchronization of coupled chaotic oscillators, Phys. rev. E 59, 6247-6250 (1999)
[27] Wang, B.; Bu, S.: Controlling the ultimate state of projective synchronization in chaos: application to chaotic encryption, Int. J. Mod. phys. B 18, 2415-2421 (2004)
[28] Xu, D.: Control of projective synchronization in chaotic systems, Phys. rev. E 63, 27201-27204 (2001)
[29] Xu, D.; Chee, C. Y.: Controlling the ultimate state of projective synchronization in chaotic systems of arbitrary dimension, Phys. rev. E 66, 046218 (2002)
[30] Xu, D.; Li, Z.: Controlled projective synchronization in nonpartially-linear chaotic systems, Int. J. Bifurcation and chaos 12, 1395-1402 (2002)
[31] Xu, D.; Li, Z.; Bishop, R.: Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems, Chaos 11, No. 3, 439-442 (2001) · Zbl 0996.37075 · doi:10.1063/1.1380370
[32] Xu, D.; Ong, W. L.; Li, Z.: Criteria for the occurrence of projective synchronization in chaotic systems of arbitrary dimension, Phys. lett. A 305, 167-172 (2002) · Zbl 1001.37026 · doi:10.1016/S0375-9601(02)01445-7
[33] Zou, Y. L.; Zhu, J.: Controlling projective synchronization in coupled chaotic systems, Chin. phys. 15, No. 9, 1965-1970 (2006)