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Anti-synchronization on autonomous and non-autonomous chaotic systems via adaptive feedback control. (English) Zbl 1197.37138

Summary: The anti-synchronization of a general class of chaotic systems is investigated. A simple adaptive feedback scheme is proposed to anti-synchronize many familiar chaotic systems, including autonomous systems and non-autonomous systems. Lyapunov analysis for the error system gives the asymptotic stability conditions based on the invariance principle of differential equations. The schemes are successfully applied to three groups of examples: the van der Pol-Duffing oscillator, the parametrically harmonically excited 4D system, and the additionally harmonically excited Murali-Lakshmanan-Chua circuit. Numerical results are presented to justify the theoretical analysis in this paper.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

37N35 Dynamical systems in control
34D20 Stability of solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93D21 Adaptive or robust stabilization
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[1] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys Rev Lett, 64, 821-824 (1990) · Zbl 0938.37019
[2] 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
[3] Huang, L. L.; Feng, R. P.; Wang, M., Synchronization of chaotic systems via nonlinear control, Phys Lett A, 320, 271-275 (2004) · Zbl 1065.93028
[4] Park, E. H.; Zaks, M. A.; Kurths, J., Phase synchronization in the forced Lorenz system, Phys Rev E, 60, 6627-6638 (1999) · Zbl 1062.37502
[5] Banerjee, S.; Saha, P.; Chowdhury, A. R., On the application of adaptive control and phase synchronization in non-linear fluid dynamics, Int J Non-Linear Mech, 39, 25-31 (2004) · Zbl 1225.76137
[6] Yang, S. S.; Juan, C. K., Generalized synchronization in chaotic systems, Chaos, Solitons & Fractals, 9, 1703-1707 (1998) · Zbl 0946.34040
[7] Hramov, A. E.; Koronovskii, A. A., Generalized synchronization: a modified system approach, Phys Rev E, 71, 067201 (2005)
[8] Rosenblum, M. G.; Pikovsky, A. S.; Kurths, J., From phase to lag synchronization in coupled chaotic oscillators, Phys Rev Lett, 78, 4193-4196 (1997)
[9] Shahverdiev, E. M.; Sivaprakasam, S.; Shore, K. A., Lag synchronization in time-delayed systems, Phys Lett A, 292, 320-324 (2002) · Zbl 0979.37022
[10] Xu, D. L.; Li, Z. G.; Bishop, S. R., Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems, Chaos, 11, 439-442 (2001) · Zbl 0996.37075
[11] Yan, J. P.; Li, C. P., Generalized projective synchronization of a unified chaotic system, Chaos Solitons & Fractals, 26, 1119-1124 (2005) · Zbl 1073.65147
[12] Lin, W.; He, Y. B., Complete synchronization of the noise-perturbed Chua’s circuits, Chaos, 15, 023705 (2005)
[13] Yoshida, K.; Sato, K.; Sugamata, A., Noise-induced synchronization of uncoupled nonlinear systems, J Sound Vib, 290, 34-47 (2006)
[14] Hasler, M.; Maistrenko, Y.; Popovych, O., Simple example of partial synchronization of chaotic systems, Phys Rev E, 58, 6843-6846 (1998)
[15] Lim, W.; Kim, S.-Y., Coupling effect on the occurrence of partial synchronization in four coupled chaotic systems, Phys Lett A, 353, 398-406 (2006) · Zbl 1181.37045
[16] Yan, Z. Y., Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems – a symbolic numeric computation approach, Chaos, 15, 023902 (2005) · Zbl 1080.93008
[17] Wang, Q.; Chen, Y., Generalized Q-S (lag, anticipated and complete) synchronization in modified Chua’s circuit and Hindmarsh-Rose systems, Chaos Solitons & Fractals, 181, 48-56 (2006) · Zbl 1145.37312
[18] Liu, W. Q., Antiphase synchronization in coupled chaotic oscillators, Phys Rev E, 73, 057203 (2006)
[19] Liu, J. B.; Ye, C. F.; Zhang, S. J.; Song, W. T., Anti-phase synchronization in coupled map lattices, Phys Lett A, 274, 27-29 (2004) · Zbl 1050.37518
[20] Uchida, A.; Liu, Y.; Fischer, I.; Davis, P.; Aida, T., Chaotic antiphase dynamics and synchronization in multimode semiconductor lasers, Phys Rev A, 64, 023801 (2001)
[21] Kim, C. M.; Rim, S. H.; Kye, W. H.; Ryu, J. W.; Park, Y. J., Anti-synchronization of chaotic oscillators, Phys Lett A, 320, 39-46 (2003) · Zbl 1098.37521
[22] Zhang, Y. P.; Sun, J. T., Chaotic synchronization and anti-synchronization based on suitable separation, Phys Lett A, 330, 442-447 (2004) · Zbl 1209.37039
[23] Li, G. H., Synchronization and anti-synchronization of colpitts oscillators using active control, Chaos Solitons & Fractals, 26, 87-93 (2005) · Zbl 1122.34320
[24] Hu, J.; Chen, S. H.; Chen, L., Adaptive control for anti-synchronization of Chua’s chaotic system, Phys Lett A, 339, 455-460 (2005) · Zbl 1145.93366
[25] Chiang TY, Lin JS, Liao TL, Yan JJ. Anti-synchronization of uncertain unified chaotic systems with dead-zone nonlinearity. Nonlinear Anal TMA 2007; in press.; Chiang TY, Lin JS, Liao TL, Yan JJ. Anti-synchronization of uncertain unified chaotic systems with dead-zone nonlinearity. Nonlinear Anal TMA 2007; in press.
[26] Huang, D. B., Simple adaptive-feedback controller for identical chaos synchronization, Phys Rev E, 58, 6843-6846 (1998)
[27] Philominathan, P.; Neelamegam, P., Characterization of chaotic attractors at bifurcations in Murali-Lakshmanan-Chua’s circuit and one-way coupled map lattice system, Chaos Solitons & Fractals, 12, 1005-1017 (2001) · Zbl 1016.37017
[28] Bowong, S.; McClintock, P. V.E., Adaptive synchronization between chaotic dynamical systems of different order, Phys Lett A, 358, 134-141 (2006) · Zbl 1142.93405
[29] Qi, G. Y.; Du, S. Z.; Chen, G. R.; Chen, Z. Q.; Yuan, Z. Z., On a four-dimensional chaotic system, Chaos Solitons & Fractals, 23, 1671-1682 (2005) · Zbl 1071.37025
[30] Hua, C. C.; Guan, X. P.; Shi, P., Adaptive feedback control for a class of chaotic systems, Chaos Solitons & Fractals, 23, 757-765 (2005) · Zbl 1061.93503
[31] Baˇdescu, C. S.; Margareta, I.; Oprişan, S., On the chaotic oscillations of Bloch walls and their control, Chaos Solitons & Fractals, 8, 33-43 (1997) · Zbl 0916.58025
[32] Hassan, K. K., Nonlinear systems (2002), Prentice Hall: Prentice Hall New Jersey · Zbl 1003.34002
[33] Fotsin, H. B.; Woafo, P., Adaptive synchronization of a modified and uncertain chaotic Van der Pol-Duffing oscillator based on parameter identification, Chaos Solitons & Fractals, 24, 1363-1371 (2005) · Zbl 1091.70010
[34] Lei, Y. M.; Xu, W.; Shen, J. W., Robust synchronization of chaotic non-autonomous systems using adaptive-feedback control, Chaos Solitons & Fractals, 31, 371-379 (2007) · Zbl 1142.93425
[35] Murali, K.; Lakshmanan, M., Synchronization through compound chaotic signal in Chua’s circuit and Murali-Lakshmanan-Chua circuit, Int J Bifur Chaos, 7, 415-421 (1997) · Zbl 0890.94041
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