Synchronization of Genesio chaotic system via backstepping approach. (English) Zbl 1091.93028

Summary: Backstepping design is proposed for synchronization of Genesio chaotic system. Firstly, the control problem for the chaos synchronization of nominal Genesio systems without unknown parameters is considered. Next, an adaptive backstepping control law is derived to make the error signals between drive Genesio system and response Genesio system with an uncertain parameter asymptotically synchronized. Finally, the approach is extended to the synchronization problem for the system with three unknown parameters. The stability analysis in this article is proved by using a well-known Lyapunov stability theorem. Note that the approach provided here needs only a single controller to realize the synchronization. Two numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.


93D15 Stabilization of systems by feedback
34C28 Complex behavior and chaotic systems of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI


[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] Lu, H.; He, Z., Chaotic behavior in first-order autonomous continuous-time systems with delay, IEEE Trans Circuit Syst, 43, 700-702 (1996)
[4] Chen, G., Chaos on some controllability conditions for chaotic dynamics control, Chaos, Solitons & Fractals, 8, 1461-1470 (1997)
[5] Pyragas, K., Continuous control of chaos by self-controlling feedback, Phys Lett A, 170, 421-428 (1992)
[6] Park, J. H.; Kwon, O. M., LMI optimization approach to stabilization of time-delay chaotic systems, Chaos, Solitons & Fractals, 23, 445-450 (2005) · Zbl 1061.93509
[7] Park, J. H., Adaptive synchronization of a unified chaotic systems with an uncertain parameter, Int J Nonlinear Sci Numer Simul, 6, 201-206 (2005) · Zbl 1401.93123
[8] Wang, C. C.; Su, J. P., A new adaptive variable structure control for chaotic synchronization and secure communication, Chaos, Solitons & Fractals, 20, 967-977 (2004) · Zbl 1050.93036
[9] Wu, X.; Lu, J., Parameter identification and backstepping control of uncertain Lü system, Chaos, Solitons & Fractals, 18, 721-729 (2003) · Zbl 1068.93019
[10] Hu, J.; Chen, S.; Chen, L., Adaptive control for anti-synchronization of Chua’s chaotic system, Phys Lett A, 339, 455-460 (2005) · Zbl 1145.93366
[11] Li, D.; Lu, J. A.; Wu, X., Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems, Chaos, Solitons & Fractals, 23, 79-85 (2005) · Zbl 1063.37030
[12] Lü, J.; Zhou, T.; Zhang, S., Chaos synchronization between linearly coupled chaotic systems, Chaos, Solitons & Fractals, 14, 529-541 (2002) · Zbl 1067.37043
[13] Park, J. H., Stability criterion for synchronization of linearly coupled unified chaotic systems, Chaos, Solitons & Fractals, 23, 1319-1325 (2005) · Zbl 1080.37035
[14] Agiza, H. N.; Yassen, M. T., Synchronization of Rossler and Chen chaotic dynamical systems using active control, Phys Lett A, 278, 191-197 (2001) · Zbl 0972.37019
[15] Wang, Y.; Guan, Z. H.; Wang, H. O., Feedback an adaptive control for the synchronization of Chen system via a single variable, Phys Lett A, 312, 34-40 (2003) · Zbl 1024.37053
[16] Bai, E. W.; Lonngren, K. E., Sequential synchronization of two Lorenz systems using active control, Chaos, Solitons & Fractals, 11, 1041-1044 (2000) · Zbl 0985.37106
[17] Lu, J.; Wu, X.; Han, X.; Lü, J., Adaptive feedback synchronization of a unified chaotic system, Phys Lett A, 329, 327-333 (2004) · Zbl 1209.93119
[18] Yang, X. S.; Chen, G., Some observer-based criteria for discrete-time generalized chaos synchronization, Chaos, Solitons & Fractals, 13, 1303-1308 (2002) · Zbl 1006.93580
[19] Huang, L.; Feng, R.; Wang, M., Synchronization of chaotic systems via nonlinear control, Phys Lett A, 320, 271-275 (2004) · Zbl 1065.93028
[20] Chen, M.; Han, Z., Controlling and synchronizing chaotic Genesio system via nonlinear feedback control, Chaos, Solitons & Fractals, 17, 709-716 (2003) · Zbl 1044.93026
[21] Genesio, R.; Tesi, A., A harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, Automatica, 28, 531-548 (1992) · Zbl 0765.93030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.