Chaos control of chaotic dynamical systems using backstepping design. (English) Zbl 1102.37306

Summary: This work presents chaos control of chaotic dynamical systems by using backstepping design method. This technique is applied to achieve chaos control for each of the dynamical systems of Lorenz, Chen and Lü systems. Based on Lyapunov stability theory, control laws are derived. We use the same technique to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. Numerical simulations are shown to verify the results.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93C10 Nonlinear systems in control theory
93D99 Stability of control systems
Full Text: DOI


[1] Chen, G.; Dong, X., From chaos to order: perspectives, methodologies and applications (1998), World Scientific: World Scientific Singapore
[2] Chen, G.; Ueta, T., Yet another chaotic attractor, Int J Bifurcat Chaos, 9, 1465 (1999) · Zbl 0962.37013
[3] Ueta, T.; Chen, G., Bifurcation analysis of Chen’s attactor, Int J Bifurcat Chaos, 10, 1917 (2000) · Zbl 1090.37531
[4] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys Rev Lett, 64, 1196 (1990) · Zbl 0964.37501
[5] Fuh, C. C.; Tung, P. C., Controlling chaos using differential geometric method, Phys Rev Lett, 75, 2952 (1995)
[6] Chen, G.; Dong, X., On feedback control of chaotic continuous time systems, IEEE Trans Circ Syst, 40, 591 (1993) · Zbl 0800.93758
[7] Yassen, M. T., Chaos control of Chen chaotic dynamical system, Chaos, Solitons & Fractals, 15, 271 (2003) · Zbl 1038.37029
[8] Yassen, M. T., Controlling chaos and synchronization for new chaotic system using linear feedback, Chaos, Solitons & Fractals, 26, 913 (2005) · Zbl 1093.93539
[9] Agiza, H. N., Controlling chaos for the dynamical system of coupled dynamos, Chaos, Solitons & Fractals, 12, 341 (2002) · Zbl 0994.37047
[10] Sanchez, E. N.; Perez, J. P.; Martinez, M.; Chen, G., Chaos stabilization: an inverse optimal control approach, Latin Am Appl Res: Int J, 32, 111 (2002)
[11] Yassen, M. T., Adaptive control and synchronization of a modified Chua’s circuit system, Appl Math Comp, 135, 113 (2001) · Zbl 1038.34041
[12] Liao, T.-L.; Lin, S.-H., Adaptive control and synchronization of Lorenz systems, J Franklin Inst, 336, 925 (1999) · Zbl 1051.93514
[13] Lü, J.; Zhang, S., Controlling Chen’s chaotic attractor using backstepping design based on parameters identificaion, Phys Lett A, 286, 148 (2001) · Zbl 0969.37509
[14] Lorenz, E. N., Deterministic non-periodic flows, J Atmos Sci, 20, 130 (1963)
[15] Stewart, I., The Lorenz attractor exists, Nature, 406, 948 (2000)
[16] Vanêĉek, A.; Ĉelikovskŷ, S., Control systems: from linear analysis to synthesis of chaos (1996), Prentice-Hall: Prentice-Hall London · Zbl 0874.93006
[17] Lü, J.; Chen, G., A new chaotic attractor coined, Int J Bifurcat Chaos, 12, 659 (2002) · Zbl 1063.34510
[18] Lü, J.; Chen, G.; Zhang, S., Dynamical analysis of a new chaotic attractor, Int J Bifurcat Chaos, 12, 1001 (2002) · Zbl 1044.37021
[19] Lü, J.; Chen, G.; Zhang, S., The compound structure of a new chaotic attractor, Chaos, Solitons & Fractals, 14, 669 (2002) · Zbl 1067.37042
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.