Title:
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Control of a class of chaotic systems by a stochastic delay method (English) |
Author:
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Zhang, Lan |
Author:
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Zhang, Chengjian |
Author:
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Zhao, Dongming |
Language:
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English |
Journal:
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Kybernetika |
ISSN:
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0023-5954 |
Volume:
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46 |
Issue:
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1 |
Year:
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2010 |
Pages:
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38-49 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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A delay stochastic method is introduced to control a certain class of chaotic systems. With the Lyapunov method, a suitable kind of controllers with multiplicative noise is designed to stabilize the chaotic state to the equilibrium point. The method is simple and can be put into practice. Numerical simulations are provided to illustrate the effectiveness of the proposed controllable conditions. (English) |
Keyword:
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random dynamical system |
Keyword:
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unified chaotic system |
Keyword:
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stochastic delay differential equations |
Keyword:
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multiplicative noise |
Keyword:
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maximal Lyapunov exponent |
MSC:
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34H15 |
MSC:
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34K50 |
MSC:
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60H10 |
MSC:
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60H30 |
MSC:
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93C23 |
MSC:
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93D15 |
MSC:
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93E15 |
idZBL:
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Zbl 1201.60061 |
idMR:
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MR2666893 |
. |
Date available:
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2010-06-02T19:40:33Z |
Last updated:
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2013-09-21 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/140052 |
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Reference:
|
[1] S. H.Chen, J. Liu, and J. A. Lu: Tracking control and synchronization of chaotic system based upon sampled-data feedback.Chinese Phys. 11(2002), 233–237. |
Reference:
|
[2] E.Ott, C.Grebogi and J.A.Yorke: Controlling chaos.Phys. Rev. Lett. 64 (1990), 1196–1199. MR 1041523 |
Reference:
|
[3] L. M. Pecora and T. L. Carrol: Synchronization in chaotic systems.Phys. Rev. Lett. 64 (1990), 8, 821–824. MR 1038263 |
Reference:
|
[4] B. R. Andrievskii and A. L. Fradkov: Control of chaos: Methods and Applications.I. Methods. Autom. Telemekh. 5 (2003), 3–45. MR 2093398 |
Reference:
|
[5] B. R. Andrievskii and A. L. Fradkov: Control of chaos: Methods and Applications.II. Appl. Autom. Telemekh. 4 (2004), 3–34. MR 2095138 |
Reference:
|
[6] L. Arnold: Random Dynamical Systems.Springer-Verlag, Berlin 1998. Chap. 1, pp. 5–6. Zbl 1092.34028, MR 1374107 |
Reference:
|
[7] C. T. H. Baker, J. M. Ford, and N. J. Ford: Bifurcation in approximate solutions of stochastic delay differential equations.Internat J. Bifurcation and Chaos 14(2004), 9, 2999–3021. MR 2099159 |
Reference:
|
[8] H. Crauel and F. Flandoli: Additive noise destroys a pitchfork bifurcation.J. Dynam. Diff. Equations 10 (1998), 2, 259–274. MR 1623013 |
Reference:
|
[9] J. K. Hale: Theory of functional differential equations.Appl. Math. Sci., Vol.3, Springer-Verlag, Berlin 1977, Chap. 1, pp. 17–18. Zbl 1092.34500, MR 0390425 |
Reference:
|
[10] D. M. Li, J. A. Lu, X. Q. Wu, and G. R. Chen: Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system.J. Math. Anal. Appl. 323 (2006), 844–853. MR 2260147 |
Reference:
|
[11] J. H. Lü and G. R. Chen: A new chaotic attractor conined.Internat. J. Bifurcation and Chaos 12(2002), 3, 659–661. MR 1894886 |
Reference:
|
[12] X. R. Mao: Stochastic Differential Equations and Their Applications.Horwood Publ. 1997, Chap. 5, pp. 179–183. Zbl 0892.60057 |
Reference:
|
[13] J. H. Lü, G. R. Chen, D. Z. Chen, and S. Čelikovský: Bridge the gap between the Lorenz system and the Chen system.Internat. J. Bifurcation and Chaos 12 (2002), 12, 2917–2926. MR 1956411 |
Reference:
|
[14] V. I. Oseledets: A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems.Trans. Moscow Math. Soc. 19 (1968), 197–231. Zbl 0236.93034 |
Reference:
|
[15] C. Zhang, W. Lin, J. Zhou: Complete and generalized synchronization in a class of noise perturbed chaotic systems.Chaos 17 (2007), 023106-1. MR 2340609 |
Reference:
|
[16] L. Zhang and C. J. Zhang: Control a class of chaotic systems by a simple stochastic method.Dynamics of Continuous, Discrete and Impulsive Systems, Series B, Special Issue on Software Engineering and Complex Networks 14 (2007), S6, 210–214. MR 2378808 |
Reference:
|
[17] W. Q. Zhu and H. L. Huang: Stochastic stability of quasi-non-integrable Hamiltonian systems.J. Sound and Vibration 218 (1998), 769–789. |
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