zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Impulsive control-induced effects on dynamics of single and coupled ODE systems. (English) Zbl 1183.70071
Summary: The dynamics of differential system can be changed very obviously after inputting impulse signals. Previous studies show that the single chaotic system can be controlled to periodic motions using impulsive control method. It was well known that the dynamics of hyper-chaotic and coupled systems are very important and more complex than those of a single system. In this paper, particular impulsive control of the hyper-chaotic Lü system was proposed, which is with outer impulsive signals. It can be seen that such impulsive strategy can generate chaos from periodic orbit or control chaos to periodic orbit etc. For the first time, impulsive control induced effects on dynamics of coupled systems are considered in this paper, where the impulse effect has outer input signals. Many interesting and useful results are obtained. The coupled system can realize synchronization and its synchronization manifold can be changed with such impulsive control signals. Strict theories are given, and numerical simulations confirm the correctness of theoretical results.

70Q05Control of mechanical systems (general mechanics)
70K55Transition to stochasticity (chaotic behavior)
Full Text: DOI
[1] Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) · Zbl 0719.34002
[2] Fu, X.L., Yan, B.Q., Liu, Y.S.: Introduction to Impulsive Differential System. Academic Press of China, Beijing (2005)
[3] Yang, T.: Impulsive Control Theory. Springer, Berlin (2001) · Zbl 0996.93003
[4] Liu, X.Z., Willms, A.R.: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Prob. Eng. 2, 277--299 (1996) · Zbl 0876.93014 · doi:10.1155/S1024123X9600035X
[5] Yang, T., Chua, L.O.: Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication. Int. J. Bifurc. Chaos 7, 645--664 (1997) · Zbl 0925.93374 · doi:10.1142/S0218127497000443
[6] Lorenz, E.N.: Deterministic non-periods flows. J. Atmos. Sci. 20, 130--141 (1963) · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[7] Celikovsky, S., Chen, G.R.: On a generalized Lorenz canonical form of chaotic systems. Int. J. Bifurc. Chaos 12, 1789--1812 (2002) · Zbl 1043.37023 · doi:10.1142/S0218127402005467
[8] Zhou, T.S., Chen, G.R., Celikovsky, S.: Sil’nikov chaos in the generalized Lorenz canonical form of dynamics systems. Nonlinear Dyn. 39, 319--334 (2005) · Zbl 1142.70012 · doi:10.1007/s11071-005-4195-8
[9] Lü, J.H., Chen, G.R., Cheng, D., Celikovsky, S.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurc. Chaos 12, 2917--2926 (2002) · Zbl 1043.37026 · doi:10.1142/S021812740200631X
[10] Stojanovski, T., Kocarev, L., Parlitz, U.: Driving and synchronizing by chaotic impulses. Phys. Rev. E 43, 782--785 (1996)
[11] Xie, W.X., Wen, C.Y., Li, Z.G.: Impulsive control for the stabilization and synchronization of Lorenz systems. Phys. Lett. A 275, 67--72 (2000) · Zbl 1115.93347 · doi:10.1016/S0375-9601(00)00584-3
[12] Li, Z.G., Wen, C.Y., Soh, Y.C.: Analysis and design of impulsive control systems. IEEE Trans. Autom. Control. 46, 894--903 (2001) · Zbl 1001.93068 · doi:10.1109/9.928590
[13] Sun, J.T., Zhang, Y.P.: Impulsive control of Rossler systems. Phys. Lett. A 306, 306--312 (2003) · Zbl 1006.37049 · doi:10.1016/S0375-9601(02)01499-8
[14] Wu, X.Q., Lu, J.A., Tse, C.K., Wang, J.J., Liu, J.: Impulsive control and synchronization of the Lorenz systems family. Chaos Solitons Fractals 31, 631--638 (2007) · Zbl 1142.93029 · doi:10.1016/j.chaos.2005.10.017
[15] Chen, A.M., Lu, J.A., Lü, J.H., Yu, S.M.: Generating hyperchaotic Lü attractor via state feedback control. Phys. A 364, 103--110 (2006) · doi:10.1016/j.physa.2005.09.039
[16] Yang, T., Yang, C.M., Yang, L.B.: Control of Rossler systems to periodic motions using control method. Phys. Lett. A 232, 356--361 (1997) · Zbl 1053.93507 · doi:10.1016/S0375-9601(97)00408-8
[17] Ruan, J., Lin, W.: Chaos in a class of impulsive differential equation. Commun. Nonlinear Sci. Numer. Simul. 4, 166--169 (1999) · Zbl 0924.35186
[18] Lin, W.: Some problems in chaotic systems and their applications. Fudan University’s doctoral dissertation (2002)
[19] Sun, J.T., Zhang, Y.P., Qiao, F., Wu, Q.D.: Some Impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach. Chaos Solitons Fractals 19, 1049--1055 (2004) · Zbl 1069.37029 · doi:10.1016/S0960-0779(03)00264-9
[20] Zhou, J., Chen, T.P., Gao, Y.H.: Synchronization dynamics of complex networks with impulsive control. In: Second National Forum on Complex Dynamical Networks, Beijing, pp. 226--230 (2005)
[21] Liu, B., Liu, X.Z., Chen, G.R., Wang, H.Y.: Robust impulsive synchronization of uncertain dynamical networks. IEEE Trans. Circuits Syst.-I 52, 1431--1440 (2005) · doi:10.1109/TCSI.2005.851708
[22] Chua, L.O., Komuro, M., Matsumoto, T.: The double scroll family. Part I: Rigorous proof of chaos. IEEE Trans. Circuits Syst.-I 33, 1073--1097 (1986)
[23] Chua, L.O., Itoh, M., Kocarev, L., Eckert, L.: Chaos synchronization in Chua’s circuit. J. Circuits Syst. Comput. 3, 93--108 (1993) · Zbl 0875.94133 · doi:10.1142/S0218126693000071
[24] Chua, L.O., Yang, T., Zhong, G.Q., Wu, C.W.: Adaptive synchronization of Chua’s oscillators. Int. J. Bifurc. Chaos 6, 189--201 (1996) · doi:10.1142/S0218127496001946