Abstract
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.
Similar content being viewed by others
References
Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
Fu, X.L., Yan, B.Q., Liu, Y.S.: Introduction to Impulsive Differential System. Academic Press of China, Beijing (2005)
Yang, T.: Impulsive Control Theory. Springer, Berlin (2001)
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)
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)
Lorenz, E.N.: Deterministic non-periods flows. J. Atmos. Sci. 20, 130–141 (1963)
Celikovsky, S., Chen, G.R.: On a generalized Lorenz canonical form of chaotic systems. Int. J. Bifurc. Chaos 12, 1789–1812 (2002)
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)
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)
Stojanovski, T., Kocarev, L., Parlitz, U.: Driving and synchronizing by chaotic impulses. Phys. Rev. E 43, 782–785 (1996)
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)
Li, Z.G., Wen, C.Y., Soh, Y.C.: Analysis and design of impulsive control systems. IEEE Trans. Autom. Control. 46, 894–903 (2001)
Sun, J.T., Zhang, Y.P.: Impulsive control of Rossler systems. Phys. Lett. A 306, 306–312 (2003)
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)
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)
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)
Ruan, J., Lin, W.: Chaos in a class of impulsive differential equation. Commun. Nonlinear Sci. Numer. Simul. 4, 166–169 (1999)
Lin, W.: Some problems in chaotic systems and their applications. Fudan University’s doctoral dissertation (2002)
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)
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)
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)
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)
Chua, L.O., Itoh, M., Kocarev, L., Eckert, L.: Chaos synchronization in Chua’s circuit. J. Circuits Syst. Comput. 3, 93–108 (1993)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Han, X., Lu, Ja. Impulsive control induced effects on dynamics of single and coupled ODE systems. Nonlinear Dyn 59, 101–111 (2010). https://doi.org/10.1007/s11071-009-9524-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-009-9524-x