Impulsive synchronization of chaotic systems Chuandong Li a and Xiaofeng Liao College of Computer Science and Engineering, Chongqing University, 400030 China Xingyou Zhang College of Mathematics and Physics, Chongqing University, 400030 China and Institute of Fundamental Science, Massey University, Private Bag 11 222, Palmerston North, New Zealand Received 20 January 2004; accepted 9 March 2005; published online 27 April 2005 The issue of impulsive synchronization of a class of chaotic systems is investigated. Based on the impulsive theory and linear matrix inequality technique, some less conservative and easily verified criteria for impulsive synchronization of chaotic systems are derived. The proposed method is applied to the original Chua oscillators, and the corresponding synchronization conditions are obtained. Moreover, the boundary of the stable region is also estimated in terms of the equidistant impulse interval. The effectiveness of our method is shown by computer simulation. © 2005 American Institute of Physics. DOI: 10.1063/1.1899823 Many synchronization schemes for chaotic systems have been reported in the literature (see Refs. 1–4 and 7–9, and the references therein). In these schemes (for driving- response systems), the continuous signals derived from the driving systems are almost always used to drive the response systems. However, the continuous signals trans- mitted along the channel may lead to several disadvan- tages such as, e.g., multichannel transmission and control complexity. Based on the impulsive control theory, 10 many researchers (e.g., Yang 10–12 and Sun 15,16 ) have in- vestigated the impulsive synchronization schemes and achieved the synchronization of several types of chaotic systems using only small control impulses. The strength of this scheme exists in the practicality in transmission and control because the synchronization signal is actually discrete. I. INTRODUCTION Over the last decade, synchronization of coupled chaotic system has attracted a great deal of attention due to its po- tential applications for secure communications 1–6 since the pioneering work of Pecora and Carroll. 7,8 Different regimes, namely, complete synchronization, phase synchronization, generalized synchronization, lag synchronization, and antici- pative synchronization, etc., have been investigated for vari- ous chaotic systems such as, e.g., chaotic circuits, chaotic laser systems, pairs of neurons, chemical oscillators. A com- plete synchronization implies coincidence of states of inter- acting systems; a generalized synchronization is defined as the presence of some functional relation between the states of response and driver for drive–response systems; phase synchronization means entrainment of phases of chaotic os- cillators; lag synchronization appears as a coincidence of shifted-in-time states of two systems; and anticipative syn- chronization is that a dissipative chaotic system with a time- delayed feedback could drive an identical system in such a way that the driven system anticipates the driver by synchro- nizing with their future states. For more information about these chaotic systems and synchronization regimes, we refer the readers to the review monograph, 9 where the authors pre- sented the main ideas involved in the field of synchronization of chaotic systems and relevant experimental applications in detail. Because impulsive control allows the stabilization and synchronization of chaotic systems using only small control impulses, it has been widely used to stabilize and synchro- nize chaotic systems, 10–16 even though the chaotic behavior may follow unpredictable patterns. The main idea of impul- sive control approach is to change the states of a system wherever some conditions are satisfied. These impulsively controlled systems are described by impulsive differential equations. Recently, several impulsive synchronization schemes have been reported in the literature. 10,15 Yang et al. 11 achieved the synchronization of two identical chaotic systems, i.e., Chua circuit, using the state variable at the fixed instant time as the impulsive signal. The authors inves- tigated the impulsive synchronization criteria for coupled chaotic systems via unidirectional linear error feedback ap- proach in Ref. 15. The stabilization and synchronization of Lorenz systems via impulsive signal are studied in Ref. 16. However, for many-dimension chaotic systems described by the ODE or DDE ordinary or delayed differential equa- tions, the less conservative synchronization criteria such as, e.g., Theorem 2 in Ref. 14, are generally difficult to verify. Motivated by the aforementioned comments, we further investigate the impulsive synchronization of chaotic systems in this paper. Based on impulsive control theory and linear matrix inequality technique, some less conservative and eas- ily verified criteria for impulsive synchronization of chaotic systems are derived. With the help of the LMI toolbox in MATLAB, we obtain the synchronization conditions conve- niently, and consequently estimate the stable region of syn- chronized systems. The proposed method is also applied to a Author to whom correspondence should be addressed. 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