Parameter identification and adaptive impulsive synchronization of uncertain complex-variable chaotic systems
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O R I G I N A L PA P E R
Parameter identification and adaptive impulsive synchronization of uncertain complex-variable chaotic systems Song Zheng
Received: 1 January 2013 / Accepted: 17 July 2013 © Springer Science+Business Media Dordrecht 2013
Abstract Synchronization of nonlinear dynamical systems with complex variables has attracted much more attention in various fields of science and engineering. In this paper, the problem of parameter identification and adaptive impulsive synchronization for a class of chaotic (hyperchaotic) complex nonlinear systems with uncertain parameters is investigated. Based on the theories of adaptive control and impulsive control, a synchronization scheme is designed to make a class of chaotic and hyperchaotic complex systems asymptotically synchronized, and uncertain parameters are identified simultaneously in the process of synchronization. Particularly, the proposed adaptive– impulsive control laws for synchronization are simple and can be readily applied in practical applications. The synchronization of two identical chaotic complex Chen systems and two identical hyperchaotic complex Lü systems are taken as two examples to verify the feasibility and effectiveness of the proposed controllers and identifiers. Keywords Complex chaotic system · Parameter identification · Synchronization · Adaptive–impulsive control
S. Zheng (B) Institute of Applied Mathematics, Zhejiang University of Finance & Economics, Hangzhou, Zhejiang 310018, P.R. China e-mail: [email protected]
1 Introduction Synchronization is a fundamental phenomenon arising in many physical, chemical, and biological systems for which there are two or more coupled oscillating systems. Since Pecora and Carrol [1] introduced a method to synchronize two identical chaotic systems with different initial conditions in 1990, synchronization in chaotic dynamic systems has received particular attention in various fields including secure communication, chemical reactions, biological systems, information science, plasma technologies, etc. Now, the notation of synchronization is extended far beyond complete synchronization, such as generalized synchronization, phase synchronization, anticipating synchronization, antisynchronization, lag synchronization, projective synchronization, etc. Until now, a wide variety of approaches have been proposed to achieve chaos synchronization in the coupled chaotic systems, such as linear and nonlinear feedback control [2], time-delay feedback control [3], adaptive control [4], impulsive control [5], and so on. All the above results only considered the synchronization of coupled dynamical systems with real variables. For describing the real world better, many complex dynamical systems have been proposed and studied [6–27]. Fowler et al. [6] firstly introduced the complex Lorenz equations. The complex Chen and Lü systems were introduced and the global synchronization of coupled identical systems were well investigated in [7] and their controls and modified pro-
S. Zheng
jective synchronizat
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