Hybrid robust modified function projective lag synchronization in two different dimensional chaotic systems
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O R I G I N A L PA P E R
Hybrid robust modified function projective lag synchronization in two different dimensional chaotic systems Chao Luo · Xingyuan Wang
Received: 3 August 2012 / Accepted: 14 January 2013 © Springer Science+Business Media Dordrecht 2013
Abstract In this article, a novel synchronization scheme, modified function projective lag synchronization (MFPLS) in two different dimensional chaotic systems with parameter perturbations, is proposed. In the proposed method, the states of two nonidentical chaotic systems with different orders are asymptotically lag synchronized up to a desired scaling function matrix by means of reduced order and increased order, respectively. Furthermore, based on the reality situation, the parameter perturbations are involved, which are assumed to appear in both drive and response systems. With the Lyapunov stability theory, an adaptive controller is designed to achieve MFPLS. Theoretical proof and numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme. Keywords Modified function projective lag synchronization (MFPLS) · Hybrid system · Robustness · Adaptive control
C. Luo · X. Wang () Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China e-mail: [email protected] C. Luo e-mail: [email protected]
1 Introduction Since the synchronization scheme of two identical chaotic systems was proposed by Pecora and Carroll [1], this researching area has been developed into wide-scope applications including secure communication, physical systems, biological network, plasma technologies, etc. In the past few years, a great deal of control and synchronization schemes in different kinds of chaotic systems have been presented and discussed [2–15]. Recently, function projective synchronization (FPS) was proposed and extensively investigated [16, 17], in which the drive system synchronizes with the response system up to a scaling function. Moreover, Du et al. [18] proposed a generalization of FPS named modified function projective synchronization (MFPS), in which a desired scaling function matrix instead of a single function is involved into the synchronization scheme between the drive and response systems. MFPS is a generalizing definition of complete synchronization (CS), projective synchronization (PS), modified projective synchronization (MPS), and FPS depending on the form of scaling function matrix. Therefore, the investigation of modified function projective synchronization is widely noticed [19–21]. Today, most researchers focus on chaotic synchronization of the same dimension of systems. However, in reality, different order is never the obstacle to achieve the synchronization between drive and re-
C. Luo, X. Wang
sponse systems, especially in chemical and biological science. In the case of thalamic neurons, lower-order neurons are always driven by the output from neurons with higher order [22] and the synchronization can be only completed by means of reduced order. And in th
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