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In 1963, Lorenz simplified the Navier-Stokes equations of modeling weather forecasting and discovered sensitivity dependence on initial conditions in a set of three ordinary differential equations. Li and Yorke first presented the name “chaos” in the sense of “period three implies chaos”. Chaos embodies three important principles: extreme sensitivity to initial conditions, cause and effect being not proportional, as well as nonlinearity. Since the discovery of Lorenz system, more chaotic systems have been constructed such as Rossler system, hyperchaotic Rossler system, Chua’s circuit, Henon attractor, logistic map, Chen system, generalized Lorenz system etc.
Chaos has been intensively studied in the last three decades. Almost every nonlinear system in chaotic state is very sensitive to its initial conditions and often practically exhibits irregular behavior. Thus, one might be wise enough to avoid and eliminate such behavior. Chaos control (Chen & Dong, 1998; Liu & Caraballo, 2006; Yan & Yu, 2007; Yassen, 2003) and synchronization (Afraimovich, Cordonet, & Rulkov, 2002; Li, Lu, & Wu, 2005; Lian, Chiang, & Chiu, 2001; Rulkov, Afraimovich, & Lewis, 2001; Rulkov, 2001; Rulkov & Lewis, 2001; Bai & Lonngren, 2000; Heagy, Carroll, & Pecora, 1994; Itoh, Yang, & Chua, 2001; Pecora & Carroll, 1990; Tanaka & Wang, 1998; Xiaofeng & Guanrong, 2007) has attracted a great deal of attention from various fields. Over the last decades, many methods and techniques have been developed, such as OGY method, active control (Bai et al., 2000), feedback and non feedback control (Khan & Singh, 2008; Yu, 1997; Just, Bernard, Ostheimer, Reibold, & Benner, 1997; Pyragas, 1992; Toa, 2006; Yan & Yu, 2007; Yassen, 2005), Impulsive synchronization (Wang, Guan, & Xiao, 2004), adaptive feedback (Huber, 1989; Sun, Tian, Jiang, & Xu, 2007; Yan & Yu, 2007) etc. Recently, linear state error feedback synchronization (Xiaofeng et al., 2007) has provoked a renewal of interest within the context of chaotic dynamical system. Some sufficient conditions of the chaos synchronization approach are derived based on Lyapunov stabilization arguments. According to these conditions and the idea of the adaptive feedback synchronization, we design a new adaptive feedback control method. Compared with some common chaos synchronization methods such as linear feedback control, linear state error feedback control, and feedback and non-feedback control, adaptive feedback control has its characteristic that it is easy to operate in practice, when controlling non-linear dynamical chaotic systems.