Coexistence of Synchronization and Anti-Synchronization for Chaotic Systems via Feedback Control

The study of chaos can be introduced in several applications as: medical field, fractal theory, electrical circuits and secure communication, essentially based on synchronization techniques. The synchronization for chaotic systems has been extended to the scope, such as generalized synchronization, phase synchronization, lag synchronization, and even anti-synchronization (Pecora & Carroll, 1990; Wu & Chua, 1993; Michael et al., 1996; Bai & Lonngsen, 1997; Yang & Duan, 1998; Taherion & Lai, 1999; Zhang & Sun, 2004; Li, 2005; Yassen, 2005; Hammami et al., 2009; Juhn et al., 2009; Hammami et al., 2010b). For an n master chaotic system coupled to an n slave one, described respectively by ( ) ( ( )) m m x t f x t = $ and by ( ) ( ( )), s s x t g x t = $ where m x and s x are phase space variables, and (.) f and (.) g the corresponding nonlinear functions, the synchronization in a direct sense implies that 0 ( ) ( ) s m x t x t − → as . t →+∞ The property of anti-synchronization constitutes a prevailing phenomenon in symmetrical oscillators, in which the state vectors have the same absolute values but opposite signs, which implies that 0 ( ) ( ) s m x t x t + → as . t →+∞ When synchronization and anti-synchronization coexist, simultaneously, in chaotic systems, the synchronization is called hybrid synchronization (Juhn et al., 2009).