Abstract
This paper derives new results for the hybrid synchronization of identical Liu systems, identical Lu systems, and non-identical Liu and Lu systems via adaptive control method. Liu system (Liu et al. 2004) and Lu system (Lu and Chen, 2002) are important models of three-dimensional chaotic systems. Hybrid synchronization of the three-dimensional chaotic systems addressed in this paper is achieved through the synchronization of the first and last pairs of states and anti-synchronization of the middle pairs of the two systems. Adaptive control method is deployed in this paper for the general case when the system parameters are unknown. Sufficient conditions for hybrid synchronization of identical Liu systems, identical Lu systems and non-identical Liu and Lu systems are derived via adaptive control theory and Lyapunov stability theory. Since the Lyapunov exponents are not needed for these calculations, the adaptive control method is very effective and convenient for the hybrid synchronization of the chaotic systems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the proposed synchronization schemes.
Highlights
A chaotic system is a very special nonlinear dynamical system, which has several properties such as the sensitivity to initial conditions as well as an irregular, unpredictable behaviour
The hybrid chaos synchronization error is defined by e1 = y1 − x1
We discuss the hybrid synchronization of identical Lü chaotic systems ([44], 2002), where the parameters of the master and slave systems are unknown
Summary
A chaotic system is a very special nonlinear dynamical system, which has several properties such as the sensitivity to initial conditions as well as an irregular, unpredictable behaviour. Projective synchronization (PS) is characterized by the fact that the master and slave systems could be synchronized up to a scaling factor, whereas in generalized projective synchronization (GPS), the responses of the synchronized dynamical states synchronize up to a constant scaling matrix α. In hybrid synchronization of chaotic systems [41,42], one part of the system is synchronized and the other part is anti-synchronized so that the complete synchronization (CS) and antisynchronization (AS) coexist in the system. We investigate the hybrid chaos synchronization of uncertain three-dimensional chaotic systems, viz.
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