Synchronization And Anti-synchronization Of Several Typical Chaotic Systems

Nonlinear science is a discipline of nonlinear fields. Particularly, Chaos theory is one of important sub-discipline of nonlinear science. This paper studies several synchronizations and anti-synchronizations of chaotic and hyper-chaotic systems. The effectiveness and feasibility of these programs can be proved by theoretical analysis and numerical simulations. The main originality in this paper can be summarized as follows:(1) Using adaptive control techniques, we investigate the multi-switching synchronization of chaotic systems with parameters unknown. Based upon the Lyapunov stability theory, we design the controllers and updating laws of different switching, and it is extended to investigate the synchronization problems with different combinations of slave states with master systems. We take Lorenz system and Chen system as example to analyze the multi-switching synchronization process of different structure of chaotic systems. Finally, numerical simulations have shown the effectiveness of the method.(2) In this paper, we use active control method and adaptive control method separately. Adaptive control method is applied when system parameters are unknown and active control method is applied when system parameters are known. Based upon the Lyapunov stability theory and the adaptive control techniques, the controllers and update laws of parameters of anti-synchronization are designed. Sufficient conditions of anti-synchronization for both cases have been analyzed. Finally, numerical simulations are performed for the hyperchaotic Lorenz system and hyperchaotic Chen system, hyperchaotic Lorenz system and hyperchaotic Lii system to demonstrate the effectiveness of the proposed control strategies.(3) In this paper, we investigate the reduced-order synchronization between unified hyper chaotic systems and unified systems. Based upon the Lyapunov stability theory and the adaptive control techniques, we design the controller of reduced-order synchronization. It is proved theoretically that the controller can make the drive system and the response system with different orders and unknown parameters asymptotically synchronized. Finally, numerical simulations have shown the effectiveness of the method.(4) This paper investigates the finite-time chaos synchronization of unified hyper chaotic system. Based on the finite-time stability theory, we design controllers to synchronize two homogeneous and heterogeneous hyper chaotic systems with disturbed parameters. It is proved theoretically that these controllers can make the drive system and response system asymptotically synchronized. Finally, numerical simulations are performed for two hyper chaotic Lorenz systems, hyper chaotic Lorenz system and hyper chaotic Chen system to demonstrate the effectiveness of the proposed control strategies.