For multi-degree-of-freedom dynamical systems, a novel control method was proposed to realize synchronization and anti-synchronization of chaos between the original and derived systems. The presented method was proved theoretically and realized by linear coupling of velocity. Furthermore, for any coupling coefficient larger than the critical value, the two systems can continue synchronization and anti-synchronization by the coupling method. This means that the synchronization and anti-synchronization can be kept within a wide range of coupling parameters. The simulation results for Duffing and Mathieu systems proved that the proposed method was correct and effective.

To realize the synchronization of nonlinear dynamical system, the general control method is unidirectional linear coupling. Research on function coupling of chaos synchronization is not enough, so there arises a question: for nonlinear dynamical system, if chaos synchronization is realized by linear coupling, whether can any type of function coupling always make the system go to chaos synchronization? In this paper, a class of nonlinear dynamical system is considered and the relation between linear coupling and function coupling is investigated. It is proved that if linear coupling can make chaos synchronization, then any function satisfying some conditions can do so too. The condition is given and proved. Finally for Duffing system, three coupling functions are used to prove the analytical result. The simulation results show that the conclusion is correct.