This Letter introduces a new method-mode decomposition-for stability analysis of periodic orbits. Using this method, the stability of a periodic solution of an autonomous system, as well as the stability of synchronization within three chaotic systems with linear coupling, can be analyzed. As an example, a rigorous sufficient condition on the coupling coefficients for achieving chaos synchronization is obtained, for the case of three-coupled identical Lorenz systems. Numerical simulations are shown for demonstration.

An ω-symmetrically coupled system consisting of identical fractional-order differential systems including chaotic and nonchaotic systems is investigated in this paper. Such a coupled system has, in its synchronous state, a mode decomposition by which the linearized equation can be decomposed into motions transverse to and parallel to the synchronous manifold. Furthermore, the decomposition can induce a sufficient condition on synchronization of the overall system, which guarantees, if satisfied, that a group synchronization is achieved. Two typical numerical examples, fractional Brusselators and the fractional Rossler system, are used to verify the theoretical prediction. The theoretical analysis and numerical results show that the lower the order of the fractional system, the longer the time for achieving synchronization at a fixed coupling strength.