Achieving Synchronization in Arrays of Coupled Differential Systems with Time-Varying Couplings

Abstract

We study complete synchronization of the complex dynamical networks described by linearly coupled ordinary differential equation systems (LCODEs). Here, the coupling is timevarying in both network structure and reaction dynamics. Inspired by our previous paper (Lu et al. (2007-2008)), the extended Hajnal diameter is introduced and used to measure the synchronization in a general differential system. Then we find that the Hajnal diameter of the linear system induced by the time-varying coupling matrix and the largest Lyapunov exponent of the synchronized system play the key roles in synchronization analysis of LCODEs with identity inner coupling matrix. As an application, we obtain a general sufficient condition guaranteeing directed time-varying graph to reach consensus. Example with numerical simulation is provided to show the effectiveness of the theoretical results.