Exponential synchronization in complex networks with a single coupling delay

This paper investigates local exponential synchronization of a general complex network with a single coupling delay. The coupling matrix and time delay in such a network are both time varying. According to the existence of the invariant synchronization manifold, we study the synchronization issues of the concerned network with k-type and non-k-type topologies, respectively. Several delay-dependent synchronization criteria for arbitrary delayed networks are derived in form of lower dimensional linear matrix inequalities by means of Lyapunov function approach. By solving the matrix inequalities, it is shown that, no matter what topology of the delayed network is, there exists a threshold of delay below which, synchronization is achieved. The effectiveness of our results is illustrated with numerical simulations in the end.