Some theoretical results of synchronization of a linearly coupled dynamical system with random perturbation on a network

Abstract

This paper attempts to investigate synchronization of a coupled dynamical system with random noise on a network (graph). Some mathematical results are obtained in this paper, first, a synchronized threshold is given for a linearly coupled dynamical system perturbed by the random noise; second, when the linearly coupled dynamical system with random noise reaches synchronization, the long time behavior of identical orbit on each node is discussed; finally, some interesting comparison between the synchronization of the coupled gradient system and its small random perturbed system is considered.