The existence of inner synchronized stationary distribution for stochastic coupled systems on networks

This paper is concerned with the existence of inner synchronized stationary distribution for stochastic coupled systems on networks (SCSNs), which is the first time to consider this problem. Compared with the existing results on inner synchronization problem, we study the problem based on Lyapunov method and Kirchhoff's Matrix Tree Theorem in graph theory without utilizing Kronecker product method and Linear matrix inequalities, which simplifies some complex analysis and avoids difficulties. Then some new sufficient conditions are presented to guarantee the existence of inner synchronized stationary distribution for SCSNs. These conditions show that the existence domain of inner synchronized stationary distribution has a close relationship with stochastic perturbation intensity. And when stochastic perturbation vanishes, inner synchronized stationary distribution will become complete synchronization. To illustrate the practicability of theoretical results, an application about stochastic coupled oscillators is given with a numerical example being carried out.