The Existence Of Stationary Distribution For Two Kinds Of Stochastic Coupled Systems

Systems in nature are inevitably affected by environmental noise.Many research results show that large environmental noise can destroy the stability or the equilibrium point of the deterministic system.This may lead to a phenomenon of weak stability,that is,the solution is fluctuating near the equilibrium point,which is called stationary distribution by some scholars.In synchronization problem,when zero equilibrium of error system is destroyed by environmental noise,it may cause stationary distribution.Here,complete synchronization of the original system is destroyed,which produces synchronized stationary distribution.On the other hand,many complex systems in our life can be transformed into general coupled systems.In recent years,coupled systems have been widely concerned.Due to the complexity of stochastic coupled systems,there are few results about synchronized stationary distribution of stochastic coupled systems.Thus,our research focuses on synchronized stationary distribution of stochastic coupled systems,which makes some sense.In this paper,Kirchhoff’s matrix tree theorem,Lyapunov method and stochastic analysis technique are used to study the existence of synchronized stationary distribution of two types of stochastic coupled systems.The method adopted in this paper avoids the difficulty of directly constructing a suitable Lyapunov function.The specific research contents are as follows:In the second chapter,we study the existence of synchronized stationary distribution of leader-follower stochastic coupled systems.The existing results on inner synchronization problem of coupled systems mainly use the combination of Lyapunov method and Kronecker product method.Different from the previous results,this chapter studies this kind of inner synchronization problem combined with Lyapunov method and Kirchhoff’s matrix tree theorem.Some sufficient conditions are obtained to guarantee the existence of synchronized stationary distribution in leader-follower stochastic coupled systems.To illustrate the practicability of theoretical results,an application about stochastic coupled oscillators is given with a numerical example carried out.In the third chapter,it mainly studies the existence of synchronized stationary distribution for a special class of hybrid stochastic coupled systems(stochastic coupled systems with Markovian switching).Using M-matrix method,Lyapunov method,Kirchhoff’s matrix tree theorem and stochastic analysis technique,two sufficient criteria are obtained,ensuring the existence of synchronized stationary distribution for hybrid stochastic coupled systems.It shows that synchronized stationary distribution region is closely related to the intensity of stochastic disturbance and feedback control.And under certain conditions,synchronized stationary distribution will become complete synchronization.Finally,the main results are applied to Chua’s circuit network model,and sufficient criteria are obtained to guarantee the existence of synchronized stationary distribution.At the same time,the numerical example is also given.