Synchronization Of Random Dynamical System Driven By Wiener Processes

This paper mainly study synchronization phenomena of random dynamical system driven by Wiener noise, it is a improvement of the synchronization phenomena of random dynamical system driven by Gaussian noises or Lévy noises. We mainly improve the prove method. This paper mainly focus on the following two aspects:Firstly, discusses the synchronization phenomena of random dynamical system with additive and multiplicative Wiener noises, when the system exists additive Wiener noises, by using the conclusions proved in the research of random dynamical system with additive Gaussian noises,at the same time we use the method with moving random items to one side of the equation and Gronwall inequality to prove the boundedness of solution, while the corresponding difference between any pair of solutions of the equation tends to zero, which proves the existence, uniqueness and boundedness of stochastic stationary solutions of dissipatively coupled stochastic differential equations, when the system exists linear multiplicative Wiener noises, the stochastic differential equations are transformed to random ordinary differential equations using a transposition of transformation which involving the corresponding Wiener processes, and then by calculating corresponding difference between any pair of solutions of the random ordinary differential equations and the distance from the origin, prove the existence, uniqueness and boundedness of stochastic stationary solution of dissipatively coupled stochastic differential equations; And then, we verify the stochastic stationary solutions of dissipatively coupled stochastic differential equations converge to the stochastic stationary solutions of "average" equation, which gives the results of synchronization of random dynamical system driven by additive and linear multiplicative Wiener noises.In this paper, simple and easy using transposed and the Ito integration to prove, unlike many other articles, which need the help of O-U process or langevin equation, random attractors etc., similarly we can use this method in the synchronization of stochastic dynamic system driven by Gaussian noises, Lévy noises, or O-U noises and so on.