Synchronization And Averaging Principal Of Stochastic Differential Equations

In this thesis,through the equivalent transformation of coupled synchronized system to the multi-scale equations,the relationship between the synchronization problem with singular perturbation and the averaging principle of multi-scale stochastic differential equations is established for the first time.Main goal is to obtain the synchronization of stochastic differential equations with nonlinear multiplicative noises,within the framework of average principle.In addition,the research on synchronization problem provides a new research perspective for the averaging principle of multi-scale equations,and provides an application background for further study of the average principle in the sense of stationary solutions.The main work of this thesis is as follows:In chapter 1,the background of the topic,the research status,and the main research content and innovation are introduced.In chapter 2,the existence and uniqueness conditions of solutions to stochastic differential equations and the concepts of stochastic stationary solutions are briefly reviewed.In chapter 3,a coupled synchronization system with nonlinear multiplicative noise is established,and the synchronization system is transformed into equivalent multi-scale stochastic differential equations(SDEs).On the one hand,a direct linear cross coupled mode of the original SDEs is adopted to obtain a synchronization system with singular perturbations.It does not take the special method of transforming SDEs into stochastic ordinary differential equations,nor introduces the complex exponential factors.On the other hand,through the equivalent transformation,the synchronization problem of the the original coupled system is transformed into the convergence problem of the stationary solutions of multi-scale equations.In chapter 4,for the equivalent multi-scale equations after transformation,the synchronization problem of the coupled system driven by nonlinear multiplicative noise is solved.Independent of the previous single attractor technology,this chapter constructs the relationship between the stationary solution and the general solution,which reduces the convergence problem of the stationary solution to that of the general solution.Therefore,based on the moment estimation of the general solution,the synchronization problem that the stationary solution of the coupled system converges to the stationary solution of the average equation is solved,and the convergence rate is obtained.Finally,some examples are given to show that the existing synchronization results of SDEs driven by additive noise and pure linear product noise are extended to that of SDEs with nonlinear multiplicative noise,by the direct linear cross coupled mode.In chapter 5,the average principle of the stationary solution of fully coupled multiscale equations is considered,as a general form of multi-scale equations studied in Chapter4.Then,the synchronization problem of SDEs is solved within the framework of the average principle,by constructing the relationship between the average principle of multi-scale equations and the synchronization problem of coupled system.Since the existing average principle argument cannot be directly applied to solve the synchronization problem of this thesis,this chapter studies the average principle from a new perspective: First,investigate the convergence of fast process and the corresponding invariant measure,which has never been mentioned in the previous studies.In addition,the average principle of the fully coupled multi-scale equations in which the diffusion term of the slow system depends on the fast variable is reconsidered,different from the existing averaging principle conclusion that the strong convergence of this case is not valid.The relationship between the convergence rate and the parameter order of the coefficients is also discussed.Finally,on the basis of the constructed relationship between the stationary solution and the general solution,the average principle of stationary solution is established for the first time.It differs from the average principle of the general solution with fixed initial value,given in previous literatures.In chapter 6,the existence of stationary solutions under nondissipative conditions and the synchronization of functional stochastic differential equations are studied.Finally,the results of this paper are summarized and the future research work is briefly introduced.