Asymptotic Behavior Of Two Classes Of Dynamical Systems With Small Random Perturbations

Random perturbations are ubiquitous in nature,and studying the asymptotic behavior of dynamical systems under random perturbations has always been one of the hotspots and difficulties in the field of mathematics and engineering.When the magnitude of the perturbation is small,but it can affect the behavior of a dynamical system,it is very meaningful to study dynamical systems under small random perturbations.This dissertation studies the asymptotic behavior of two classes of dynamical systems with small random perturbations.One is to study the effect of coupling strength and small noise strength on the synchronized system,and the other is to study the effect of small mass and small noise strength on the second-order Mc Kean-Vlasov stochastic systems,including their central limit theorem,large deviations principle and moderate deviations principle,etc.The main results and findings are briefly concluded as follows:(1)We study the average principle and central limit theorem of synchronized system with small noise.In this paper,we convert the synchronized system equivalently into a new type of slow-fast system,establish the connection between the synchronization problem and the asymptotic behavior of the slow-fast system,and then study the asymptotic behavior of the slow-fast system.According to the interaction between noise strength and coupling strength,the average principle of slow process is obtained in three different cases,and then the central limit theorem of slow process in continuous function space is obtained by martingale method,so that the average principle and central limit theorem of synchronized system are obtained through equivalence relation.(2)We study the large deviations principle of synchronized system.In this paper,the weak convergence method is used to prove that the slow process in cases 1 and 2 satisfies the large deviations principle,a unifying expression of the rate function is given,and an explicit expression formula of the rate function in case 1 is obtained,and then the large deviations principle of the synchronized system is obtained through the equivalence relation.Finally,for a special system,the representation of the rate function in case 3 is given,and the synchronization of quasipotential for a linear system is studied.This paper assumes that the fast and slow processes take values in the entire Euclidean space and the coefficients satisfy the linear growth condition,which makes the analysis of tightness and the existence and uniqueness of invariant measures of operators involving control variables more complicated.(3)We study the asymptotic behavior of a class of second-order Mc Kean-Vlasov stochastic differential equations.This chapter first proves that the Smoluchowski-Kramers approximation of the second-order Mc Kean-Vlasov stochastic differential equations holds.Then,using the weak convergence method,the large and moderate deviations principle is established for such equations under the interactive influence of small mass and small noise strength.These results generalize the corresponding results of Smoluchowski-Kramers approximation of classical stochastic differential equations to the case where the coefficients are dependent on the distribution,and reveal the interaction mechanism of noise strength and small mass.