Dynamic Analysis Of A Class Of Stochastic Differential Equations

In this thesis,stochastic rumor propagation model with white noise and nonlinear pertur-bation are studied,respectively.The existence and uniqueness of solutions of the model are proved,stability and instability of equilibrium point are discussed by using Lyapunov function-al analysis method,and the existence and uniqueness of ergodicity stationary distribution are also considered.This thesis is divided into the following two parts:The first part is a stochastic rumor propagation model with white noise.It is proved that equilibrium point is global asymptotic stable in probability.Furthermore,in the case of R0<1,the crowd got infected by rumor will converge to 0 exponentially almost surely.Sufficient conditions for the instable in probability of the solution of the rumor propagation model are given.We obtain the limiting behavior of solution in the case of R0>1.In particular,in the case of small noise intensity,the solution of random rumor propagation model will oscillate near E0.The second part is a stochastic rumor propagation model with nonlinear perturbation.With the help of the Khasminskii s lemma,(i.e.,Lemma 2.5 of the second chapter),the sufficient con-ditions for the unique ergodicity stationary distribution of stochastic rumor propagation model are given by constructing suitable Lyapunov functional.