Brownian Noise Suppressing Explosion And Its Effects On The Dynamical Behaviors Of Equation

This paper disscusses a nonlinear system dx(t)= axm(t)dt+bxn(t)dB(t) (1.1.1) where a> 0, b> 0, t≥0, m> 0, n≥1,5(t)is a standard one-dimensional Brownian motion.To explain the aim of this paper,let us consider the ordinary equation x(t)= axm(t).Obviously,the solution of this equation will explode to infinity at the finite time when m> 1.In this paper,we will use bx" to perturb the system into Ito form,and show that the solution of equation (1.1.1) with probability one that may not explode.Then we will reveal some important asymptotic properties.This paper is composed of four parts. In the first chapter, we introduce the historical background of the problems which will be investigated, the main results of this paper and some useful tools. In the second chapter, first,we use Lyapunov function to prove the existence and uniqueness of the solution of the stochastic diffenential equation with single perturbation,it is fundamental to study our next main result. Second,we get the a series of asymptotic properties of the solution.In the third chapter, the dynamical behaviors of stochastic differental equation with two-perturbation are given.In the last chapter, we give the summary of this article.