The Analysis And Study Of The Influence Of Noise On Differential Systems

As is well known, noise has a great influence on deterministic systems. Actually, it can be used to not only suppress fluctuation of a unstable system, but also keep stability of a stable system. In this paper, we introduce a new polynomial growth condition. Under this condition and local Lipschitz condition, we study two problems: one is noise suppresses fluctuation of stochastic differential system, and the other is noise expresses mean square globally exponential stability of stochastic differential time delay system.Now, we introduce our main results as follow:Firstly, we deal with the problem of noise suppresses fluctuation of stochastic differential system.In this part, we introduce a new polynomial growth condition, which not only greatly relaxes the form of drift term, but also overcomes the boundedness of stochas-tic feedback form and limitation of application scope of traditional growth condition. Under this condition and local Lipschitz condition, we prove that even the state of deterministic differential system is unknown, noise can suppresses fluctuation of cor-responding stochastic differential system from following three aspects:(1) The existence and uniqueness of stochastic differential system solution, i.e., we can construct a suitable stochastic feedback term to make sure the stochastic dif-ferential system exists a unique global solution even the corresponding deterministic differential system may explode in a finite time;(2) The global solution is limited in the sense of moment;(3) The global solution grows at most polynomially.Besides, we can make above three results still apply to the time delay system by add time delay terms in the new polynomial growth condition.Secondly, we study the problem of noise expresses mean square globally expo- nential stability of stochastic differential time delay system.In this part, we show that when the determinate time delay system is globally exponentially stable, noise can make the corresponding stochastic differential delay system be mean square globally exponentially stable. In particular, we remove the traditional linear growth condition and introduce a new polynomial growth condition with time delay for diffusion term on the basis of the new polynomial growth condi-tion. It is clear to see that, this condition overcomes the boundedness of diffusion term form and makes more systems can be studied.Finally, we present some unsolved problems.