在现实生活中，动力系统经常会受到随机因素的影响，这样的问题可由相应的随机偏微分方程描述，随着学科的发展，和实际生活更贴切的随机偏微分方程和随机动力系统受到广泛关注。在本项目中，将考虑几类随机偏微分方程的动力学，主要考虑噪声对于动力学性质的影响，包括吸引子的存在性，不变流形的存在性与逼近，随机分岔的存在性与类型，分岔解的逼近等问题。具体问题包括①具有乘性噪声项的随机Swift-Hohenberg方程的动力学性质，研究它的随机中心流形和随机分岔的细致刻画。②分数阶Brown运动驱动的随机Swift-Hohenberg方程的不变流形的存在性，及分数阶噪声对于分岔结构的影响。③加性噪声对于随机Kuramoto-Sivashinsky方程的随机吸引子稳定性的影响。④一类慢-快随机发展方程的随机慢不变流形的存在性，构造的它的逼近，并考虑慢流形上的随机分岔。

In real life, the dynamical system is often affected by random factors, such problems can be described by the corresponding stochastic partial differential equations, with the development of sciences, stochastic partial differential equations and stochastic dynamical systems which are more appropriate to real life have attracted the attention of more and more researchers. In this project, we will consider the dynamics of several classes of stochastic partial differential equations, mainly considering the influence of noise on dynamic properties, including the existence of attractors, existence and approximation of invariant manifold, existence and type of stochastic bifurcation, approximate solutions of stochastic bifurcation problems. The specific problems include ① the dynamical properties of the stochastic Swift-Hohenberg equation with a multiplicative noise term, and the detailed description of its random center manifold and stochastic bifurcation；②the existence of invariant manifolds for stochastic Swift-Hohenberg equation driven by a fractional Brownian motion, and the effect of fractional noise on bifurcation structure; ③ the effect of additive noise on the stability of random attractor for stochastic Kuramoto-Sivashinsky equation; ④the existence and approximation of random slow invariant manifold for a class of slow fast stochastic evolution equation, and stochastic bifurcation on the slow manifold.