科学与工程中的动力系统常受到随机因素的影响,包括随机涨落、随机边界作用、随机风力以及系统的内在非高斯扰动等。随机因素也许幅度很小或持续时间很短，但对系统的长期影响可能很微妙或很深远。随机因素有“不利”的影响但也可能有“有利”的作用。因此，非常有必要刻画随机因素对动力学的影响。我们将研究如下问题: 1. 随机偏微分方程在谱裂口条件不满足情况下的 “修正的慢流形”及约化; 2. 通过研究随机慢系统亚稳态现象来定量刻画随机因素对动力学的影响; 3. 将理论研究应用于大气海洋耦合系统。 具体研究包括修正随机慢流形、随机慢系统逼近和预估, 特别是作为稀有事件的亚稳态之间的迁移规律。研究特色是通过分析具体的动力学性质（如不变集, 逃逸概率，极大似然快慢分解) 和估计一些宏观刻划量(如迁移概率, 平均逗留时, 随机不变流形逼近精度) 来揭示随机影响下的动力学性态，以及高斯和非高斯噪声对迁移的不同影响。

Dynamical systems in science and engineering are often under the influences of random fluctuations, such as random forcing, random boundary interactions and nternal agitations. Although random fluctuations may seem small in amplitude or fast in time scale, their impact on overall system evolution may be delicate or even profound. It is thus desirable to characterize the impact of noise on dynamics. We propose to investigate the following inter-related topics:.1. Approximate random slow manifold reduction for stochastic partial differential equations with no gap condition;.2. Metastable phenomena for stochastic partial differential equations and their characterization on random slow manifolds;.3. Applications to geophysical fluid flow systems..The specific research topics include random slow manifold reduction, approximation and numerical validation, modified random slow systems, and especially rare events such as the dynamical transitions between metastable states. Dynamical properties (such as random invariant sets, escape phenomena) are analyzed and macroscopic quantities (such as escape probability, mean residence time near the metastable states, and approximation errors for random slow manifolds) are estimated, in order to achieve better scientific understanding of stochastic dynamics of atmospheric and oceanic fluid flows under random boundary conditions.