本项目研究Morse-Smale系统在非退化随机扰动下的稳定性，即，Morse-Smale系统在非退化随机扰动下，研究当噪声强度趋于零时，平稳测度族的极限测度及其支撑结构，探索什么样的回复运动被保持下来。我们的研究目标是证明极限测度是某些稳定奇点的Delta测度和某些稳定周期轨道的Haar测度的凸组合。为实现这一目标我们需要完成如下三个任务：1. 证明极限测度在（双曲）排斥子上的测度为零；2. 证明极限测度在鞍型奇点和鞍型周期轨道上的测度为零； 3. 分析极限测度在哪些稳定奇点和周期轨道上的测度为零。同时，我们将研究结构稳定系统和非结构稳定系统在非退化随机扰动下的稳定性具有本质差别。

Our main focus is on studying the stability of the Morse-Smale systems under nondegenerate stochastic perturbations, that is, it studies that, when the noise intensity goes to zero, weak* limits of the stationary measures of the Morse-Smale systems with stochastic perturbations and supports of limit measures. To find which kind of recurrent motions are preserved by such perturbations. Our ultimate goal will be that each limit measure is a convex combination of the Delta measures of some stable equilibria and the Haar measures of some stable periodic orbits. To achieve this goal, it is divided into three steps: 1. To prove that each limit measure on (hyperbolic) repellers is zero; 2. To prove that each limit measure at saddles is zero; 3. To find out which stable equilibrium and periodic orbit is a null set with respect to limit measures. At the same time, we will show that there is a big difference between the structurally stable systems and nonstructurally stable systems under nondegenerate stochastic perturbations.