近可积随机哈密顿系统的稳定性

本项目主要研究近可积随机哈密顿系统拟周期环面的保持性，轨道的长时间稳定性和确定型近可积哈密顿系统的统计系综。自Poincaré研究N体问题开始，到KAM理论的建立和发展，人们对近可积哈密顿系统的动力学行为已经有了较为清晰的认识。然而，自然界中的动力学行为往往还会受到不确定因素的干扰，关于随机哈密顿系统的课题一直受到人们的高度关注。本项目将研究可积哈密顿系统在随机扰动下的动力学稳定性问题。在系统满足非退化条件的假设下探究拟周期环面是否能在某种意义下保持下来。在可积部分满足一定的“陡性”条件时探究轨道在关于扰动参数倒数的指数长时间内的稳定性。此外，本项目还将探究以随机变量为初值在近可积哈密顿流作用下形成的随机过程的收敛性问题。

In this project, we will study the persistence of quasi-periodic tori and the long time stability of nearly integrable stochastic Hamiltonian systems. Moreover, we will study the statistical ensembles of deterministic nearly integrable Hamiltonian systems. From Poincaré's study of N-body problems to the development of KAM theory, people have a clear understanding of the dynamic behavior of nearly integrable Hamiltonian systems. However, the phenomena in nature are almost always subject to random noise and stochastic Hamiltonian systems have been highly concerned. This project will study the dynamic stability of integrable Hamiltonian systems under random perturbations. Under the nondegenerate conditions, we want to prove the persistence of quasi-periodic tori in some sense. When the integrable part satisfies certain "steepness" conditions, the stability of the orbit in exponential long time will be explored. In addition, the project will also study the convergence of the stochastic process, which is formed by the nearly integrable Hamiltonian flow with random initial value.