随机动力系统的复杂性

This project is an interdisciplinary research in the area of dynamical systems, stochastic analysis, stochastic differential equations, and geometry. It is directed towards the development of the fundamental theory for the complicated dynamics of random dynamical systems and the stochastic geodesic flows. The proposed topics include the theory of chaotic dynamics for infinite dimensional random dynamical systems; the ergodic theory of stochastic geodesic flows; the complicated dynamics of differential equations driven by random/stochastic forcing; the theory of normally hyperbolic random invariant manifolds for infinite dimensional random dynamical systems and their applications; and the stochastic flow generated by stochastic partial differential equations.This project is directed to both advancing the boarders of knowledge in critical areas of mathematics and also relating those advances to problems arising at the forefront of other sciences. It is also expected that the techniques developed in this project will have an impact on the advancement of mathematics beyond the vision outlined in this proposal.

本项目位于动力系统，随机分析、随机微分方程、微分几何的交叉领域,有很强的应用背景，涉及随机动力系统的两个方面：随机动力系统的复杂性和随机测地流。具体课题包含无穷维随机动力系统的混沌理论；随机测地流的遍历理论；确定性微分方程在随机外力作用下的复杂动力学行为；无穷维随机动力系统法向双曲流形理论及应用； 随机偏微分方程所生成的随机流的存在性问题。这些课题是当前国际上随机动力系统研究的热门方向。项目组全体成员力争在上述各个方向的研究中在理论和方法上作出创新性成果，以推动学科的发展。

本项目位于动力系统，随机分析、随机微分方程、微分几何的交叉领域,有很强的应用背景。在项目执行期间，研究了随机动力系统的混沌理论，部分双曲动力系统以及无穷维动力系统中SRB测度理论, 随机动力系统不变流形理论、随机动力系统吸引子的存在性,动力系统随机熵、线性漂移和空间几何性质之间的联系和刚性性质等课题的研究。这些课题是当前国际上随机动力系统研究的热门方向。对随机动力系统的混沌理论，开展随机动力系统正熵蕴涵混沌这一公开问题的研究。对度量空间中的无穷维随机动力系统在不假设任何双曲性的条件下证明了正熵蕴含弱马蹄的存在，同时也证明了正拓扑熵蕴涵Li-Yorke混沌。