时滞随机动力系统不变叶层的相关性质

Invariant manifolds and invariant foliations for dynamical systems have long been a important problems. They are the basic tools for studying structural stability, linearization, homoclinic (heteroclinic) bifurcation, etc. However, few papers have been published on random dynamical systems with delays. In this project, we study some properties of invariant foliations for delay random dynamical systems, including existence, uniqueness, smoothness, etc. By modified Hadamard’s methods or Lyapunov-Perron’s methods, we shall prove existence and uniqueness of invariant foliations, and obtain gap condition of smoothness. Moreover, we shall provide condition of intersection of invariant manifolds and invariant foliations.

不变流形和不变叶层一直以来都是动力系统研究的重要内容，它们是研究系统结构稳定、线性化以及同异宿分岔等问题的重要工具。然而，带时滞随机动力系统在此方面的研究很少。本项目研究的带时滞随机动力系统不变叶层的有关性质，包括存在唯一性、N阶光滑性等。通过修改后的Lyapunov-Perron方法，得到不变叶层的存在唯一性，以及叶层N阶光滑性的间隙条件。最后，我们将给出不变流形和不变叶层相交的条件。

本项目研究了希尔伯特空间中的非一致伪双曲的无穷维随机动力系统不变流形和不变叶层的光滑性。我们得到了此系统关于各种情形不稳定流形，稳定流形以及稳定叶层光滑性所需的条件，同时我们还证明了不变流形和不变叶层的可测性。