对于正规形理论中的若干问题的研究及其应用

In recent years, the research of normal form theory has a rapid development with larger scope and depth. On one hand, it extends its research area from quasi(almost)-periodic systems and systems with time delay or control terms to random dynamical systems, non-autonomous systems admitting non-uniformly exponential dichotomy spectrum and some others. On the other hand, it goes into the thorough investigation of the related conceptions of the normal form itself. From the classical definitions to the generalized ones, such as exponentially small remainder, almost reducibility, ultra-differential topology, we have more and more new research objects. . .According to such new situations and based on our previous work about quasi-periodic systems and random dynamical systems, the key of this project is expected to build the convergent normal form criteria for the non-autonomous system admitting non-uniformly exponentially dichotomy spectrum and find the connections between those and invariant manifold theorems. Meanwhile, we also want to extend the Siegel type theorems from classical analytic topology to the ultra-differential topology and establish the relationship between different types of small divisor conditions and the convergence of normal forms. Moreover, we shall do a few works about other common problems.

近年来，正规形理论蓬勃发展，不仅研究的范围逐渐扩大，而且研究的深度亦不断增加。前者主要体现在研究的对象从拟(概)周期系统、带有时滞或控制项系统，扩展到包括随机动力系统,乃至更广的满足非一致指数二分谱的种种非自治系统;后者主要表现在对正规形概念本身内涵的不断挖掘：从经典定义，到带有指数小余项，几乎可约，甚至是超可微（ultra-differential）拓扑正规形等相关概念日益丰富。..为了应对这些新情况，本项目的主要目的就是在原有工作积累的基础上，借鉴处理拟周期和随机系统正规形时的经验，对于更广的系统，即线性部分满足非一致指数二分谱的非自治系统，建立相应的正规形收敛性定理并研究它们和不变流形定理之间的关系。同时，推广经典的Siegel定理，在拟解析甚至超可微拓扑下，对于正规形的收敛性和不同小除数条件的关系开展研究。另外，我们还将就其它的一些同行们比较关心的问题开展一些讨论。

近年来，正规形理论蓬勃发展，不仅其研究的范围逐渐扩大，而且研究的深度亦不断增加。本项目主要研究了超可微拓扑下正规形的新性态，并建立了其与向量场层（foliation）上的几何性态之间的关联。主要接受和发表了如下结果：1，我们发现了在Gevrey光滑性下，即使有Siegel小除数，也可以通过加权模技术来精确的把非线性项的特性通过共轭不变量表示出来。2，通过考察平面鞍点附近的monodromy来刻画附近层上的叶子的分形维数，3，参于讨论解决一些NLS方程，反转系统的拟周期解的存在性，Liouville小除数下线性方程的约化等问题。目前我们取得的结果，充分说明了超可微拓扑在正规形研究中的重要性，它不仅能处理更多的小除数条件，而且还能更好的刻画非线性项，从而可以把动力系统的不同分支联系起来。