在多尺度系统中具有稳定性交替的空间对照结构研究

Study of the contrast structure for multi-time scale systems with exchange of stabilities can be quite difficult. Currently, the existing results are very few. Therefore, it has been one of the main research topics in singular perturbation theory. When different degenerate roots intersect at one point, causing the exchange of stabilities, the system may generate turning point (non-hyperbolic point). The dynamical behavior of the trajectory around the turning point is complicated, and the study of such phenomenon can be challenging and fruitful. From the application point of view, such complicated phenomenon exists widely in various fields of sciences and technology. Moreover, theoretical research in singular perturbation problems has become an important motivation to study the complicated behavior in dynamical systems theory. This project is concerned with the study on contrast structure caused by the exchange of stabilities due to the intersection of different degenerate roots. The purpose of this research is to find out the dynamical behavior of trajectory around the non-hyperbolic point. We will combine the geometric singular perturbation method, the dynamical system's method, and traditional asymptotic expansion method and use them to study the basic theory. For the proposed problems, we will consider the existence of solutions, construct uniformly valid asymptotic expansions and obtain estimates for remainders. This project will provide new approaches and new methods to treat other complicate phenomena in singularly perturbed problems.

在多尺度系统中，具有稳定性交替空间对照结构的研究是相当困难的，目前已有的结果也很少，它已成为奇摄动研究领域的主要研究对象。 一方面， 当不同的退化根交于一点引起稳定性交替时必然会产生转点（非法向双曲点）， 其周围轨线的动力学行为复杂多变， 对它的研究极具挑战性和富有成果； 另一方面， 这一复杂的空间对照结构现象越来越多地存在于自然科学研究的各个领域， 现已成为近代奇摄动理论中复杂动力学行为的重要源头之一。 本项目将以不同退化根相交引起稳定性交替的空间对照结构为研究对象， 搞清楚在非法向双曲点附近轨线的动力学行为，把几何奇摄动方法和近代动力系统方法揉合在一起再加上传统的渐近展开法进行基础理论研究，我们将讨论这类问题解的存在性， 构造一致有效渐近展开式， 并给出余项估计， 为奇摄动问题中其它复杂现象的研究提供新思路和新方法。

本课题主要研究了若干奇摄动问题中具有稳定性交替的空间对照结构。从二阶非线性奇摄动方程着手，逐步推广到了具有快慢变量的高维 Tikhonov 系统。按要求已完成了计划书中列出的研究内容，在 SCI 杂志上共计发表学术论文13篇，获得了专家和同行的好评。.本项目获得的主要结果为：解决了计划书中提出的两个关键问题，在非法向双曲点附近采用多尺度的“多元缝接法”、“多尺度匹配法”以及“吹胀法”，构造了一致有效的多尺度解，刻画了在非双曲平衡点附近解的动力学行为，获得了一系列原创性的成果。.本课题主要的创新之处在于把传统的边界层函数法和现代几何奇摄动方法有机结合起来，既有渐近展开也有定性分析，它们之间相辅相成，在整个定义域上完整刻画了解流的动力学性质，得到了一致有效的多尺度解，以及给出了解的余项估计。所做的工作推进了前人的研究结果，为进一步研究奇摄动问题中的复杂现象提供了理论依据。