在多尺度系统中具有稳定性交替的空间对照结构研究

项目名称： 在多尺度系统中具有稳定性交替的空间对照结构研究

项目编号： No.11471118

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 倪明康

作者单位： 华东师范大学

项目金额： 60万元

中文摘要： 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。

中文关键词： 多尺度系统；稳定性交替；空间对照结构；非法向双曲点；渐近解

英文摘要： 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.

英文关键词： Multi-time scale systems;Exchange of stabilities;Contrast structure;Non-hyperbolic;Asymptotic solution