高维奇异奇摄动系统的空间对照结构

In recent years, the study on the contrast structure becomes a hot topic in the theory of singularly perturbed problems. As far as our known,the contrast structure in singularly perturbed problems is mainly classified as a step-type contrast structure or spike-type contrast structure . Its fundamental characteristics is that there is t*(or multiple t*) within the domain of interest, which is called as an internal transition point. The position of t* is unknown in advance and it needs to be determined thereafter. In the neighborhood of t*, the solution y(t,u) will have an abrupt structure change. In the different sides of t*, if y(t,u) approaches different reduced solutions, we call it step-type contrast structure. If y(t,u) approaches to the same reduced solution, we call it spike-type contrast structure. .In Russian,the most study on contrast structure is by the method of boundary function and concentrated on the second order singular perturbed equations . Only little research on the third order equations with the specific boundary value condition. In Western, the study on this issue is by the method of dynamic systems or geometric method. A further study found that the existence of a step-type(spike-type) contrast structure is closely related to the existence of a heteroclinic orbit(holoclinic orbit) of its auxiliary system(its fast system) in its corresponding phase space. However, how to find and construct such an orbit in a higher dimensional dynamic system(n>=3) is itself difficult in general in the theory of qualitative analysis(geometric method). This is why it is not a easy case from the issue of contrast structure in plane in extension to the higher dimensional case. On the other hand, the contrast structure in the singular singularly perturbed problem is a piece of virgin land. In this field, we should not only seek out two isolated reduced solutions from a family of reduced solutions but also search the heteroclinic orbit connecting nonhyperbolic equilibria. All of these matters are very difficult..In this project, we take the contrast structure as the researched object and we use the stitched method which is proposed by the professor NI Mingkang to our theoretical study. Combining the boundary function method with the theory of geometric singular perturbation, we not only prove the the existence of a contrast structure solution in these two cases, but also the uniformly valid asymptotic expansion of a solution with a contrast structure is constructed when such a contrast structure is available. When carrying out this project, we may find some new ideas for solving other complex singular perturbed problems and lay a solid foundation for the further theoretical research.

近年来，奇摄动问题中的空间对照结构成为非常活跃又重要的研究领域，主要分为阶梯状空间对照结构和脉冲状空间对照结构两大类，其在辅助系统的相空间分别对应于异宿轨道和同宿轨道。随着奇摄动理论和方法在自然科学领域的广泛应用，人们越来越多的遇到了出现空间对照结构的困难。.本项目以空间对照结构为研究对象，采用倪明康教授独创的“缝接法”进行基础理论研究，把边界函数法和几何奇摄动理论相结合，不但可以证明奇异奇摄动问题解的存在性，构造它们一致有效的渐近解，而且还可以进行余项估计，为进一步推动奇摄动的基础理论研究奠定了坚实的基础，也为奇摄动问题中的其他复杂现象的研究提供了新的思路。

本课题主要研究了高维奇异奇摄动系统的空间对照结构。从低维的奇异奇摄动方程着手，逐步推广到全为快变量的高维系统和具有快慢变量的高维Tikhonov系统，已基 本完成了项 目申请书列出的所有内容，共完成学术论文7篇，其中4篇在SCI杂志上发表，2篇在CSCD杂志上发表，1篇被CSSCI录用。. 本课题获得的主要结果解决了申请书中提出的拟解决的关键问题，主要是在非双曲平衡点附近引进不同的尺度构造定义域上的形式渐近解，利用高维相空间非双曲平衡点的异宿轨道的光滑缝接, 得到确定转移点位置的等式，再结合几何奇摄动理论，证明了解的存在性。获得的一系列原创性的成果，得到专家和同行的认可。. 本课题主要的创新之处在于把边界层函数法、缝接法、k+σ交换引理有机结合起来。在定性分析的基础上，构造多尺度的形式渐近解，证明了形式渐近解的一致有效性并给出余项估计。此外，我们对所讨论的大多数问题都给出了具体的算例并进行了数值模拟。本课题推广了前人的研究工作，对现有的空间对照结构理论进行了深化和完善，也为奇摄动问题中的其他复杂现象的研究提供了新的思路。