几何奇异摄动理论及其应用  

Geometric Singular Perturbation Theory and Its Applications

作　　者：张祥 ZHANG Xiang(School of Mathematical Science,Shanghai Jiao Tong University,Shanghai 200240)

机构地区：[1]上海交通大学数学科学学院,上海200240

出　　处：《四川师范大学学报(自然科学版)》2025年第4期466-476,F0002,共12页Journal of Sichuan Normal University(Natural Science)

基　　金：国家自然科学基金(12471169、12071284和12161131001)。

摘　　要：奇异摄动系统理论因工程应用而诞生,对于其动力学的研究方法早期主要是非标准分析和渐近展开法.随着动力系统几何理论的发展,奇异摄动系统动力学的研究形成了一套完整的几何理论.该理论的基础是Fenichel关于法向双曲的不变流形理论和后续不断发展起来的处理转向点的理论和技术.主要介绍Fenichel的3个不变流形理论,以及折点的跳跃和鸭点理论,转向点的进出函数理论,和从中心稳定流形到中心不稳定流形的流的演化交换引理等.同时简要介绍这些理论在生物数学模型中的应用和遗留问题.Singular perturbation theory was initiated from engineering and its applications.The research methods to studying their dynamics are nonstandard analysis and asymptotic expansion method in past.With development of geometric theory for dynamical systems,the researches formulated a series of geometric theories.This theory is based on Fenichel’s normally hyperbolic invariant manifold theory,then there appear the theories and techniques to handling turning point.This paper summarizes Fenichel’s three invariant manifold theories,together with jump and canard point theories,entry and exit function theory on turning point,and the exchange lemma dealing with the evolution of the flow from center stable manifold to center unstable manifold.Some applications of these theories to biological mathematical models and so on are also presented.

关 键 词：奇异摄动系统 Fenichel不变流形理论 折点和转向点理论 交换引理 生物数学 

分 类 号：O175.1[理学—数学] O175.12[理学—基础数学] O175.14