Mathematical Sciences: Conley Index Theory and Its Applications

数学科学：康利指数理论及其应用

• 批准号: 9101412
• 负责人: Konstantin Mischaikow
• 金额: $ 5.77万
• 依托单位: Georgia Tech Research Corporation - GA Tech Research Institute
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-15 至 1993-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9101412&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Conley; Index; Theory

The investigator is developing methods for studying the structure of isolated invariant sets that are applicable to a wide variety of problems, e.g., travelling waves for reaction-diffusion systems, dynamic phase transition, magnetohydrodynamic shock waves, delay equations, to name a few. More specifically, it is concerned with deriving algebraic invariants that describe invariant sets, determining how they can change through bifurcations, and determining which geometric topological structures are imposed on the invariant sets by the algebraic invariants. The guiding philosophy behind these methods comes from the fact that many differential equations that arise as models of biological, chemical, or physical systems share the following three characteristics: (1) they are derived from idealized descriptions of complicated phenomena, (2) they are parameterized systems and exact values of the parameters are not known, (3) they are nonlinear. Thus, from the point of view of applications, describing the global and robust structure of the solutions to these differential equations is of great importance. At the same time, it is well known that nonlinear systems undergo tremendously complicated bifurcations (e.g., the appearance of chaotic dynamics). Therefore, any general theory must rely on tools that are indifferent to the fine structure of the invariant sets, e.g. the Conley index theory.

研究人员正在开发研究 结构的孤立不变集，适用于一个 各种各样的问题，例如，行波 反应扩散系统，动态相变， 磁流体动力学冲击波，延迟方程，仅举几例。 更具体地说，它涉及到推导代数 描述不变集的不变量，确定它们如何 通过分叉改变，并确定哪些几何 拓扑结构被强加在不变集上， 代数不变量 这些方法背后的指导思想来自于 事实上，许多微分方程作为模型出现， 生物、化学或物理系统共享以下内容 三个特点：（1）源于理想化 复杂现象的描述，（2）它们是参数化的 系统和参数的精确值是未知的， (3)它们是非线性的。 因此，从 应用程序，描述了全球和强大的结构， 解这些微分方程是非常重要的。 同时，众所周知，非线性系统经历 非常复杂的分叉（例如，外观 混沌动力学（Chaotic Dynamics） 因此，任何一般理论都必须依赖于 工具是无关紧要的精细结构的不变量 Conley指数理论等。