Mathematical Sciences: Qualitative Analysis of Nonlinear Dynamical Systems

当两个或多个简单行为的弛豫振荡器 新的和复杂的定性行为在许多 例本研究旨在了解这些影响， 探索相流的几何形状， 开发分析见解和估计。 在另一个方向， 研究者将绝热不变性理论扩展到 慢变遍历系统， 几何概念 当两个简单的振荡器，如电路或 生物细胞，每个都以简单的周期性方式行为， 耦合，由此产生的系统可以获得全新的功能 例如混沌行为、分叉等，其中有 似乎在各个组成部分中没有任何暗示。 目的 这项研究的目的是描述和解释这种现象， 在微分方程中， 电气或机械系统。 在另一个方向， 绝热不变性的现象具有根本的重要性 在动力系统理论中;当一个系统 在很长一段时间里经历着缓慢的变化。 有些 出乎意料的是，系统可能会产生“记忆”：一些量， 称为绝热不变量，在整个过程中几乎没有变化。 系统的演变。 (The最简单的例子是一个摆在 一个字符串的长度，比如说，慢慢减半;在这种情况下， 能量与频率的比率变化很小;这 比率是绝热不变量）。 绝热不变量产生于 粒子中电磁约束等离子体研究 加速器，在行星运动和许多其他情况下。 虽然受试者年龄约80岁，但没有 对这一现象的满意分析对于一个重要的类 混沌系统。 这是该学科的一个严重空白 考虑到绝热不变性是一个基本方面， 哈密顿动力系统的一部分。 建议制定一个 绝热不变性现象的严格理论 混沌系统 这样的理论将是一个基本的 动力系统应用的意义 也作为一个例子，这将是一个小，但重要的 在长期开放的问题提供了一个严格的积木 理想气体理论的基础--用 一个由许多硬球在一个不断变化的容器中碰撞而成的系统 尺寸和形状。

When two or more simply behaving relaxation oscillators are coupled, new and complex qualitative behavior arises in many cases. This research aims at understanding these effects by exploring the geometry of the phase flow using previously developed analytic insights and estimates. In another direction, the investigator extends the theory of adiabatic invariance to slowly varying ergodic systems by using combinatorial and geometric ideas. When two simple oscillators such as an electric circuit or a biological cell, each behaving in a simple periodic fashion, are coupled, the resulting system can acquire totally new features such as chaotic behavior, bifurcations, etc., of which there seems to have been no hint in the individual components. The aim of this research is to describe and to explain phenomena of this sort in differential equations that model coupled biological, electric, or mechanical systems. In a different direction, the phenomenon of adiabatic invariance is of fundamental importance in the theory of dynamical systems; it arises when a system undergoes a slow change over a long period of time. Somewhat unexpectedly, a system may develop a "memory": some quantities, called the adiabatic invariants, change little over the course of the system's evolution. (The simplest example is a pendulum on a string whose length is, say, halved slowly; in this case the ratio of the energy to the frequency changes very little; this ratio is an adiabatic invariant). Adiabatic invariants arise in the study of elecromagnetically confined plasmas, in particle accelerators, in planetary motion and in many other cases. Although the subject is about 80 years old, there is no satisfactory analysis of this phenomenon for an important class of chaotic systems. This is a serious gap in the subject considering the fact that adiabatic invariance is a basic aspect of Hamiltonian dynamical systems. It is proposed to develop a rigorous theory of the phenomenon of adiabatic invariance for chaotic systems. Such a theory would be of fundamental significance for applications of dynamical systems as well. As an example, it would be a small but important building block in the long-open problem of providing a rigorous foundation for the theory of ideal gases -- modeling a gas by a system of many colliding hard spheres in a vessel of changing size and shape.