Mathematical Sciences: Qualitative Analysis of Nonlinear Dynamical Systems

当两个或多个简单行为的弛豫振荡耦合时，在许多情况下会产生新的和复杂的定性行为。本研究旨在通过使用先前开发的分析见解和估计来探索相流的几何形状来理解这些影响。在另一个方向上，研究者利用组合和几何思想将绝热不变性理论扩展到缓慢变化的遍历系统。当两个简单的振荡器，如电路或生物细胞，每一个行为在一个简单的周期方式，是耦合的，由此产生的系统可以获得全新的特征，如混沌行为，分岔等，这似乎是没有在单独的组件暗示。这项研究的目的是描述和解释这类现象的微分方程模型耦合的生物，电，或机械系统。另一方面，绝热不变性现象在动力系统理论中具有根本的重要性；当一个系统在很长一段时间内经历缓慢的变化时，它就会出现。出乎意料的是，一个系统可能会发展出一种“记忆”：一些被称为绝热不变量的量，在系统的演化过程中几乎没有变化。（最简单的例子是一根弦上的钟摆，它的长度慢慢地减半；在这种情况下，能量与频率的比例变化很小；这个比例是绝热不变量）。绝热不变量出现在电磁约束等离子体、粒子加速器、行星运动和许多其他情况的研究中。虽然这门学科大约有80年的历史，但对于一类重要的混沌系统，对这一现象的分析还没有令人满意。考虑到绝热不变性是哈密顿动力系统的一个基本方面，这是本学科的一个严重缺陷。提出了建立混沌系统绝热不变性现象的严格理论。这一理论对动力系统的应用也具有重要意义。举个例子，在为理想气体理论提供严格基础这一长期悬而未决的问题中，它将是一个很小但很重要的组成部分。理想气体理论是通过在一个大小和形状变化的容器中由许多相互碰撞的硬球体组成的系统来模拟气体的。

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.