Mathematical Sciences: Nonlinear Dynamics and its Application in Physical Systems

[406022]这项研究的目的是发现和解释物理系统数学模型中的新现象。更具体地说，我建议在两个不同的方向上工作：(1)耦合松弛振子的动力学和(2)哈密顿系统中不变集的持久性。第一个方向旨在理解当简单振荡器耦合在一起时会发生什么。这种系统出现在许多物理环境中，特别是在生物学和电路中。虽然单个振荡器以一种简单而可预测的方式工作，但几个这样简单的“细胞”放在一起却可以以一种完全意想不到的神秘方式工作。我建议研究这种简单的“细胞”相互作用的“解剖学”，这种相互作用产生了“有机体”复杂而丰富的行为。与第一个主题相反，第二个主题涉及哈密顿系统，该系统模拟了许多可以忽略能量耗散的情况，例如粒子加速器中的粒子运动，几何光学和天体力学。尽管在20世纪60年代（Kolmogorov-Arnold-Moser理论）和80年代（Aubry-Mather理论）取得了显著的进展，但由于这些系统的丰富性和复杂性，在这些系统的理论中仍然存在深刻的基本未解问题。我们建议在一类特殊的系统（首先由Hedlund提出）上解决其中的一些问题，这种系统一方面比一般情况更容易处理，但另一方面又足够丰富，可以提供新的见解和提示。提出了两个研究方向：一是耦合弛豫振子的动力学，二是哈密顿系统中不变集的持久性。在第一个领域，我建议发展耦合弛豫振子的定性理论，扩展Cartwright—Littlewood的经典工作以及最近关于周期性强迫弛豫振子的工作。在过去的几年里，在这个领域发现了两个有趣的现象，我希望有更多的发现。第二个方向涉及理解R^4辛映射在可积情况下的强扰动下不变集的行为。我打算考虑一类特殊的系统，即三维环面上的测地线流，带有Hedlund引入的一种特殊的黎曼度量。一方面，这足够简单，易于处理，但又足够重要，可以深入了解一般情况。

9406022 LevI The goal of this research is to find and to explain new phenomena in mathematical models of physical systems. More specifically, I propose to work in two different directions: (1) Dynamics of coupled relaxation oscillators and (2) Persistence of invariant sets in Hamiltonian systems. The first of these directions aims at understanding what happens when simple oscillators are coupled together. Systems of this kind arise in numerous physical settings, particularly in biology and electric circuits. While an individual oscillator behaves in a simple and predictable way, several such simple "cells" put together can behave in a totally unexpected and mysterious way. I propose to study the "anatomy" of this interaction of simple "cells" which produces complex and rich behavior of the "organism". The second topic, in contrast to the first, deals with Hamiltonian systems which model many situation where energy dissipation can be neglected, such as motion of particles in particle accelerators, geometric optics and celestial mechanics. There are still deep fundamental unanswered questions in the theory of such systems due to their richness and complexity, although remarkable progress has been made in the 1960s (the Kolmogorov-Arnold-Moser theory) and in 1980's (the Aubry-Mather theory). It is proposed to address some of these questions on a particular class of systems (first introduced by Hedlund) which is more tractable than the general case on the one hand, but is sufficiently rich to give new insights and hints on the other. Two research directions are proposed: first, the dynamics of coupled relaxation oscillators, and second, persistence of invariant sets in Hamiltonian systems. In the first of these areas I propose to develop qualitative theory of coupled relaxation oscillators, extending the classical work of Cartwright--Littlewood as well as the more recent work on periodically forced relaxation oscillators. In the last couple of years two int eresting phenomena were discovered in this area, and I hope that there are more to be found. The second of the two directions deals with understanding the behavior of invariant sets in symplectic maps of R^4 under strong perturbations from the integrable case. I plan to consider a special class of systems, namely, geodesic flows on a 3D torus with a particular kind of Riemannian metric introduced by Hedlund. This is sufficiently simple to be tractable on the one hand and yet sufficiently nontrivial to give insight into the general situation.