Nonlinear Dynamics with Applications to Physical Systems

提出的研究旨在发现具有基本数学兴趣和物理相关性的新现象，并尽可能以最简单的方式解释它们。提出的三个主要项目是：(i)研究在许多物理应用中产生的共振的稳健性或脆弱性，例如在固态物理中；（ii）解释最近发现的磁动学的几何机制（在许多物理实验中观察到），并建立平均理论和微分几何之间的联系-两个看似无关的领域。（iii）利用最近发现的希尔方程与轮胎轨迹问题之间的联系，从而统一两个看似无关的领域。这个项目的一些结果可能会对一些基本感兴趣的现象给出一个新的、更简单的理解，并可能被写入教科书。该项目由三个部分组成，统一的愿望是发现和理解动力学中的新现象。几年前，J. B. Keller和V. Arnold发现并分析了两类不同问题中共振的一个基本特征：圆映射和马蒂厄型方程。项目第一部分的目标是在Arnold和Keller研究的两个类的基础上再增加一个类，即保持面积的柱面图。除了它的基本兴趣之外，后一个例子在许多其他设置中也出现在固态物理中。该项目的第二部分旨在理解有些神秘的磁动机制，并旨在探索两个基本对象之间的联系，一个来自力学（陀螺仪效应），另一个来自微分几何（雅可比场）。第三个项目建议利用最近发现的两个看似无关的物体之间的联系：(i)希尔方程，被广泛研究了几个世纪；（ii）轮胎痕迹，一个最近被研究的物体。希望一个领域的研究结果能给另一个领域带来新的见解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The proposed research aims at discovering new phenomena of fundamental mathematical interest as well as physical relevance, and at explaining them in the simplest possible way. The three main proposed projects are (i) study of the robustness or fragility of resonances arising in many physical applications, e.g. in solid state physics, (ii) explaining the geometrical mechanism of the recently discovered ponderomotive magnetism (observed in many physical experiments) and developing a connection between averaging theory and differential geometry – two seemingly unrelated fields, and (iii) exploiting the recently discovered connection between Hill's equation on the one hand and the tire track problem on the other, thus unifying two seemingly unrelated areas. Some results of this project are likely to give a new and simpler understanding of some phenomena of fundamental interest, and may make their way into textbooks. The project consists of three parts, unified by the desire to discover and understand new phenomena in dynamics. Some years ago J. B. Keller and V. Arnold discovered and analyzed a fundamental feature of resonances in two different classes of problems: circle maps and Mathieu-type equations. The goal of the first part of the project is to add a yet one more class to the two studied by Arnold and Keller, namely the area-preserving cylinder maps. Besides of its basic interest, the latter example comes up in solid-state physics among many other settings. The second part of the project aims at understanding the somewhat mysterious mechanism of ponderomotive magnetism, and also aims to explore a connection between two fundamental objects, one from mechanics (the gyroscopic effect) and the other from differential geometry (Jacobi fields). The third project proposes to exploit the recently found connection between two seemingly unrelated objects: (i) Hill's equation, studied extensively over a couple of centuries, and (ii) tire tracks, a more recently studied object. It is hoped that the results from one area will give new insights into the other.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.