Nonlinear Dynamics with Applications to Physical Systems

该项目涉及三个方向。第一个是所谓的扭转图，这是研究几乎所有运动或静止的物理系统的基本组成部分，在这些系统中摩擦可以忽略不计。举个例子，扭曲图可以由控制晶格模型中电子分布的方程产生。研究者先前的工作表明了一个意想不到的数学效应；对晶格的情况进行解释，这个一般的数学结果相当于一个意想不到的低电阻。这种效应的机制尚不清楚，研究者致力于发展一种数学理论来解释它。该项目的第二部分特别涉及最近发现的“有源磁”。回想一下，电场的任何变化都会引起磁场，这是自法拉第以来所知道的。令人惊讶的是，表面上类似的效应也发生在力学中（没有电场存在），这是研究者和合作者最近才发现的。例如，在旋转（非球形）小行星的引力场中，粒子的行为就像它们在磁场存在的情况下带电一样——实际上既没有电荷，也没有磁场。这是一个神秘而基本的数学现象。这个项目的目标是利用微分几何工具对这种效应进行几何解释。项目的第三部分探讨了最近发现的希尔方程（在无数物理和工程问题中无处不在）与“轮胎轨迹”问题之间的联系。这个主题的主要价值在于统一和简化两个看似不同的数学领域，从而使人们对两者都有更丰富的理解，也许还会有新的发现。该项目涉及三个方向，通过使用几何和物理动机的分析来统一。第一个方向涉及圆柱的保面积扭转映射；这样的图出现在许多情况下，例如晶格的Frenkel-Kontorova （F-K）模型。研究者研究周期轨道的鲁棒性是否相对于某些扰动（例如，在F-K例子中增加一个倾斜的周期势）随着图中谐波数的增加而增加。已知有两种完全不同（且稍微简单一些）的对象显示出类似的效果：圆映射和马修型希尔方程。如果成功，该项目将在后两个类中添加一个新的系统类。第二个方向的研究目标包括从几何上解释最近发现的“有源磁”，其中保守力场的快速时间周期振荡引起了研究者和合作者通过正常形式的计算发现的法拉第效应：在这样一个场中的粒子表现得好像它们在磁场存在时具有电荷，尽管既没有电荷也没有磁场。其中一个目标是解释这种效应背后的几何原理。第三个方向探讨了希尔方程（在数学和应用的许多领域中出现）与几何“轮胎轨迹”问题之间的联系所提出的问题。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 project involves three directions. The first one deals with so-called twist maps -- a basic building block in studying almost every kind of moving or stationary physical system where friction is negligible. As an illustrating example, a twist map can arise from the equation governing the distribution of electrons in models of crystal lattices. Prior work by the investigator suggests an unexpected mathematical effect; interpreted for the case of a crystal lattice, this general mathematical result amounts to an unexpectedly low electric resistance. The mechanism of this effect is not understood, and the investigator works to develop a mathematical theory explaining it. The second part of the project deals in particular with the recently discovered "ponderomotive magnetism." Recall that any change in an electric field gives rise to a magnetic field, as has been known since Faraday. Surprisingly, a superficially similar effect takes place in mechanics (without electric fields present), as was discovered only recently by the investigator and collaborators. For example, particles in the gravitational field of a spinning (non-spherical) asteroid behave as if they were charged and in the presence of a magnetic field -- with neither charge nor the field actually present. This is a mysterious and fundamental mathematical phenomenon. A goal of this project is to develop a geometrical explanation of this effect using tools of differential geometry. The third part of the project explores the recently discovered connection between Hill's equation (ubiquitous in innumerable problems in physics and engineering) on the one hand, and the "tire track" problem on the other. The main value of this topic lies in unifying and simplifying two seemingly distinct areas of mathematics, resulting in richer understanding of both, and perhaps in new discoveries.The project involves three directions unified by use of geometry and analysis with physical motivation. The first direction involves area-preserving twist maps of the cylinder; such maps arise in many settings, e.g. the Frenkel-Kontorova (F-K) model of a crystal lattice. The investigator studies whether the robustness of periodic orbits with respect to certain perturbations (e.g., addition of a tilt to the periodic potential in the F-K example) increases with the number of harmonics in the map. Two entirely different (and slightly simpler) objects are known to show an analogous effect: circle mappings and Mathieu-type Hill's equations. If successful, the project would add a new class of systems to the latter two classes. The second direction of research includes the goal to explain geometrically the recently discovered "ponderomotive magnetism," in which rapid time-periodic oscillation of a conservative force field gives rise to a Faraday-like effect discovered by the investigator and collaborators through a normal-form computation: particles in such a field behave as if they possessed electric charge in the presence of a magnetic field, though neither an electric charge nor a magnetic field is present. One of the goals is to explain the geometry underlying this effect. The third direction explores questions raised by the connection between Hill's equation (arising in many areas of mathematics and applications) on the one hand and the geometrical "tire track" problem on 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.