Nonlinear Dynamics with Applications to Physical Systems

该项目涉及三个方向。 第一个涉及所谓的扭曲映射--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。 作为一个说明性的例子，扭曲映射可以从控制晶格模型中电子分布的方程中产生。 研究者先前的工作提出了一个意想不到的数学效应;对于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。 这种效应的机制还不清楚，研究人员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“有质动力磁性"。“回想一下，电场的任何变化都会产生磁场，这是自法拉第以来就知道的。 令人惊讶的是，表面上类似的效果发生在力学（没有电场存在）中，正如研究人员和合作者最近发现的那样。 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是带电的，并且存在磁场-既没有电荷也没有磁场。 这是一个神秘而基本的数学现象。 这个项目的一个目标是开发一个几何解释这种效果使用微分几何的工具。 该项目的第三部分探讨了最近发现的希尔方程（在物理学和工程学的无数问题中无处不在）与“轮胎痕迹”问题之间的联系。 本课题的主要价值在于统一和简化两个看似不同的数学领域，从而对两者有更丰富的理解，并可能有新的发现。该项目涉及三个方向，通过使用几何和分析与物理动机相统一。 第一个方向涉及圆柱体的面积保持扭曲映射;这种映射出现在许多设置中，例如晶格的Frenkel-Kontorova（F-K）模型。 研究者研究周期轨道对于某些扰动的鲁棒性（例如，在F-K示例中，向周期势添加倾斜）随着映射中谐波的数量而增加。 已知两个完全不同的（稍微简单一点的）对象表现出类似的效果：圆映射和马修型希尔方程。 如果成功，该项目将在后两类系统中增加一类新的系统。 第二个研究方向包括从几何学上解释最近发现的“有质动力磁性”，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规形式计算发现的法拉第效应：粒子在这样的场中的行为就好像它们在磁场存在的情况下具有电荷，尽管既没有电荷也没有磁场。 目标之一是解释这种效应背后的几何结构。 第三个方向探索希尔方程（在数学和应用的许多领域中出现）与几何“轮胎痕迹”问题之间的联系所提出的问题。该奖项反映了NSF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.