Invariant Objects and Their Connections in Dynamical Systems: Rigorous Results, Computations, and Applications

自然界中的许多系统都受确定性规则的支配。然而，即使是简单的规则多次应用的结果也可能是复杂和难以理解的（想想一杯咖啡中勺子的简单动作；重复几次后，液体就完全搅拌了）。众所周知，小的扰动可以迅速增长，例如气象学中的“蝴蝶效应”。其他发生复杂效应的系统包括受太阳和行星变化力影响的小天体的运动。在某些情况下（例如，在卫星的运动中），设计利用干扰的自然放大来以小努力取得大效果的机动是有意义的。为了理解这种复杂的行为，本研究项目寻找以有序方式移动的碎片，并作为所有其他运动的锚点和参考点。这样就可以对可能的运动进行系统的分类。为了实现这些目标，本项目将采用数学几个部分的方法。一个物体行为简单的事实可以被表述为一个函数方程，可以用严谨的数学方法使用泛函分析和拓扑学来研究。在这些研究中，令人惊讶的是微妙的数论性质（例如，频率是否满足整数系数的多项式方程）发挥了实际作用。功能分析的工具必须辅以很强的几何直觉。该项目还旨在开发实用的计算算法，使理论中获得的知识具体化。工具的混合（分析、拓扑、几何和数值算法）使参与项目的学生能够有效地训练他们随后从事学术或工业职业。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。

Many systems in nature are governed by deterministic rules. Nevertheless, the results of applying even simple rules many times may be complicated and hard to understand (think of a simple motion of a spoon in a cup of coffee; after a few repetitions the liquid is completely scrambled). It is well known that small disturbances can have rapid growth, as for example in the "butterfly effect" in meteorology. Other systems where complicated effects take place include the motions of small celestial bodies subject to the changing forces of the sun and the planets. In some cases (for example, in the motion of satellites) it is of interest to devise maneuvers that take advantage of the natural amplification of disturbances to accomplish big effects with small efforts. To understand such complicated behavior, this research project searches for pieces that move in an orderly way and that act as anchors and reference points for all the other motions. This allows systematic classification of the possible motions. To accomplish these goals, this project will employ methods from several parts of mathematics. The fact that an object behaves simply can be formulated as a functional equation that can be studied with rigorous mathematics using functional analysis and topology. One surprise in these studies is that delicate number-theoretic properties (for instance whether a frequency satisfies a polynomial equation with integer coefficients) play a practical role. The tools from functional analysis must be supplemented by a very strong geometric intuition. The project also aims to develop practical algorithms for calculation, so that the knowledge acquired in the theory can be made concrete. The mixture of tools (analysis, topology, geometry, and numerical algorithms) makes involvement in the project an effective training ground for students who can subsequently pursue either academic or industrial careers.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.