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

动态系统中的不变对象及其连接：严格的结果、计算和应用

基本信息

批准号：
1800241
负责人：
Rafael de la Llave
金额：
$ 27万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800241&HistoricalAwards=false
关键词：
Invariant Objects Their Connections Dynamical

项目摘要

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.

自然界中的许多系统都是由确定性规则控制的。然而，即使是简单的规则多次应用的结果也可能是复杂的，难以理解的（想想勺子在一杯咖啡中的简单运动;重复几次之后，液体完全被搅乱了）。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”。其他发生复杂影响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），设计利用扰动的自然放大以实现小努力大效果的机动是有意义的。为了理解这种复杂的行为，这个研究项目寻找以有序的方式移动的部件，并作为所有其他运动的锚和参考点。这允许对可能的运动进行系统分类。为了实现这些目标，这个项目将采用数学的几个部分的方法。事实上，一个对象的行为简单，可以制定为一个功能方程，可以研究与严格的数学使用功能分析和拓扑。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。从功能分析的工具，必须由一个非常强大的几何直观的补充。该项目还旨在开发实用的计算算法，以便将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）使参与该项目的学生成为一个有效的培训基地，他们可以随后从事学术或工业职业。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。