Invariant objects in dynamical systems: Analysis and numerics

自然科学中的许多问题都是由一个具有简单规则的系统来模拟的，该规则给出了系统的下一状态作为当前状态的函数。越来越明显的是，其中许多规则在重复应用时会导致复杂的行为。这种复杂的行为通常被称为“混乱”。理解一个系统的行为并做出有用的预测是一个深刻的数学问题。一种可能的方法是寻找“地标”。里程碑是一个小的子系统，它以一种简单的方式运行，并锚定系统的行为。如果能够找到足够多的这样的地标，它们可以为动力学提供一个骨架，让人们对系统本身有一个全面的了解。该计划是开发系统和准确的方法来计算地标。另一个目标是证明定理，证明满足某些条件的计算是正确的。计划获得的结果表明，如果在某些配置中发现了某些地标(可以通过有限精度计算进行验证)，则可以得到所有时间的结论。希望能接触到由固态物理和化学中的问题引发的具体问题。该项目中的工作也将被用作研究生、博士后和参观者的训练场。50多年来，动力系统中最常用的标志是通常的双曲流形和准周期轨道，以及它们的稳定和不稳定流形。在这里，建议发展一种系统的方法，从理论上和数值上计算这些物体。另一个目标是证明验证近似解的构造性存在定理，并开发和实现快速而准确的算法。统一的原则是寻找描述不变性的函数方程，然后使用各种方法试图求解它们。从几何学到泛函分析，方法会有所不同。进一步的工作是开始研究一些无限维问题的特解，包括具有状态依赖时滞的偏微分方程组和时滞微分方程解。

Many problems in the natural sciences are modeled by a system with a simple rule that gives the next state of the system as a function of the present one. It has become increasingly apparent that many of these rules lead to complicated behavior when applied repeatedly. This complicated behavior is often called "chaos". To understand the behavior of a system and make useful predictions is a deep mathematical problem. One possible approach is to find "landmarks". A landmark is a small subsystem that behaves in a simple manner and which anchors the behavior of the system. If enough of these landmarks can be found, they can provide a skeleton for the dynamics which gives a global understanding of the system itself. The plan is to develop systematic and accurate methods for the calculation of landmarks. Another goal is to prove theorems which show that the calculations satisfying some conditions are correct. It is planned to obtain results which show that if certain landmarks are found in some configurations (which can be verified with finite accuracy calculations) then conclusions can be obtained for all times. The hope is to make contact with concrete problems motivated by questions in solid state physics and chemistry. The work in the project will also be used as a training ground for graduate students, postdocs and visitors. For over 50 years, the most commonly used landmarks in dynamical systems have been normally hyperbolic manifolds and quasi-periodic orbits as well as their stable and unstable manifolds. Here, it is proposed to develop a systematic way of computing these objects theoretically as well as numerically. Another goal is to prove constructive existence theorems that validate approximate solutions and also to develop and implement fast and accurate algorithms. The unifying principle is to look for functional equations that describe the invariance and then use a variety of methods to try to solve them. The methods will vary from geometry to functional analysis. A further part of the proposed work is to begin studying special solutions in some infinite dimensional problems including partial differential equations and delay differential equations with state dependent delays.