Invariant objects in dynamical systems: Analysis and numerics

自然科学中的许多问题都是用一个简单的规则来模拟的，这个规则给出了系统的下一个状态作为当前状态的函数。 越来越明显的是，这些规则中的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“混沌”。 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。 一种可能的方法是找到“地标”。 地标是一个小的子系统，它以简单的方式运行，并锚定系统的行为。 如果能找到足够多的这些标志，它们就能为动态提供一个骨架，从而对系统本身有一个全面的了解。计划是制定系统和准确的地标计算方法。 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某些地标（可以通过有限精度计算进行验证），则可以获得所有时间的结论。 希望能与固体物理和化学中的问题所激发的具体问题联系起来。 该项目的工作也将被用作研究生，博士后和游客的培训基地。 50多年来，动力系统中最常用的地标是通常的双曲流形和拟周期轨道以及它们的稳定和不稳定流形。 在这里，它建议开发一个系统的方法来计算这些对象的理论和数值。 另一个目标是证明建设性的存在定理，验证近似解，并开发和实现快速和准确的算法。 统一原则是寻找描述不变性的函数方程，然后使用各种方法来解决它们。这些方法将从几何分析到功能分析而有所不同。 进一步的工作是开始研究一些无穷维问题的特解，包括偏微分方程和具有状态依赖时滞的时滞微分方程。

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.