Invariant objects in dynamical systems: Analysis and numerics

自然科学中的许多问题都是用一个系统来建模的，这个系统有一个简单的规则，它把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奥的数学问题。一种可能的方法是寻找“地标”。地标是一个小的子系统，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些地标，它们就能够为动态提供一个框架，从而提供对系统本身的整体理解。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某些地标（可以用有限精度计算验证），则可以获得所有时间的结论。希望能接触到固态物理和化学问题引发的具体问题。该项目的工作也将被用作研究生、博士后和访客的培训基地。50多年来，动力系统中最常用的标志是双曲流形和准周期轨道，以及它们的稳定流形和不稳定流形。在此，提出了一种系统的方法来计算这些对象的理论和数值。另一个目标是证明证明近似解的建设性存在性定理，并开发和实现快速准确的算法。统一的原则是寻找描述不变性的泛函方程，然后用各种方法尝试求解它们。方法从几何到功能分析各不相同。下一步的工作是开始研究一些无限维问题的特殊解，包括状态相关时滞的偏微分方程和时滞微分方程。

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.