Computational Intersection Theory for Infinite Dimensional Dynamical Systems

本研究的目的是发展数学上严格的计算方法来研究无穷维动力系统的稳定流形和不稳定流形的交叉点。 这个问题自然分为两个不同的技术挑战。 首先，有必要将现有的计算相交理论的方法扩展到比目前更高的维度。 这个问题将通过降低到低维慢稳定不变流形来解决。 为了研究连接动力学，计算这些约化流形的线性丛也很重要。这需要将经典Floquet理论扩展到慢流形环境中。第二个主要挑战是发展后验技术来证明无限维中连接轨道的存在。问题是：一旦在足够高的有限维投影中建立了相应的连接，我们是否可以得出无限维系统中连接轨道存在的结论？解决这个问题需要扩展现有的方法来研究无穷维平衡和周期轨道的设置的边值问题，描述连接轨道。该项目还将考虑计算机辅助技术的可行性，以研究关于参数的连续性，以及连接轨道的分叉。这项研究将产生新的方法，以确保科学计算的正确性。 该项目的重点是应用数学的无限维模型，如偏微分方程，延迟方程和重整化算子。 除了提供数学上严格的误差界限，这些问题的近似数值解，计算机辅助证明的技术，从这项工作中产生的能够提供答案的理论问题的非线性系统的全局动力学。 例如，通过建立一些横向连接轨道的存在性，可以证明在偏微分方程的相空间中存在湍流、时空混沌或正拓扑熵。其他理论问题，可能会接近计算一旦技术的这个项目成为可用包括研究组合动力学的重整化算子，以及一些问题，在非线性分析涉及的应用弗洛尔的同调理论。该项目的一个中心主题是，在每个阶段的进步，理论和计算工具的发展将被应用到应用数学和动力系统理论的既定问题。

The purpose of this research is to develop mathematically rigorous computational methods for studying intersections of stable and unstable manifolds of infinite dimensional dynamical systems. The problem splits naturally into two distinct technical challenges. First it is necessary to extend existing methods of computational intersection theory to higher dimensions than currently accessible. This problem will be addressed via reductions to lower dimensional slow stable invariant manifolds. In order to study connecting dynamics it is also important to compute the linear bundles of these reduced manifolds. This requires an extension of classical Floquet theory into the slow manifold setting. The second major challenge is to develop a-posteriori techniques for proving the existence of connecting orbits in infinite dimensions. The question is: can we conclude the existence of connecting orbits in the infinite dimensional system once the existence of corresponding connections have been established in a projection of high enough finite dimension? Answering this question requires extending existing methods for studying infinite dimensional equilibria and periodic orbits to the setting of the boundary value problems which describe connecting orbits. The project will also consider the plausibility of computer assisted techniques for studying continuation with respect to parameter, as well as bifurcations of connecting orbits.This research will yield new methods for insuring the correctness of scientific computations. The focus of the project is on infinite dimensional models of applied mathematics such as partial differential equations, delay equations, and renormalization operators. In addition to providing mathematically rigorous error bounds for approximate numerical solutions of these problems, the techniques of computer assisted proof resulting from this work are able to provide answers to theoretical questions about the global dynamics of nonlinear systems. For example by establishing the existence of some transverse connecting orbits it is possible to prove the existence of turbulence, spatiotemporal chaos, or positive topological entropy in the phase space of a partial differential equation. Other theoretical problems which might be approached computationally once the techniques of this project become available include studying the combinatorial dynamics of renormalization operators, as well as some problems in nonlinear analysis involving the application of Floer's Homology theory. A central theme of this project is that at each stage of advancement the theoretical and computational tools developed will be applied to established problems of applied mathematics and dynamical systems theory.