Computational Intersection Theory for Infinite Dimensional Dynamical Systems

这项研究的目的是发展数学严格的计算方法来研究无限维动力系统的稳定流形和不稳定流形的交集。这个问题自然而然地分裂为两个截然不同的技术挑战。首先，有必要将现有的计算交集理论方法扩展到比目前可访问的更高的维度。这个问题将通过降维到低维的慢稳定不变流形来解决。为了研究连通动力学，计算这些约化流形的线性丛也很重要。这就需要将经典的弗洛奎特理论推广到慢流形环境中。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联系，我们能得出无限维系统中存在连接轨道的结论吗？回答这个问题需要将现有的研究无限维平衡和周期轨道的方法推广到描述连通轨道的边值问题的背景下。该项目还将考虑计算机辅助技术研究参数连续性的可能性，以及连接轨道的分叉。这项研究将为确保科学计算的正确性提供新的方法。该项目的重点是应用数学的无限维模型，如偏微分方程组、延迟方程和重整化算子。除了为这些问题的近似数值解提供严格的数学误差界外，由这项工作产生的计算机辅助证明技术还能够为有关非线性系统的全局动力学的理论问题提供答案。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相空间中存在湍流、时空混沌或正拓扑熵。一旦这个项目的技术可用，可能会在计算上解决的其他理论问题包括研究重整化算子的组合动力学，以及涉及Floer同调理论应用的非线性分析中的一些问题。该项目的一个中心主题是，在发展的每个阶段，所开发的理论和计算工具将应用于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.