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.
