Topological Methods for Evolution Equations
Date
2016
Authors
Stephens, Thomas Dean
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Abstract
In this dissertation we develop a framework for rigorously computing the Conley index of isolated invariant sets for flows generated by finite-dimensional systems of ordinary dif- ferential equations x ̇ = f(x), where f : Rn → Rn. Our main contribution in this area is the characterization of isolating blocks in terms of the level sets and superlevel sets of two real- valued functions, u,v : Rn → R. The functions u and v incorporate geometric quantities computed on the boundary of proposed isolating blocks and relate them to local behavior of the vector field f. In order to obtain numerically rigorous results in this area, we have developed a new tool for computing superlevel sets of real-valued functions u : Rn → R that guarantees our superlevel set approximations are homotopy equivalent to the actual superlevel sets we are interested in. This new tool is presented as the first logical half of this document, as it is a significant advance in its own right.
Our work makes use of basic differential geometry on piecewise smooth manifolds, ex- ploits the interplay between flows and the topology of the underlying phase space (provided by the Waz ̇ewski theorem), and employs interval arithmetic and automatic differentiation. We provide full details for a collection of algorithms which enable the practitioner to easily apply our framework to a wide variety of problems in the theory of dynamical systems.
Several examples are provided showing the relative simplicity of our approach over earlier
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Keywords
Mathematics, Computer-assisted proof, Conley index, Dynamical systems, Isolated invariant set, Isolating block, Superlevel sets