Topological Methods for Evolution Equations

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dc.contributor.advisorWanner, Thomas
dc.contributor.authorStephens, Thomas Dean
dc.creatorStephens, Thomas Dean
dc.date.accessioned2016-09-28T10:23:06Z
dc.date.available2016-09-28T10:23:06Z
dc.date.issued2016
dc.description.abstractIn 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
dc.format.extent149 pages
dc.identifier.urihttps://hdl.handle.net/1920/10461
dc.language.isoen
dc.rightsCopyright 2016 Thomas Dean Stephens
dc.subjectMathematics
dc.subjectComputer-assisted proof
dc.subjectConley index
dc.subjectDynamical systems
dc.subjectIsolated invariant set
dc.subjectIsolating block
dc.subjectSuperlevel sets
dc.titleTopological Methods for Evolution Equations
dc.typeDissertation
thesis.degree.disciplineMathematics
thesis.degree.grantorGeorge Mason University
thesis.degree.levelPh.D.

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