Abstract
OF THE DISSERTATION Shift equivalence and a combinatorial-topological approach to discrete-time dynamical systems by Justin Bush Dissertation Director: Konstantin Mischaikow Given a parameterized family of discrete-time dynamical systems, we aim to investigate how the global dynamics depends on the parameters in a way that is meaningful for applications. The discrete Conley index is an algebraic topological invariant of recurrent dynamics that is robust to small changes in parameters. Its definition, however, is given in terms of shift equivalence, which is not straightforward to compute in the category of abelian groups. We discuss the challenge of interpreting shift equivalnce, and give a construction that for every square integer matrix produces an interval map that giving rise to dynamics represented by that matrix. We conclude with applications of this approach to dynamical systems to the logistic map, Newton’s method in the plane, and to population models in biology.
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