Cataloging the global behavior of dynamical systems : adaptively searching parameter space using the Conley-Morse database

Abstract

The aim of this thesis is to build upon a combinatorial-topological framework to global dynamics of multiparameter dynamical systems. A combinatorial multivalued map of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented via a directed graph. The dynamics is then decomposed into the recurrent and gradient-like parts by graph theoretic algorithms using an adaptive computation. The novelty of this thesis is to introduce a similar adaptive scheme in parameter space. Furthermore, it is proven that this scheme produces an output which is naturally coarser than the output of an original computation. Incorporating previous results, we make an estimate for the savings achieved by this adaptive scheme in the setting of a saddle-node bifurcation. Furthermore, we make an empirical comparison of how well our scheme approximates an original computation.