A Restricted Conley Index and Robust Dynamics of Coupled Oscillator Systems

Abstract

In this thesis, we explore the robustness of heteroclinic cycles which can appear as solutions of dynamical systems subject to certain constraints. We develop a method, using topological notions, to inspect the dynamics of simple heteroclinic cycles; in particular, in the first part of the thesis we develop a “restricted Conley index”. This tool is defined by restricting the general Conley index to specific invariant subspaces associated with constraints on the vector field. The resulting restricted Conley index allows us to find connections that are robust to perturbations that respect these constraints. In the second part of this thesis we study the dynamical problem of designing a system of globally coupled oscillator systems with a specific structure. We extend some known results on conditions for stability of cluster states in these systems and, as an example, we give sufficient stability conditions for cluster states with three non-trivial clusters, (2, 2, 2)−cluster states, in a system of six globally coupled oscillators. We show that robust heteroclinic cycles connecting these states can appear as a result of our investigation on dynamics of a systems of six globally coupled oscillators, and we use the restricted Conley index to investigate the robustness of heteroclinic cycles between three nontrivial clusters.