Complex Mixed-Mode Oscillations and a Search for Oscillator Glass

Abstract

In this thesis, we consider problems across two families of dynamical systems: low-dimensional systems that have multiple timescales, and high-dimensional systems of coupled oscillators. We study a three-dimensional system in two slow and one fast variables, which has been used to model electrochemical oscillations. We demonstrate the existence of a manifold of elusive Shilnikov homoclinic orbits in the parameter space. Each of these orbits organizes a complex structure of mixed-mode oscillations (MMOs). We then study the dynamics near a tangency bifurcation between an unstable manifold and a slow manifold. We find bifurcations and define a dynamical partition to analyze some of the complicated MMOs which arise immediately after the tangency. Finally, we consider a variant of a Kuramoto system which has been used as a model of spin glass. We apply an ansatz of Ott and Antonsen to effectively reduce the dimension of the system, and derive a phase diagram of the system.