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Complex Mixed-Mode Oscillations and a Search for Oscillator Glass

Author
Lizarraga, Ian Malcolm
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.
Date Issued
2017-08-30Subject
bifurcation; Koper; Kuramoto; manifold; oscillations; timescale; Physics; Mathematics; Applied mathematics
Committee Chair
Guckenheimer, John Mark
Committee Member
Rand, Richard Herbert; Strogatz, Steven H.
Degree Discipline
Applied Mathematics
Degree Name
Ph. D., Applied Mathematics
Degree Level
Doctor of Philosophy
Rights
Attribution-NonCommercial 4.0 International
Type
dissertation or thesis
Except where otherwise noted, this item's license is described as Attribution-NonCommercial 4.0 International