Ph.D. Thesis Defense: Wai Shing Lee
Friday, December 7, 2012
4:00 p.m. AVW 2328
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Announcement: Ph.D. Thesis Defense
Name: Wai Shing Lee
Date: Friday, December 7, 2012 at 4pm
Location: AVW 2328
Professor Edward Ott, Chair
Professor Thomas M. Antonsen Jr.
Professor P. S. Krishnaprasad
Professor Michelle Girvan
Professor Rajarshi Roy (Deans Representative)
Title: Some Theoretical Questions of Large Coupled Oscillator Network
The interaction of many coupled dynamical units is a common theme across a broad range of scientific disciplines. Examples include coupled lasers, Josephson junction circuits, interacting yeast cells, pacemaker cells in the heart, pedestrian induced oscillation of footbridges, chemically reacting systems, circadian rhythm, and many others. A very useful simplified framework for beginning to understanding phenomena observed in these situations is the coupled oscillator network description. In this thesis, we study a few theoretical problems related to this.
The first part of the thesis studies generic effects of heterogeneous communication delays on the dynamics of large systems of coupled oscillators. In this part, we study a modification of the Kuramoto model (phase oscillator model) incorporating a distribution of interaction delays. Corresponding to the case of a large number of oscillators, we consider the continuum limit. By focusing attention on the reduced dynamics on an invariant manifold of the original system, we derive governing equations for the system, which we use to study stability of the incoherent states and the dynamical transitional behavior from stable incoherent states to stable coherent states. We find that spread in the distribution function of delays can greatly alter the system dynamics.
The second part of this thesis is a sequel to the first part. Here, we consider systems of many spatially distributed phase oscillators that interact with their neighbors, and each oscillator is allowed to have a different natural frequency, as well as a different response time to the signals it receives from other oscillators in its neighborhood. By first reducing the microscopic dynamics of these systems to a macroscopic partial-differential-equation description, we then numerically find that finite oscillator response time leads to many interesting spatio-temporal dynamical behaviors including propagating fronts, spots, target patterns, chimerae, spiral waves, etc., and we study interactions and evolutionary behaviors of these spatio-temporal patterns.
The last part of this thesis addresses the behavior of large systems of heterogeneous, globally coupled oscillators each of which is described by the generic Landau-Stuart equation, which incorporates both phase and amplitude dynamics of individual oscillators. One goal of the present study is to investigate the effect of a spread in the amplitude growth parameter of the oscillators and the effect of a homogeneous nonlinear frequency shift. Both of these effects are of potential relevance to recently reported experiments. Our second goal is to gain further understanding of the observation that, at large coupling strength, the macroscopic nonlinear behavior of the global order parameter is often a simple constant amplitude sinusoidal oscillation (referred to as a ``locked state").