ABSTRACT Title of dissertation: THE EFFECTS OF COUPLING DELAY AND AMPLITUDE / PHASE INTERACTION ON LARGE COUPLED OSCILLATOR NETWORKS

The interaction of many coupled dynamical units is a theme across many scien-tific disciplines. A useful framework for beginning to understanding such phenomena is the coupled oscillator network description. In this dissertation, we study a few problems related to this. The first part of the dissertation studies generic effects of heterogeneous in-teraction delays on the dynamics of large systems of coupled oscillators. Here, we modify the Kuramoto model (phase oscillator model) to incorporate a distribution of interaction delays. Corresponding to the continuum limit, we focus on the reduced dynamics on an invariant manifold of the original system, and derive governing equations for the system, which we use to study stability of the incoherent state 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 dissertation is a sequel to the first part. Here, we con-sider systems of many spatially distributed phase oscillators that interact with thei