The interaction of many coupled dynamical units is a theme across many scientific 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 interaction 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 consider systems of many spatially distributed phase oscillators that interact with their neighbors, and each oscillator can have a different natural frequency, and a different response time to the signals it receives from other oscillators in its neighborhood. By first reducing the microscopic dynamics to a macroscopic partial-differential-equation description, we then numerically find that finite oscillator response time leads to many interesting spatio-temporal dynamical behaviors, and we study interactions and evolutionary behaviors of these spatio-temporal patterns.

The last part of this dissertation 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. Our first goal is to investigate the effect of a spread in the amplitude growth parameter of the oscillators and that 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, a simple constant-amplitude sinusoidal oscillation is always a solution for the dynamics of the global order parameter when the system has constant nonlinear characteristics.