Synchronization of coupled simple harmonic oscillators is a well-studied problem in advanced undergraduate mechanics courses and the solution amounts to solving an eigenvalue problem. Synchronization of populations of auto-oscillators is a comparatively new field of study. The first scientists to consider such problems were mathematical biologists, but applied mathematicians and physicists have made significant contributions as well. The chief model of synchronization of distinct auto-oscillators is due to Kuramoto. The most striking feature of the model is the presence of a phase transition from an unsynchronized to a partially synchronized state at a critical value of the inter-oscillator coupling. Also, in spite of being a microscopic model that describes the interactions between individual oscillators, Kuramoto's model can be recast exactly as a mean field model. A great deal of work has focused on predicting the behavior of the mean field.
The first part of this dissertation describes my work exploring the Kuramoto model. Most physicists have approached the problem by analyzing the behavior of infinitely sized systems. I focus instead on making precise predictions for specific, finitely sized populations of oscillators. In particular, I demonstrate that the assumption of a constant mean field leads to surprisingly good self-consistent predictions for the mean field, particularly if the frequency of synchronization is made a tunable parameter. However, I find that the discontinuities in the self-consistent predictions do not exhibit critical scaling, in contradiction with the known critical behavior exhibited by the Kuramoto model.
The second part of this dissertation describes laboratory work and modeling of a mechanical system that exhibits synchronization. I examine the synchronization of 16 cell-phone vibrators coupled through a resonant plate. In light of the Kuramoto model, the interactions between the motors and the plate give somewhat unexpected results including bistability as well as ranges of frequencies in which the system never synchronize. I show, by starting with a first-principles model of the motors interacting with the plate, that the motors' interaction is similar to Kuramoto's model with two key differences: frequency-dependent coupling and a frequency-dependent phase delay
