Two problems about coupled oscillators: transient dynamics and swarming

Abstract

The conduct of coupled oscillators has long beguiled scientists. Here we study two models of such oscillators. The first is Peskin’s integrate-and-fire model. We focus on the transitory behavior, showing that in its infancy, synchrony looks much like aggregation. In the second model, we consider oscillators which ad- just their positions in space as well as their phases. We show the coaction of these two effects produces novel spatiotemporal patterns, which we study both analytically and numerically.