Designing Pulse Coupled Oscillators To Synchronize

Abstract

Oscillators exhibit some of the simplest dynamic behavior, yet systems of interacting oscillators are capable of intricate and complex behaviors. It is partly this elegance that has made them a frequent object of study, but also their ubiquity. In particular, pulse coupled oscillators (PCOs), where oscillator interactions occur at discrete times are particularly applicable. Not only do PCOs have applications in biological systems-such as the heart's pacemaker, flashing fireflies, and biological neural networks-PCOs have recently been adapted to control clocks inside wireless sensor networks [33, 21]. However, since traditional PCO models were designed to study synchronization abstractly, it is not surprising that they are no longer able to synchronize when exposed to real world settings. This thesis investigates the classic dynamical systems question of the synchronization of pulse coupled oscillators when classic dynamical systems tools no longer apply. Instead, utilizing a variety of nonstandard techniques, we design a new PCO model and prove strong theoretical results, which have applications in biology and the engineering of wireless sensor networks.