Synchronization of Oscillators

Abstract

In this thesis we consider a system of coupled oscillators: 'cells' that progress through a cycle and neither reproduce nor die. The cycle the oscillators progress through is divided into a region in which oscillators can send signals and a region in which they can receive and react to those signals by speeding up or slowing down their progression, as a generalization of the integrate-and-fire model of Mirollo & Strogatz.

We say oscillators are synchronized if they have identical phase and we are interested in the stability of the synchronized population. Does a population that is split in two return to the synchronized state or not. Instead of solving a system of coupled ODE's exactly, we analyze the dynamics of the oscillators by a Poincare-like map on the unit interval, called full cycle map, that gives the position of a group when the other has completed one cycle.

Since the synchronized state corresponds to the two boundary fixed points of the full cycle map, we can analyze the (local) stability of the synchronized state by considering the derivative of the full cycle map. Also conjectures can be made about the full cycle map itself. In this way we investigate the influence of the sign and amount of feedback on the stability of synchronization and the existence and stability of cycle phase-locked configurations: two oscillators that neither converge nor diverge after one cycle completion.

Another approach focuses on the result of feedback on oscillators going from signalling to receiving or vice versa. This method gives intuitive results on synchronization that can easily be generalized to any number of oscillators, but lacks the precision to calculate the existence and stability of cycle phase-locked configurations.