Coupled Oscillators for Orientation Dynamics

Abstract

In this thesis we study a system of coupled phase oscillators that models animals group decision making for direction of travel. Each oscillator is an individual and its phase represents the orientation of the individual. We study the scenario where some individuals have preferred directions. Since the model has a gradient structure we can exclude oscillatory solutions. We also show that in stable equilibria individuals with the same preferred direction and preference strength must have identical orientation if the coupling between oscillators is harmonic. An arbitrarily small perturbation of harmonic coupling may lead to a split of groups with identical preferences. We produce bifurcation diagrams for the case where the population consists of three groups: two groups with conflicting preferred directions and one group without preference. We locate symmetry breaking points branches of equilibria and observe different bistability regions for three different values of the coupling coefficients: weak, moderate and strong.