Dynamics and Synchronization of Weak Chimera States for a Coupled Oscillator System

Abstract

This thesis is an investigation of chimera states in a network of identical coupled phase oscillators. Chimera states are intriguing phenomena that can occur in systems of coupled identical phase oscillators when synchronized and desynchronized oscillators coexist. We use the Kuramoto model and coupling function of Hansel for a specific system of six oscillators to prove the existence of chimera states. More precisely, we prove analytically there are chimera states in a small network of six phase oscillators previously investigated numerically by Ashwin and Burylko [8]. We can reduce to a two-dimensional system within an invariant subspace, in terms of phase differences. This system is found to have an integral of motion for a specific choice of parameters. Using this we prove there is a set of periodic orbits that is a weak chimera. Moreover, we are able to confirmthat there is an infinite number of chimera states at the special case of parameters, using the weak chimera definition of [8]. We approximate the Poincaré return map for these weak chimera solutions and demonstrate several results about their stability and bifurcation for nearby parameters. These agree with numerical path following of the solutions. We also consider another invariant subspace to reduce the Kuramoto model of six coupled phase oscillators to a first order differential equation. We analyse this equation numerically and find regions of attracting chimera states exist within this invariant subspace. By computing eigenvalues at a nonhyperbolic point for the system of phase differences, we numerically find there are chimera states in the invariant subspace that are attracting within full system.