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.Republic of Iraq,
Ministry of Higher Education and Scientific Research
