Existence and stability of chimera states in a minimal system of phase oscillators

Abstract

We examine partial frequency locked weak chimera states in a network of six identical and indistinguishable phase oscillators with neighbour and next-neighbour coupling and two harmonic coupling of the form g(φ) = − sin(φ − α) + r sin 2φ. We limit to a specific partial cluster subspace, reduce to a two dimensional system in terms of phase differences and show that this has an integral of motion for α = π/2 and r = 0. By careful analysis of the phase space we show there is a continuum of neutrally stable weak chimera states in this case. We approximate the Poincar´e return map for these weak chimera solutions and demonstrate several results about the stability and bifurcation of weak chimeras for small β = π/2 − α and r that agree with numerical path following of the solutions.