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

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 $$ . We limit to a specific partial cluster subspace, reduce to a two dimensional system in terms of phase differences, and show that this has an integral of motion for $$ and $$ . By careful analysis of the phase space, we show that there is a continuum of neutrally stable weak chimera states in this case. We approximate the Poincaré return map for these weak chimera solutions and demonstrate several results about the stability and bifurcation of weak chimeras for small $$ and $$ that agree with numerical path-following of the solutions.