  Chimera states occur in networks of coupled oscillators, and are
characterized by having some fraction of the oscillators perfectly
synchronized, while the remainder are desynchronized. Most chimera states have
been observed in networks of phase oscillators with coupling via a sinusoidal
function of phase differences, and it is only for such networks that any
analysis has been performed. Here we present the first analysis of chimera
states in a network of planar oscillators, each of which is described by both
an amplitude and a phase. We find that as the attractivity of the underlying
periodic orbit is reduced chimeras are destroyed in saddle-node bifurcations,
and supercritical Hopf and homoclinic bifurcations of chimeras also occur.
