Chimeras in Two-Dimensional Domains: Heterogeneity and the Continuum Limit

We consider three different two-dimensional networks of nonlocally coupled heterogeneous phase oscillators. These networks were previously studied with identical oscillators, and a number of spatiotemporal patterns were found, mostly as a result of direct numerical simulation. Here we take the continuum limit of an infinite number of oscillators and use the Ott-Antonsen ansatz to derive continuum level evolution equations for order parameter-like quantities. Most of the patterns previously found in these networks correspond to relative fixed points of these evolution equations, and we show the following results of extensive numerical investigations of these fixed points: their existence and stability, and the bifurcations involved in their loss of stability as parameters are varied. Our results answer a number of questions posed by previous authors who studied these networks, and provide a better understanding of these networks' dynamics.