Stability of Spiral Chimera States on a Torus

We study destabilization mechanisms of spiral coherence-incoherence patterns known as spiral chimera states that form on a two-dimensional lattice of nonlocally coupled phase oscillators. For this purpose we employ the linearization of the Ott--Antonsen equation that is valid in the continuum limit and perform a detailed two-parameter stability analysis of a $D_4$-symmetric chimera state, i.e., a four-core spiral state. We identify fold, Hopf, and parity-breaking bifurcations as the main mechanisms whereby spiral chimeras can lose stability. Beyond these bifurcations we find new spatio-temporal patterns, in particular quasiperiodic chimeras and $D_2$-symmetric spiral chimeras, as well as drifting states