Chimera states for repulsively coupled phase oscillators

Chimera states represent remarkable spatio-temporal patterns, where phase-locked oscillators coexist with drifting ones. Surprisingly, they can arise in arrays of coupled identical oscillators without any sign of asymmetry as a manifestation of internal nonlinear nature of dynamical networks.

We discuss the appearance of chimera states for repulsively coupled phase oscillators of the Kuramoto-Sakaguchi type, i.e., when the phase lag parameter $\alpha > \pi/2$ and hence the network coupling works against synchronization. We find that chimeras exist in wide domain of the parameter space as a cascade of the states with increasing number of regions of irregularity---the so-called chimera's heads.

We also study the origin of the chimera states and show that they grow from so-called multi-twisted states. Three typical scenarios of the chimera birth are reported: 1) via saddle-node bifurcation on invariant circle, also known as SNIC or SNIPER, 2) via blue-sky catastrophe, when two periodic orbits – stable and saddle - approach each other creating a saddle-node periodic orbit, and 3) via homoclinic transition, when the unstable manifold comes back crossing the stable manifold of a saddle.