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Chimera state is a recently discovered dynamical phenomenon in arrays ofnonlocally coupled oscillators, that displays a self-organized spatial patternof co-existing coherence and incoherence. We discuss the appearance of thechimera states in networks of phase oscillators with attractive and withrepulsive interactions, i.e. when the coupling respectively favorssynchronization or works against it. By systematically analyzing the dependenceof the spatiotemporal dynamics on the level of couplingattractivity/repulsivity and the range of coupling, we uncover that differenttypes of chimera states exist in wide domains of the parameter space ascascades of the states with increasing number of intervals of irregularity,so-called chimera's heads. We report three scenarios for the chimera birth: 1)via saddle-node bifurcation on a resonant invariant circle, also known as SNICor SNIPER, 2) via blue-sky catastrophe, when two periodic orbits, stable andsaddle, approach each other creating a saddle-node periodic orbit, and 3) viahomoclinic transition with complex multistable dynamics including an"eight-like" limit cycle resulting eventually in a chimera state.

Cascades of Multi-headed Chimera States for Coupled Phase Oscillators