  Chimera state is a recently discovered dynamical phenomenon in arrays of
nonlocally coupled oscillators, that displays a self-organized spatial pattern
of co-existing coherence and incoherence. We discuss the appearance of the
chimera states in networks of phase oscillators with attractive and with
repulsive interactions, i.e. when the coupling respectively favors
synchronization or works against it. By systematically analyzing the dependence
of the spatiotemporal dynamics on the level of coupling
attractivity/repulsivity and the range of coupling, we uncover that different
types of chimera states exist in wide domains of the parameter space as
cascades 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 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 with complex multistable dynamics including an
"eight-like" limit cycle resulting eventually in a chimera state.
