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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.

Riddled basins denote a characteristic type of fractal domain of attraction that can arise when a chaotic motion is restricted to an invariant subspace of total phase space. An example is the synchronized motion of two identical chaotic oscillators. The paper examines the conditions for the appearance of such basins for a system of two symmetrically coupled logistic maps. We determine the regions in parameter plane where the transverse Lyapunov exponent is negative. The bifurcation curves for the transverse destabilization of lowperiodic orbits embedded in the chaotic attractor are obtained, and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations. It is shown that the appearance of transversely unstable orbits does not necessarily lead to an observable basin riddling, and that the loss of weak stability ͑when the transverse Lyapunov exponent becomes positive͒ does not necessarily destroy the basin of attraction. Instead, the symmetry of the synchronized state may break, and the attractor may spread into two-dimensional phase space. ͓S1063-651X͑98͒05303-3͔ PACS number͑s͒: 05.45.ϩb

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