We consider the dynamical behavior of coupled oscillators with robust heteroclinic cycles between saddles that may be periodic or chaotic. We differentiate attracting cycles into types that we call phase resetting and free running depending on whether the cycle approaches a given saddle along one or many trajectories. At loss of stability of attracting cycling, we show in a phase-resetting example the existence of an infinite family of stable periodic orbits that accumulate on the cycling, whereas for a free-running example loss of stability of the cycling gives rise to a single quasiperiodic or chaotic attractor.