Chunking Rhythmic Synchronization: Bellerophon States and Quantized Clusters of Globally Coupled Phase Oscillators

The emergence of phase coherence in interacting oscillators is one of the main phenomena for the coordination of events that make a system to behave cooperatively. Examples range from rhythmic physiological processes to the collective behaviors of technological and natural networks. We concentrate here on Bellerophon states, which are coherent states of rhythmic synchrony occurring in globally coupled oscillators close to the point where the transition from disorder to phase order converts from abrupt to continuous. Within Bellerophon states, oscillators form quantized clusters, where their instantaneous phases and frequencies are unlocked. Within each cluster, the oscillators’ instantaneous frequencies form a characteristic cusped pattern and, more importantly, they behave periodically in time, so that their long-time average values are the same. Along the manuscript, we give analytical and numerical description of these states, and we discuss their general appearance behind the collective rhythms reported in other systems of interacting oscillators.