Emergent dynamics of Winfree oscillators on locally coupled networks

The Winfree model is the first mathematical model for synchronization of weakly coupled oscillators. Compared to the well-known Kuramoto model, the Winfree model does not conserve the total phase. This leads to rich dynamic features compared to those produced by other phase models. In this paper, we study the emergent dynamics of the Winfree model on a locally coupled static network. A randomly chosen phase configuration undergoes several dynamic phase transitions such as incoherence, partial locking, complete locking, partial oscillator death, and complete oscillator death, as the coupling strength increases. We provide several rigorous analytical results on the emergence of these dynamic features. We also provide several numerical simulations and compare their results to the analytical results.