Dynamical origin of the explosive synchronization with partial adaptive coupling

Explosive synchronization underlying many abrupt dynamical phenomena has attracted a great deal of attention in various fields. Here, we untangle the dynamical origin of the explosive synchronization in a system of coupled phase oscillators, incorporating partial adaptation encoded by feedback between the coupling and the global rhythm of the population. We mathematically argue that there exists no a critical adaptive fraction for the synchronization transition converting from the second to first order. Explosive synchronization, which is much easier to achieve than previously deemed, takes place as long as the adaptive fraction is nonzero. In particular, we uncover that an atypical scaling of the order parameter near criticality, manifesting the metastable state in the hysteresis region, is responsible for the emergence of explosive synchronization. Our work thus provides a profound insight into the dynamical nature of the abrupt transitions induced by the adaptation, a dynamical phenomenon of continuous interest and subject to intense investigation.