Exact dynamics of phase transitions in oscillator populations with nonlinear coupling

The Kuramoto model consisting of a population of coupled phase oscillators has served as an idealized tool for studying synchronization transitions in diverse self-sustained systems. It has been a common brief that the model exhibits a first-order discontinuous (hybrid) phase transition towards synchrony with the absence of partial locking by assuming that the natural frequencies are uniformly distributed. In this paper, we consider a variant of this model by modifying its global coupling to depend on a power law function of the macroscopic order parameter of the population via an exponent α. The generalization retains the uniform coupling and mean-field character of the system, where there is an interplay between the coupling and oscillator dynamics. Surprisingly, we show that the partial locking with the coexistence of the phase locked and drifting populations indeed exists for α<1, whereas it can never occur as long as α≥1. Through a remarkable ansatz of the frequency-dependent version of Ott–Antonsen manifold, we reveal that the long term macroscopic dynamics of the resulting model, as well as their corresponding critical properties can be analytically described. More importantly, we construct the characteristic function to give intuitive interpretations of a variety of dynamical phenomena occurring in the system, such as the emergence of the partial locking, vanishing synchronization onset, and the irreversibly abrupt desynchronization transition.