Exact dynamics of phase transitions in oscillator populations with nonlinear coupling (Q2074165)

The article investigates the dynamics of an ensemble of coupled phase oscillators, which is governed by a modified Kuramoto equation using a power law function of the macroscopic order parameter. The natural frequencies of the oscillators are assumed to be uniformly distributed. It is numerically demonstrated, that depending on the exponent \(\alpha\) of the coupling law it is possible, that a partially locked state can exist for \(\alpha < 1\), whereas it cannot occur for \(\alpha \ge 1\). Using an ansatz proposed by Ott and Antonsen the authors are able to derive a partial differential equation describing the collective behaviour of the oscillators and study it's steady state solutions and the stability properties. These results are in perfect agreement with the numerically observed synchronization behaviour.