The Ott–Antonsen Ansatz in Globally Coupled Phase Oscillators

The Ott–Antonsen ansatz provides a powerful tool in investigating synchronization among coupled phase oscillators. However, previous works using the ansatz only focused on the evolution of the order parameter and the information on desynchronized oscillators is less discussed. In this work, we show that the Ott–Antonsen ansatz can also be applied to investigate the desynchronous dynamics in coupled phase oscillators. Studying the original Kuramoto model and two of its variants, we find that the dynamics of \alpha(\omega), the coefficient in the Fourier series of the probability density, can give most of the information on the synchronization, for example, the threshold of natural frequency delimiting the oscillators synchronized and desychronized by the mean field, the formulation of the effective frequency \omega_\rm e(\omega) of desynchronous oscillators, and the structure of the graph \omega_\rm e(\omega).