Bifurcations in the Sakaguchi–Kuramoto model

•We explain the appearance of non-universal synchronization transitions.•We introduce a frequency dependent version of the Ott–Antonsen method.•This method is suitable for any type of natural frequency distributions.•We provide a detailed bifurcation analysis of partially synchronized states.

We analyze the Sakaguchi–Kuramoto model of coupled phase oscillators in a continuum limit given by a frequency dependent version of the Ott–Antonsen system. Based on a self-consistency equation, we provide a detailed analysis of partially synchronized states, their bifurcation from the completely incoherent state and their stability properties. We use this method to analyze the bifurcations for various types of frequency distributions and explain the appearance of non-universal synchronization transitions.