Synchronization of phase oscillators under asymmetric and bimodal distributions of natural frequencies

We report on the setting of synchronization in ensembles of phase oscillators under an asymmetric and bimodal distribution of initial frequencies. In particular, by both the Ott–Antonsen reduction and direct numerical simulations we give evidence that the system exhibits four different transition routes towards synchronization, including continuous, discontinuous, and hybrid transitions, depending on the choice of the parameters of the frequency distribution. As the coupling strength increases, several coherent states emerge, such as two types of travelling wave states and the Bellerophon state, which are carefully characterized.