Synchronization in phase-coupled oscillator with attractive-repulsive frequencies

We investigate the synchronization behavior of a simple but quite useful mode of emergent collective behavior in ensembles of interacting dynamical elements, the Kuramoto model with attractive-repulsive frequencies features. Here, we derive a series of phase-locked (PL) states and identify the significant synchronization transition points analytically with exact boundary conditions. A detailed stability study of the model is also presented, as well as the bifurcation of the PL states set. Extremely, we show that these frequencies do not influence the stability of the system model, while the synchronization ability is considerably changed.