Periodic synchronization in a system of coupled phase oscillators with attractive and repulsive interactions

In this study we investigate the collective behavior of the generalized Kuramoto model with an external pinning force in which oscillators with positive and negative coupling strengths are conformists and contrarians, respectively. We focus on a situation in which the natural frequencies of the oscillators follow a uniform probability density. By numerically simulating the model, it is shown that the model supports multistable synchronized states such as a traveling wave state, pi state and periodic synchronous state: an oscillating pi state. The oscillating pi state may be characterized by the phase distribution oscillating in a confined region and the phase difference between conformists and contrarians oscillating around pi periodically. In addition, we present the parameter space of the oscillating pi state and traveling wave state of the model.