Twisted states in nonlocally coupled phase oscillators with bimodal frequency distribution

Twisted states, referred to traveling waves in an array of nonlocally coupled phase oscillators, have drawn some attention in recent years. In this work, we study a one-dimensional ring of phase oscillators with nonlocal coupling and a bimodal natural frequency distribution. We show that twisted standing waves and stationary twisted states appear successively with the increase of the coupling strength. In the continuum limit, we derive a low-dimensional reduced equation using the Ott-Antonsen ansatz, which verifies the twisted states in the simulations of finite networks of oscillators. We also theoretically investigate the stationary twisted states and their stabilities by using the reduced equation.