Dynamics in the Kuramoto model with a bi-harmonic coupling function

•We study a variant of the Kuramoto model with a bi-harmonic coupling function.•The model displays different synchronous dynamics. Such as the oscillating π state and travelling wave state are found.•The transitions between different dynamical states are explored by both forward continuation and backward continuation.

We study a variant of the Kuramoto model with a bi-harmonic coupling function, in which oscillators with positive first harmonic coupling strength are conformists and oscillators with negative first harmonic coupling strength are contrarians. We show that the model displays different synchronous dynamics and different dynamics may be characterized by the phase distributions of oscillators. There exist stationary synchronous states, travelling wave states, π state and, most interestingly, another type of nonstationary state: an oscillating π state. The phase distribution oscillates in a confined region and the phase difference between conformists and contrarians oscillates around π with a constant amplitude and a constant period in oscillating π state. Finally, the bifurcation diagram of the model in the parameter space is presented.