Periodic external driving of transmission-delay-coupled phase oscillator system: Switching between different twisted states

We consider the effects of an external periodic forcing on a spatially extended system that consists of identical phase oscillators coupled with transmission delays on a ring. Analyzing the continuum limit N→∞ of the model system along the Ott-Antonsen invariant manifold, we obtain the stability diagram for two regimes, called the forced and drifting entrainments. The former exhibits a spatially homogeneous solution trying to lock onto the drive, of which the stability boundary is rigorously determined. The latter represents a spatially organized group of oscillators that entrain one another at a frequency different from that of the drive. We show that in the drifting entrainment the external driving triggers the occurrence of unusual twisted states, characterized by nonuniform phase gradient as well as by the traveling wave of the order parameter amplitude. Moreover, it is found that by increasing or decreasing the forcing strength one can effectively switch between twisted states with different winding numbers. Our theoretical and numerical results for the reduced system are supported by the direct numerical simulations of the model system.