Resonance rotations of a pendulum with a vibrating suspension

Resonance rotations of the form s:n (where s and n are integers), for which the pendulum executes exactly s rotations during n oscillations of the suspension point are investigated for a plane pendulum with a vibrating suspension point, when there is external and internal damping. It is shown analytically that asymptotically stable resonance rotations of the form 1:2, 1:3, 2:1, 2:3 and 3:1 exist. Many other resonance rotations of higher orders are found from the results of a numerical experiment.