Areas of attraction of the pendulum under the influence of oblique vibration of the suspension point

The inverse problem of stabilizing a spherical pendulum in a given position by means of high-frequency vibration of the suspension point is posed. The position of the pendulum is determined by the angle between the pendulum rod and the vertical. For any given position of the pendulum, a one-parameter series of oblique vibration characteristics (the amplitude of the vibration velocity and the angle between the vibration velocity vector and the vertical) is found to stabilize the pendulum in this position. For the obtained series, the regions of attraction are determined (the initial points from which a given stable position of the pendulum will be established under the influence of vibration).