Distribution functions of argumental oscillations of the Duboshinskiy pendulum

In this paper, we study the so-called argumental oscillations, i.e., vibrational processes in a classical system with quantized amplitudes, on the example of the Duboshinskiy pendulum. We derive the exact equation of motion of the pendulum and find fast-oscillating solutions with various amplitudes. We show theoretically and prove by numeric modeling that the values of the stable amplitudes depend on the parity of the force field. We derive the envelope equations and show that they can be used for predicting the evolution of the amplitude. We consider the phase plane of the pendulum and show that it contains attraction basins, each of which corresponds to a stable amplitude. By averaging over the fast-oscillating phase of the field, we obtain the distribution function over the energy levels, i.e., the probabilities of stabilization of a particular level.