Subharmonic bifurcations in a pendulum parametrically excited by a non-harmonic perturbation

A pendulum whose support is subjected to a periodic non-harmonic oscillation in the vertical direction is considered. The subharmonic Melnikov functions for the oscillating and for the rotating motions are explicitly constructed. It is shown that both functions converge towards the homoclinic Melnikov function and furthermore all the results for the harmonic perturbation are recovered.