Phase lock and rotational motion of a parametric pendulum
  The effect of noise on a rotational mode of a pendulum excited kinematically in vertical direction has been analyzed. We have shown that for a weak noise transitions from oscillations to rotations and vice versa are possible. For a moderate noise level dynamics of the system is governed by a combination of the excitation amplitude and stochastic component. Consequently for stronger noise the rotational solution as an independent sychonized mode has vanished. 
Grzegorz Litak, Marek Borowiec, and Marian Wiercigroch