Resonance and stochastic layer in a parametrically excited pendulum

The (2M:1)-librational and (M:1)-rotational resonances are discovered in the stochastic layer of a parametrically excited pendulum. The analytical conditions for the onset of a resonance in the stochastic layer are derived. Numerical predictions of the appearance of resonance in the stochastic layer are also completed. Illustrations of the stochastic layer in the parametrically excited pendulums are given through the Poincare mapping sections. This methodology can be used for resonant layers in nonlinear Hamiltonian systems. However, the analytical approaches need to be improved for the better predictions of the resonant characteristics in the stochastic layer.