Resonant Layers in a Parametrically excited Pendulum

The energy increment spectrum method is developed for the numerical prediction of a specific primary resonant layer, and the width of the resonant layer can be estimated through the energy increment spectrum. This numerical approach is applied to investigate the (2M:1)-librational and (M:1)-rotational, resonant layers in a parametrically excited pendulum, and the corresponding analytical conditions for such resonant layers are developed. The numerical approach predicts the appearance and disappearance of resonant layers in nonlinear Hamiltonian systems rather than the conventional Poincare mapping method. Illustrations of the analytical and numerical results for the appearance and disappearance of the resonant layers are given. The width of the resonant layers in the paremetric pendulum is computed. The analytical method should be further improved through renormalization.