Stochastic and Resonant Layers in a Periodically Driven Pendulum

The stochastic and resonant layers in a periodically driven pendulum are investigated in this paper. The analytical solutions and nonlinear natural frequencies of the non-driven pendulum are first derived, and then the energy increments in the stochastic and resonant layers for the driven pendulum are approximated by the work done by the external excitation along the heteroclinic and resonant orbits, respectively. From these results, the approximate analytical conditions for the predictions of stochastic and resonant layers are constructed. Based on these conditions, input parameters are computed for the numerical simulations of the stochastic and resonant layers. The stochastic and resonant layers associated with the librational and rotational motions are illustrated via the Poincare mapping sections.