Rotating solutions and stability of parametric pendulum by perturbation method

Rotating solutions of a parametrically driven pendulum are studied via a perturbation method by assuming the undamped unforced Hamiltonian case as basic solution, and damping and excitation as small perturbations. The existence and stability of the harmonic solution are determined analytically. First-order terms are mainly considered, but the extensions to higher-order terms, as well as to subharmonic rotations, are straightforward. Setting up the analysis in the phase space instead of in the physical space has facilitated development of simple but comprehensive formulas. Comparison with numerical simulations shows that the analytical results are very effective in accurate predictions, even for fairly large amplitudes, and in detecting the saddle–node bifurcation where the rotating solution is born. A limited accuracy is observed, to first order, in detecting the period–doubling bifurcation where it loses stability, and an explanation is proposed.