Nonlinear dynamics of a classical rotating pendulum system with multiple excitations*

Abstract

We report an attempt to reveal the nonlinear dynamic behavior of a classical rotating pendulum system subjected to combined excitations of constant force and periodic excitation. The unperturbed system characterized by strong irrational nonlinearity bears significant similarities to the coupling of a simple pendulum and a smooth and discontinuous (SD) oscillator, especially the phase trajectory with coexistence of Duffing-type and pendulum-type homoclinic orbits. In order to learn the effect of constant force on this pendulum system, all types of phase portraits are displayed by means of the Hamiltonian function with large constant excitation especially the transitions of complex singular closed orbits. Under sufficiently small perturbations of the viscous damping and constant excitation, the Melnikov method is used to analyze the global structure of the phase space and the feature of trajectories. It is shown, both theoretically and numerically, that this system undergoes a homoclinic bifurcation and then bifurcates a unique attracting rotating limit cycle. Finally, the estimation of the chaotic threshold of the rotating pendulum system with multiple excitations is calculated and the predicted periodic and chaotic motions can be shown by applying numerical simulations.