Order and chaos around resonant motion in librating spring–mass–spherical pendulum

Abstract

Using both analytical and numerical techniques, we investigate the 3-degree-of-freedom, autonomous, autoparametric, Hamiltonian spring–mass–pendulum system in libration and reveal the order–chaos–order transition with the change in the ratio of frequencies of corresponding independent normal modes. In the process, we also present an integrable limit of it and find all the three independent constants of motion. Furthermore, we study the possibility of the precession of the swing plane of the constituent spherical pendulum and the related energy exchanges between the modes at the autoparametric resonance. We use the method of the fast Lyapunov indicators to characterise and distinguish the order and the chaos in the system.