Understanding the order-chaos-order transition in the planar elastic pendulum

Abstract

A planar elastic pendulum can be thought of as a planar simple pendulum and a one-dimensional Hookian spring carrying a point mass coupled together nonlinearly. This autonomous nonintegrable Hamiltonian system shows autoparametric resonance that corresponds to the primary resonance in the nearly integrable Hamiltonian approximating the planar elastic pendulum’s full Hamiltonian. The system is also known to exhibit the phenomenon of the order-chaos-order in which the system transits from a predominantly ordered state to a chaotic state and then back to a predominantly regular state. Although there are well-documented numerical experiments reporting that the system is most chaotic around the condition of autoparametric resonance, the exact mechanism behind the order-chaos-order transition sandwiching the aforementioned chaotic state is not completely understood. In this paper, by employing a combination of analytical and numerical methods, we establish that the order-chaos-order transition occurs due to the interaction between two resonances—one primary and another secondary.