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

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 2 : 1 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 2 : 1 resonances-one primary and another secondary.