Dynamics of the Nearly Parametric Pendulum

TL;DR

This paper investigates how slight ellipticity in the driving force affects the dynamics of a parametric pendulum, revealing an expanded rotation region and merged resonance tongues, thus advancing understanding of driven nonlinear oscillators.

Contribution

It introduces the effect of small ellipticity on the classical parametric pendulum, showing increased rotation regions and merged resonance tongues, which was not previously documented.

Findings

Rotation region expands with ellipticity

Resonance tongues merge into a single instability region

Enhanced understanding of driven pendulum dynamics

Abstract

Dynamically stable periodic rotations of a driven pendulum provide a unique mechanism for generating a uniform rotation from bounded excitations. This paper studies the effects of a small ellipticity of the driving, perturbing the classical parametric pendulum. The first finding is that the region in the parameter plane of amplitude and frequency of excitation where rotations are possible increases with the ellipticity. Second, the resonance tongues, which are the most characteristic feature of the classical bifurcation scenario of a parametrically driven pendulum, merge into a single region of instability.