On the Equilibria and Bifurcations of a Rotating Double Pendulum

TL;DR

This paper investigates the equilibrium configurations and bifurcations of a generalized rotating double pendulum system, employing algebraic elimination techniques to analyze complex polynomial equations across a broad parameter space.

Contribution

It introduces a comprehensive analysis of equilibria and bifurcations in a physically generalized double pendulum using advanced algebraic algorithms, expanding understanding beyond previous simplified models.

Findings

Identification of equilibrium configurations across parameters

Visualization of bifurcation phenomena in the system

Demonstration of the DixonEDF algorithm's effectiveness in complex computations

Abstract

The double pendulum, a simple system of classical mechanics, is widely studied as an example of, and testbed for, chaotic dynamics. In 2016, Maiti et al. studied a generalization of the simple double pendulum with equal point-masses at equal lengths, to a rotating double pendulum, fixed to a coordinate system uniformly rotating about the vertical. In this paper, we study a considerable generalization of the double pendulum, constructed from physical pendula, and ask what equilibrium configurations exist for the system across a comparatively large parameter space, as well as what bifurcations occur in those equilibria. Elimination algorithms are employed to reduce systems of polynomial equations, which allows for equilibria to be visualized, and also to demonstrate which models within the parameter space exhibit bifurcation. We find the DixonEDF algorithm for the Dixon resultant, written in the computer algebra system (CAS) Fermat, to be capable to complete the computation for the challenging system of equations that represents bifurcation, while attempts with other algorithms were terminated after several hours.