Damped bead on a rotating circular hoop - a bifurcation zoo

TL;DR

This paper explores the complex bifurcation phenomena of a damped bead on a rotating hoop, revealing rich dynamics and stability intricacies through phase portraits and novel analytical techniques.

Contribution

It introduces a comprehensive analysis of bifurcations in the damped bead-hoop system, including a new method for stability analysis in complex cases.

Findings

Multiple bifurcation types identified

Phase portraits illustrating diverse motion modes

New stability analysis technique developed

Abstract

The evergreen problem of a bead on a rotating hoop shows a multitude of bifurcations when the bead moves with friction. This motion is studied for different values of the damping coefficient and rotational speeds of the hoop. Phase portraits and trajectories corresponding to all different modes of motion of the bead are presented. They illustrate the rich dynamics associated with this simple system. For some range of values of the damping coefficient and rotational speeds of the hoop, linear stability analysis of the equilibrium points is inadequate to classify their nature. A technique involving transformation of coordinates and order of magnitude arguments is presented to examine such cases. This may provide a general framework to investigate other complex systems.