Singularly perturbed dynamics of the tippedisk

TL;DR

This paper analyzes the global dynamics of the tippedisk, a friction-induced inversion phenomenon, using singular perturbation theory to reveal bifurcation scenarios and derive conditions for inversion.

Contribution

It provides a comprehensive global analysis of the tippedisk's dynamics, including bifurcation scenarios and a closed-form condition for inversion, advancing understanding of friction-induced instability.

Findings

Identification of slow-fast dynamics due to friction

Bifurcation analysis involving homoclinic and Hopf bifurcations

Derivation of a critical spinning speed condition for inversion

Abstract

The tippedisk is a mathematical-mechanical archetype for a peculiar friction-induced instability phenomenon leading to the inversion of an unbalanced spinning disk, being reminiscent to (but different from) the well-known inversion of the tippetop. A reduced model of the tippedisk, in the form of a three-dimensional ordinary differential equation, has been derived recently, followed by a preliminary local stability analysis of stationary spinning solutions. In the current paper, a global analysis of the reduced system is pursued using the framework of singular perturbation theory. It is shown how the presence of friction leads to slow-fast dynamics and the creation of a two-dimensional slow manifold. Furthermore, it is revealed that a bifurcation scenario involving a homoclinic bifurcation and a Hopf bifurcation leads to an explanation of the inversion phenomenon. In particular, a closed-form condition for the critical spinning speed for the inversion phenomenon is derived. Hence, the tippedisk forms an excellent mathematical-mechanical problem for the analysis of global bifurcations in singularly perturbed dynamics.