Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?

TL;DR

This paper revisits the historical and mathematical explanation of dissipation-induced destabilization, highlighting the role of Whitney umbrella singularities in understanding Ziegler's paradox and related instabilities.

Contribution

It uncovers the early, overlooked explanation of Ziegler's paradox via Whitney umbrella singularity from 1956, predating later research.

Findings

Historical analysis of Bottema's 1956 work

Connection between Whitney umbrella singularity and dissipation-induced instability

Review of subsequent developments in stability analysis

Abstract

The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler's paradox (1952), has stimulated an ever growing interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. Since the last decade it has been widely accepted that dissipation-induced instabilities are closely related to singularities arising on the stability boundary. What is less known is that the first complete explanation of Ziegler's paradox by means of the Whitney umbrella singularity dates back to 1956. We revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. We discuss subsequent developments of the perturbation analysis of dissipation-induced instabilities and applications over this period, involving structural stability of matrices, Krein collision, Hamilton-Hopf bifurcation and related bifurcations.