Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?
Article first published online: 12 MAY 2010
Copyright © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
Volume 90, Issue 6, pages 462–488, June 2010
How to Cite
Kirillov, O.N. and Verhulst, F. (2010), Paradoxes of dissipation-induced destabilization or who opened Whitney's umbrella?. Z. angew. Math. Mech., 90: 462–488. doi: 10.1002/zamm.200900315
- Issue published online: 12 MAY 2010
- Article first published online: 12 MAY 2010
- Manuscript Accepted: 15 FEB 2010
- Manuscript Revised: 17 DEC 2009
- Manuscript Received: 18 JUN 2009
- DFG. Grant Number: HA 1060/43-1
- Dissipation-induced instabilities;
- destabilization paradox;
- Ziegler's pendulum;
- Whitney's umbrella.
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.