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

Kirillov, O. N. and Verhulst, F. (2010):
Paradoxes of dissipation-induced destabilization or who opened Whitney’s umbrella?
90, In: ZAMM - Journal of Applied Mathematics and Mechanics, (6), WILEY-VCH, pp. 462-488, [Article]

Item Type: Article
Erschienen: 2010
Creators: Kirillov, O. N. and Verhulst, F.
Title: Paradoxes of dissipation-induced destabilization or who opened Whitney’s umbrella?
Language: English
Journal or Publication Title: ZAMM - Journal of Applied Mathematics and Mechanics
Volume: 90
Number: 6
Publisher: WILEY-VCH
Divisions: 16 Department of Mechanical Engineering > Dynamics and Vibrations
16 Department of Mechanical Engineering
Date Deposited: 19 Sep 2012 14:19
Identification Number: doi:10.1002/zamm.200900315
Alternative keywords:
Alternative keywordsLanguage
Dissipation-induced instabilities, destabilization paradox, Ziegler’s pendulum, Whitney’s umbrellaEnglish
Alternative Abstract:
Alternative abstract Language
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.English
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