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

• Published in: Zeitschrift für angewandte Mathematik und Mechanik 2010-06, Vol.90 (6), p.462-488
• Main Authors: Kirillov, O.N., Verhulst, F.
• Format: Article
• Language: English
• Subjects: Calculus of variations and optimal control; destabilization paradox; Dissipation-induced instabilities; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Whitney's umbrella; Ziegler's pendulum
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.200900315

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. 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. The authors revisit this undeservedly forgotten pioneering result by Oene Bottema that outstripped later findings for about half a century. They 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.