Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella?
Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella?
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DOI：
10.1002/zamm.200900315
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发表时间：
2009-06
期刊：
影响因子：
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通讯作者：
Oleg N. Kirillov;F. Verhulst
中科院分区：
文献类型：
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作者：
Oleg N. Kirillov;F. Verhulst
保守或非保守系统因小耗散而失稳的悖论，或齐格勒悖论（1952），激发了人们对可逆系统和哈密顿系统相对于耗散扰动的敏感性的兴趣。近十年来，人们普遍认为耗散引起的不稳定性与稳定边界上的奇点密切相关。鲜为人知的是，通过惠特尼伞奇点对齐格勒悖论的第一个完整解释可以追溯到1956年。我们将重新审视奥内•博特玛（Oene Bottema）这一不应该被遗忘的开创性成果，它在大约半个世纪的时间里超越了后来的发现。我们讨论了耗散诱导不稳定性的微扰分析的后续发展及其在此期间的应用，包括矩阵的结构稳定性、Krein碰撞、Hamilton - Hopf分岔以及相关的分岔。
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.