Dissipation-Induced Instability Phenomena in Infinite-Dimensional Systems

Krechetnikov, Rouslan; Marsden, Jerrold E.

doi:10.1007/s00205-008-0193-6

Cited by 18 publications

(18 citation statements)

References 79 publications

(118 reference statements)

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“…The larger magnitudes of circulatory forces, the lower |Ω| at the onset of instability. This is a typical example of dissipation-induced instability in the sense of [11,[62][63][64] when only non-Hamiltonian perturbations can cause the destabilizing movements of eigenvalues with definite Krein signature [58]. As κ > 0 decreases, the hypersurfaces forming the dihedral angle approach each other so that, at κ = 0, they temporarily merge along the line ν = 0 and a new configuration originates for κ < 0, Fig.…”

Section: Gyroscopic Systems Of Rotor Dynamicsmentioning

confidence: 94%

“…It is remarkable that the unexpected destabilizing effect due to the introduction of dissipation was discovered in the linear stability analyses of this hydrodynamical problem by Holopainen (1961) [37] and Romea (1977) [87] at the very same period of active research on the destabilization paradox in structural mechanics. Recently these studies were revisited by Krechetnikov and Marsden [64] with the aim to handle rigorously the treatment of dissipation-induced instability.…”

Section: Near-hamiltonian Case: the Instability Of Baroclinic Zonal Cmentioning

confidence: 99%

“…The linearized equations for each layer near the basic state, characterized by the geostrophic streamfunctions −U 1 y and −U 2 y, are according to [64,87]:…”

Section: Near-hamiltonian Case: the Instability Of Baroclinic Zonal Cmentioning

confidence: 99%

“…In the inviscid case when the Ekman layer dissipation is set to zero, the transition to instability occurs through the Krein collision that occurs at U c := U 1 − U 2 = U cI , where [64,87] …”

Section: Near-hamiltonian Case: the Instability Of Baroclinic Zonal Cmentioning

confidence: 99%

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

Kirillov¹,

Verhulst

2010

Z Angew Math Mech

100

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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.

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Section: Gyroscopic Systems Of Rotor Dynamicsmentioning

confidence: 94%

Section: Near-hamiltonian Case: the Instability Of Baroclinic Zonal Cmentioning

confidence: 99%

“…The linearized equations for each layer near the basic state, characterized by the geostrophic streamfunctions −U 1 y and −U 2 y, are according to [64,87]:…”

Section: Near-hamiltonian Case: the Instability Of Baroclinic Zonal Cmentioning

confidence: 99%

Section: Near-hamiltonian Case: the Instability Of Baroclinic Zonal Cmentioning

confidence: 99%

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

Kirillov¹,

Verhulst

2010

Z Angew Math Mech

100

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“…Therefore, infinitesimal imperfections in the loss/gain balance and in the potential, destroying the PT -symmetry, can significantly decrease the interval of asymptotic stability with respect to the marginal stability interval. Such a paradoxical finite jump in the instability threshold caused by a tiny variation in the damping distribution, typically occurs in dissipatively perturbed autonomous Hamiltonian or reversible systems [29,35] of structural and contact mechanics [30,32,34] and hydrodynamics [36][37][38], as well as in periodic nonautonomous ones [39]. We have just described a similar effect when the marginally stable system is dissipative but obeys PTsymmetry.…”

Section: Potential System With Indefinite Dampingmentioning

confidence: 64%

PT-symmetry, indefinite damping and dissipation-induced instabilities

Kirillov

2012

Physics Letters A

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Citation: Kirillov, Oleg (2012) PT-symmetry, indefinite damping and dissipation-induced instabilities. Physics Letters A, 376 (15 Northumbria University has developed Northumbria Research Link (NRL) to enable users to access the University's research output. Copyright © and moral rights for items on NRL are retained by the individual author(s) and/or other copyright owners. Single copies of full items can be reproduced, displayed or performed, and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided the authors, title and full bibliographic details are given, as well as a hyperlink and/or URL to the original metadata page. The content must not be changed in any way. Full items must not be sold commercially in any format or medium without formal permission of the copyright holder. The full policy is available online: http://nrl.northumbria.ac.uk/policies.html This document may differ from the final, published version of the research and has been made available online in accordance with publisher policies. To read and/or cite from the published version of the research, please visit the publisher's website (a subscription may be required.)This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues.Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. With perfectly balanced gain and loss, dynamical systems with indefinite damping can obey the exact PT -symmetry being marginally stable with a pure imaginary spectrum. At an exceptional point where the symmetry is spontaneously broken, the stability is lost via passing through a non-semi-simple 1 : 1 resonance. In the parameter space of a general dissipative system, marginally stable PT -symmetric ones occupy singularities on the boundary of the asymptotic stability. To observe how the singular surface governs dissipation-induced destabilization of the PT -symmetric system when gain and loss are not matched, an extension of recent experiments with PT -symmetric LRC circuits is proposed.

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Authors' reply to “A remark to the paper by O. N. Kirillov and F. Verhulst “Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella?” [Zamm 90, No. 6, 462–488 (2010)]

Kirillov¹,

Verhulst²

2012

Z Angew Math Mech

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We are glad to see from the remark that the paradoxical influence of small dissipation emphasized in the seminal work by Hans Ziegler of 1952 and resolved in 1956 by Oene Bottema continues to attract an attention of researchers, confirming that our efforts in writing the survey were not useless.The remark states that the work by Bottema is "a study of the singularity in a simple specific case" while the authors of the remark "provided a final step in the qualitative understanding of Ziegler's paradox as a singularity". Note, however, that the "final step" was already taken by Bottema in 1956; since then it was re-iterated independently by a number of researchers.Indeed, we quote from Bottema (1956) where he studies the stability of a general two-degrees-of-freedom non-conservative system with damping and compares its domain of asymptotic stability (A) with the domain of marginal stability in the absence of damping (B) in the space of the coefficients of the characteristic polynomial (a1, a2, a3); he directly points out that a1 and a3 depend on damping:"One could expect B to be a limit of A, so that for a1 → 0, a3 → 0 the set A would continuously tend to B. That is not the case. . . . Here is the discontinuity we mentioned above. It plays a part in questions regarding the stability of equilibrium. The coefficients a1 and a3 depend on the linear damping forces and it is well known that the stability condition may change in a discontinuous way if a very small damping vanishes at all [1]. The phenomenon may be illustrated by a geometrical diagram.[1] Ziegler, Die Stabilitätskriterien der Elastomechanik, Ing. Arch. 20, 49-56 (1952); Bottema, On the stability of the equilibrium of a linear mechanical system, Journal of Appl. Math. Phys. (ZAMP) 6, 97-104 (1955)."The geometrical diagram plotted by Bottema in 1956 is exactly the Whitney umbrella surface, one half of which bounds the domain of asymptotic stability. Bottema, an established geometer, correctly classifies it as a ruled surface and describes in detail its properties in terms of the generators. Therefore, it was Bottema who in 1956 resolved the Ziegler's paradox by discovering the Whitney umbrella singularity on the stability boundary of two-degrees-of freedom non-conservative systems that include Ziegler's pendulum as a particular case.The results of Arnold on singularity theory of 1970s were quickly known in the Western dynamical systems community. For example, already in 1995, in the work [I. Hoveijn and M. Ruijgrok, The stability of parametrically forced coupled oscillators in sum resonance, Z. Angew. Math. Phys. 46, 384-392 (1995)] the Arnold's method of versal deformation of matrix families was used in order to describe the destabilizing effect of damping in a general four dimensional dynamical system in sum resonance with application to rotating shafts. In the space of parameters of the versal deformation, Hoveijn and Ruijgrok found a generic Whitney umbrella singularity from the list of Arnold and plotted the corresponding singular stability threshold....

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Dissipation-Induced Instability Phenomena in Infinite-Dimensional Systems

Cited by 18 publications

References 79 publications

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

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

PT-symmetry, indefinite damping and dissipation-induced instabilities

Authors' reply to “A remark to the paper by O. N. Kirillov and F. Verhulst “Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella?” [Zamm 90, No. 6, 462–488 (2010)]

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