Le Canard de Painlevé

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Abstract

We consider the problem of a slender rod slipping along a rough surface. Painleve [C. R. Seances Acad. Sci., 121 (1895), pp. 112-115; C. R. Seances Acad. Sci., 141 (1905), pp. 401-405; C. R. Seances Acad. Sci., 141 (1905), pp. 546-552] showed that the governing rigid body equations for this problem can exhibit multiple solutions (the indeterminate case) or no solutions at all (the inconsistent case), provided the coefficient of friction mu exceeds a certain critical value mu(p). Subsequently Genot and Brogliato [Eur. J. Mech. A Solids, 18 (1999), pp. 653-677] proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Instead the rod will either stop slipping and stick or it will lift off from the surface. Between these two cases is a special solution for mu > mu(c) > mu(p), where mu(c) is a new critical value of the coefficient of friction. Physically, the special solution corresponds to the rod slipping until it reaches a singular "0/0" point P. Even though the rigid body equations cannot describe what happens to the rod beyond the singular point P, it is possible to extend the special solution into the region of indeterminacy. This extended solution is very reminiscent of a canard [E. Benoit et al., Collect. Math., 31-32 (1981), pp. 37-119]. To overcome the inadequacy of the rigid body equations beyond P, the rigid body assumption is relaxed in the neighborhood of the point of contact of the rod with the rough surface. Physically this corresponds to assuming a small compliance there. It is natural to ask what happens to both the point P and the special solution under this regularization, in the limit of vanishing compliance. In this paper, we prove the existence of a canard orbit in a reduced four-dimensional slow-fast phase space, connecting a two-dimensional focus-type slow manifold with the stable manifold of a two-dimensional saddle -type slow manifold. The proof combines several methods from local dynamical system theory, including blowup. The analysis is not standard, since we only gain ellipticity rather than hyperbolicity with our initial blowup.
Original languageEnglish
JournalS I A M Journal on Applied Dynamical Systems
Volume17
Issue number1
Pages (from-to)859-908
ISSN1536-0040
DOIs
Publication statusPublished - 2018

Keywords

  • Painlevé paradox
  • Impact without collision
  • Compliance
  • Regularization

Cite this

Kristiansen, K. Uldall ; Hogan, S. J. / Le Canard de Painlevé. In: S I A M Journal on Applied Dynamical Systems. 2018 ; Vol. 17, No. 1. pp. 859-908.
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keywords = "Painlev{\'e} paradox, Impact without collision, Compliance, Regularization",
author = "Kristiansen, {K. Uldall} and Hogan, {S. J.}",
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Le Canard de Painlevé. / Kristiansen, K. Uldall; Hogan, S. J.

In: S I A M Journal on Applied Dynamical Systems, Vol. 17, No. 1, 2018, p. 859-908.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Le Canard de Painlevé

AU - Kristiansen, K. Uldall

AU - Hogan, S. J.

PY - 2018

Y1 - 2018

N2 - We consider the problem of a slender rod slipping along a rough surface. Painleve [C. R. Seances Acad. Sci., 121 (1895), pp. 112-115; C. R. Seances Acad. Sci., 141 (1905), pp. 401-405; C. R. Seances Acad. Sci., 141 (1905), pp. 546-552] showed that the governing rigid body equations for this problem can exhibit multiple solutions (the indeterminate case) or no solutions at all (the inconsistent case), provided the coefficient of friction mu exceeds a certain critical value mu(p). Subsequently Genot and Brogliato [Eur. J. Mech. A Solids, 18 (1999), pp. 653-677] proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Instead the rod will either stop slipping and stick or it will lift off from the surface. Between these two cases is a special solution for mu > mu(c) > mu(p), where mu(c) is a new critical value of the coefficient of friction. Physically, the special solution corresponds to the rod slipping until it reaches a singular "0/0" point P. Even though the rigid body equations cannot describe what happens to the rod beyond the singular point P, it is possible to extend the special solution into the region of indeterminacy. This extended solution is very reminiscent of a canard [E. Benoit et al., Collect. Math., 31-32 (1981), pp. 37-119]. To overcome the inadequacy of the rigid body equations beyond P, the rigid body assumption is relaxed in the neighborhood of the point of contact of the rod with the rough surface. Physically this corresponds to assuming a small compliance there. It is natural to ask what happens to both the point P and the special solution under this regularization, in the limit of vanishing compliance. In this paper, we prove the existence of a canard orbit in a reduced four-dimensional slow-fast phase space, connecting a two-dimensional focus-type slow manifold with the stable manifold of a two-dimensional saddle -type slow manifold. The proof combines several methods from local dynamical system theory, including blowup. The analysis is not standard, since we only gain ellipticity rather than hyperbolicity with our initial blowup.

AB - We consider the problem of a slender rod slipping along a rough surface. Painleve [C. R. Seances Acad. Sci., 121 (1895), pp. 112-115; C. R. Seances Acad. Sci., 141 (1905), pp. 401-405; C. R. Seances Acad. Sci., 141 (1905), pp. 546-552] showed that the governing rigid body equations for this problem can exhibit multiple solutions (the indeterminate case) or no solutions at all (the inconsistent case), provided the coefficient of friction mu exceeds a certain critical value mu(p). Subsequently Genot and Brogliato [Eur. J. Mech. A Solids, 18 (1999), pp. 653-677] proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Instead the rod will either stop slipping and stick or it will lift off from the surface. Between these two cases is a special solution for mu > mu(c) > mu(p), where mu(c) is a new critical value of the coefficient of friction. Physically, the special solution corresponds to the rod slipping until it reaches a singular "0/0" point P. Even though the rigid body equations cannot describe what happens to the rod beyond the singular point P, it is possible to extend the special solution into the region of indeterminacy. This extended solution is very reminiscent of a canard [E. Benoit et al., Collect. Math., 31-32 (1981), pp. 37-119]. To overcome the inadequacy of the rigid body equations beyond P, the rigid body assumption is relaxed in the neighborhood of the point of contact of the rod with the rough surface. Physically this corresponds to assuming a small compliance there. It is natural to ask what happens to both the point P and the special solution under this regularization, in the limit of vanishing compliance. In this paper, we prove the existence of a canard orbit in a reduced four-dimensional slow-fast phase space, connecting a two-dimensional focus-type slow manifold with the stable manifold of a two-dimensional saddle -type slow manifold. The proof combines several methods from local dynamical system theory, including blowup. The analysis is not standard, since we only gain ellipticity rather than hyperbolicity with our initial blowup.

KW - Painlevé paradox

KW - Impact without collision

KW - Compliance

KW - Regularization

U2 - 10.1137/17M1122256

DO - 10.1137/17M1122256

M3 - Journal article

VL - 17

SP - 859

EP - 908

JO - S I A M Journal on Applied Dynamical Systems

JF - S I A M Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 1

ER -