Non oscillating solutions of analytic gradient vector fields

Fernando Sanz

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 4, page 1045-1067
  • ISSN: 0373-0956

Abstract

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Let γ be an integral solution of an analytic real vector field ξ defined in a neighbordhood of 0 ∈ ℝ 3 . Suppose that γ has a single limit point, ω ( γ ) = { 0 } . We say that γ is non oscillating if, for any analytic surface H , either γ is contained in H or γ cuts H only finitely many times. In this paper we give a sufficient condition for γ to be non oscillating. It is established in terms of the existence of “generalized iterated tangents”, i.e. the existence of a single limit point for any transform property for the solutions of a gradient vector field ξ = ∇ g f of an analytic function f of order 2 at 0 ∈ ℝ 3 , where g is an analytic riemannian metric.

How to cite

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Sanz, Fernando. "Non oscillating solutions of analytic gradient vector fields." Annales de l'institut Fourier 48.4 (1998): 1045-1067. <http://eudml.org/doc/75308>.

@article{Sanz1998,
abstract = {Let $\gamma $ be an integral solution of an analytic real vector field $\xi $ defined in a neighbordhood of $0\in \{\Bbb R\}^3$. Suppose that $\gamma $ has a single limit point, $\omega (\gamma )=\lbrace 0\rbrace $. We say that $\gamma $ is non oscillating if, for any analytic surface $H$, either $\gamma $ is contained in $H$ or $\gamma $ cuts $H$ only finitely many times. In this paper we give a sufficient condition for $\gamma $ to be non oscillating. It is established in terms of the existence of “generalized iterated tangents”, i.e. the existence of a single limit point for any transform property for the solutions of a gradient vector field $\xi =\nabla _g f$ of an analytic function $f$ of order 2 at $0\in \{\Bbb R\}^3$, where $g$ is an analytic riemannian metric.},
author = {Sanz, Fernando},
journal = {Annales de l'institut Fourier},
keywords = {vector field; gradient; tangent; oscillation; blowing-up; desingularization; center manifold},
language = {eng},
number = {4},
pages = {1045-1067},
publisher = {Association des Annales de l'Institut Fourier},
title = {Non oscillating solutions of analytic gradient vector fields},
url = {http://eudml.org/doc/75308},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Sanz, Fernando
TI - Non oscillating solutions of analytic gradient vector fields
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 4
SP - 1045
EP - 1067
AB - Let $\gamma $ be an integral solution of an analytic real vector field $\xi $ defined in a neighbordhood of $0\in {\Bbb R}^3$. Suppose that $\gamma $ has a single limit point, $\omega (\gamma )=\lbrace 0\rbrace $. We say that $\gamma $ is non oscillating if, for any analytic surface $H$, either $\gamma $ is contained in $H$ or $\gamma $ cuts $H$ only finitely many times. In this paper we give a sufficient condition for $\gamma $ to be non oscillating. It is established in terms of the existence of “generalized iterated tangents”, i.e. the existence of a single limit point for any transform property for the solutions of a gradient vector field $\xi =\nabla _g f$ of an analytic function $f$ of order 2 at $0\in {\Bbb R}^3$, where $g$ is an analytic riemannian metric.
LA - eng
KW - vector field; gradient; tangent; oscillation; blowing-up; desingularization; center manifold
UR - http://eudml.org/doc/75308
ER -

References

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  14. [14] F. SANZ, Trajectoires non-oscillantes des champs de vecteurs gradients. C. R. Acad. Sci. Paris, t. 325, Série I (1997), 429-432. Zbl0889.58049MR98h:58153
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