Instability of equilibria in dimension three

Marco Brunella

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 5, page 1345-1357
  • ISSN: 0373-0956

Abstract

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In this paper we show that if v is an analytic vector field on ℝ 3 having an isolated singular point at 0, then there exists a trajectory of v which converges to 0 in the past or in the future. The proof is based on certain results concerning desingularizaton of vector fields in dimension three and on index-type arguments à la Poincaré-Hopf.

How to cite

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Brunella, Marco. "Instability of equilibria in dimension three." Annales de l'institut Fourier 48.5 (1998): 1345-1357. <http://eudml.org/doc/75321>.

@article{Brunella1998,
abstract = {In this paper we show that if $v$ is an analytic vector field on $\{\Bbb R\}^3$ having an isolated singular point at 0, then there exists a trajectory of $v$ which converges to 0 in the past or in the future. The proof is based on certain results concerning desingularizaton of vector fields in dimension three and on index-type arguments à la Poincaré-Hopf.},
author = {Brunella, Marco},
journal = {Annales de l'institut Fourier},
keywords = {singularities of vector field; invariant manifolds; blow-ups; stability problems; analytic vector fields; isolated singular point; irreducible analytic subvariety; Euler characteristic; unstable point},
language = {eng},
number = {5},
pages = {1345-1357},
publisher = {Association des Annales de l'Institut Fourier},
title = {Instability of equilibria in dimension three},
url = {http://eudml.org/doc/75321},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Brunella, Marco
TI - Instability of equilibria in dimension three
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 5
SP - 1345
EP - 1357
AB - In this paper we show that if $v$ is an analytic vector field on ${\Bbb R}^3$ having an isolated singular point at 0, then there exists a trajectory of $v$ which converges to 0 in the past or in the future. The proof is based on certain results concerning desingularizaton of vector fields in dimension three and on index-type arguments à la Poincaré-Hopf.
LA - eng
KW - singularities of vector field; invariant manifolds; blow-ups; stability problems; analytic vector fields; isolated singular point; irreducible analytic subvariety; Euler characteristic; unstable point
UR - http://eudml.org/doc/75321
ER -

References

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  1. [AI] V.I. ARNOL'D, JU. IL'YASHENKO, Ordinary differential equations, Enc. Math. Sci. vol. 1, Springer Verlag (1988). Zbl0718.34070
  2. [BD] P. BONCKAERT, F. DUMORTIER, Smooth invariant curves for germs of vector fields in R3 whose linear part generates a rotation, Jour. Diff. Eq., 102 (1980), 95-116. Zbl0584.58009
  3. [Ca1] F. CANO, Desingularization strategies for three-dimensional vector fields, Springer Lecture Notes 1259 (1987). Zbl0645.14005MR90i:32020a
  4. [Ca2] F. CANO, Final forms for a three dimensional vector field under blowing-up, Ann. Inst. Fourier, 37-2 (1987), 151-193. Zbl0607.58027MR88j:58105
  5. [Pa] V.P. PALAMODOV, Stability of equilibria in a potential field, Funct. Anal. Appl., 11 (1977), 42-55. Zbl0415.70006
  6. [Sa] F. SANCHO DE SALAS, Desingularization, residuos y solucion de las ecuaciones diferenciales, Thesis, Univ. de Salamanca (1996). 

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