Instability of equilibria in dimension three
Annales de l'Institut Fourier, Tome 48 (1998) no. 5, p. 1345-1357
Dans cet article on démontre que si v est un champ de vecteurs analytique sur 3 ayant une singularité isolée en 0, alors il existe une trajectoire de v qui tend vers 0 dans le passé ou dans le futur. La preuve repose sur certains résultats concernant la désingularisation des champs de vecteurs en dimension trois et sur des arguments d’indice à la Poincaré-Hopf.
In this paper we show that if v is an analytic vector field on 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.
@article{AIF_1998__48_5_1345_0,
     author = {Brunella, Marco},
     title = {Instability of equilibria in dimension three},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {48},
     number = {5},
     year = {1998},
     pages = {1345-1357},
     doi = {10.5802/aif.1657},
     zbl = {0930.37025},
     mrnumber = {99i:34040},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1998__48_5_1345_0}
}
Brunella, Marco. Instability of equilibria in dimension three. Annales de l'Institut Fourier, Tome 48 (1998) no. 5, pp. 1345-1357. doi : 10.5802/aif.1657. http://www.numdam.org/item/AIF_1998__48_5_1345_0/

[AI] V.I. Arnol'D, Ju. Il'Yashenko, Ordinary differential equations, Enc. Math. Sci. vol. 1, Springer Verlag (1988). | Zbl 0718.34070

[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. | Zbl 0584.58009

[Ca1] F. Cano, Desingularization strategies for three-dimensional vector fields, Springer Lecture Notes 1259 (1987). | MR 90i:32020a | Zbl 0645.14005

[Ca2] F. Cano, Final forms for a three dimensional vector field under blowing-up, Ann. Inst. Fourier, 37-2 (1987), 151-193. | Numdam | MR 88j:58105 | Zbl 0607.58027

[Pa] V.P. Palamodov, Stability of equilibria in a potential field, Funct. Anal. Appl., 11 (1977), 42-55. | Zbl 0415.70006

[Sa] F. Sancho De Salas, Desingularization, residuos y solucion de las ecuaciones diferenciales, Thesis, Univ. de Salamanca (1996).