Ordinary Differential Equations
On the normal form of a system of differential equations with nilpotent linear part
Comptes Rendus. Mathématique, Volume 336 (2003) no. 2, pp. 129-134.

We consider prenormal forms associated to generic perturbations of the system $\stackrel{˙}{x}=2\mathrm{y},\phantom{\rule{3.30002pt}{0ex}}\stackrel{˙}{y}=3{\mathrm{x}}^{2}$. It is known that they have a formal normal form $\stackrel{˙}{x}=2\mathrm{y}+2\mathrm{x}\phantom{\rule{1.69998pt}{0ex}}{\Delta }^{*},\phantom{\rule{3.30002pt}{0ex}}\stackrel{˙}{y}=3{\mathrm{x}}^{2}+3\mathrm{y}\phantom{\rule{1.69998pt}{0ex}}{\Delta }^{*}$, where ${\Delta }^{*}=\mathrm{x}+{\mathrm{A}}_{0}\left({\mathrm{y}}^{2}-{\mathrm{x}}^{3}\right)$ [Differential Equations 158 (1) (1999) 152–173]. We show that the series A0 and the normalizing transformations are divergent, but 1-summable.

On considère des formes prénormales associées à des perturbations génériques du système $\stackrel{˙}{x}=2\mathrm{y},\phantom{\rule{3.30002pt}{0ex}}\phantom{\rule{3.30002pt}{0ex}}\stackrel{˙}{y}=3{\mathrm{x}}^{2}$. Il est connu qu'elles admettent une forme normale formelle $\stackrel{˙}{x}=2\mathrm{y}+2\mathrm{x}\phantom{\rule{1.69998pt}{0ex}}{\Delta }^{*},\phantom{\rule{3.30002pt}{0ex}}\phantom{\rule{3.30002pt}{0ex}}\stackrel{˙}{y}=3{\mathrm{x}}^{2}+3\mathrm{y}\phantom{\rule{1.69998pt}{0ex}}{\Delta }^{*}$, où ${\Delta }^{*}=\mathrm{x}+{\mathrm{A}}_{0}\left({\mathrm{y}}^{2}-{\mathrm{x}}^{3}\right)$ [Differential Equations 158 (1) (1999) 152–173]. Nous démontrons que A0 et les transformations normalisantes sont divergentes, mais 1-sommable.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(02)00022-5

Mireille Canalis-Durand 1; Reinhard Schäfke 2

1 GREQAM, Université d'Aix-Marseille III, 13002 Marseille, France
2 IRMA, Université Louis Pasteur, 67084 Strasbourg cedex, France
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Mireille Canalis-Durand; Reinhard Schäfke. On the normal form of a system of differential equations with nilpotent linear part. Comptes Rendus. Mathématique, Volume 336 (2003) no. 2, pp. 129-134. doi : 10.1016/S1631-073X(02)00022-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)00022-5/

[1] Handbook of Mathematical Functions (M. Abramowitz; I.A. Stegun, eds.), Dover, New York, 1964

[2] M. Canalis-Durand; F. Michel; M. Teisseyre Algorithms for formal reduction of vector fields singularities, J. Dynamical Control Systems, Volume 7 (2001) no. 1, pp. 101-125

[3] D. Cerveau; R. Moussu Groupes d'automorphismes de $\left(ℂ,0\right)$ et équations différentielles $y\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}\mathrm{y}+\cdots =0$, Publ. Soc. Math. France, Volume 116 (1988), pp. 459-488

[4] J. Ecalle Les fonctions résurgentes. III : L'équation du pont et la classification analytique des objets locaux, Publ. Math. Orsay 85-05, 1985

[5] F. Loray Réduction formelle des singularités cuspidales de champs de vecteurs analytiques, Differential Equations, Volume 158 (1999) no. 1, pp. 152-173

[6] F.W.J. Olver Asymptotics and Special Functions, Academic Press, New York, 1974

[7] J.-P. Ramis Les séries k-sommables et leurs applications, Complex Analysis, Microlocal Calcul and Relativistic Quantum Theory, Lecture Notes in Phys., 126, 1980, pp. 178-199

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