Generalised power series solutions of sub-analytic differential equations

Mickaël Matusinski; Jean-Philippe Rolin

doi:10.1016/j.crma.2005.11.005

Ordinary Differential Equations

Generalised power series solutions of sub-analytic differential equations

Presented by: Étienne Ghys

Mickaël Matusinski ¹; Jean-Philippe Rolin ¹

¹ I.M.B., université de Bourgogne, 9, avenue Savary, B.P. 47870, 21078 Dijon cedex, France

Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 99-102.

Abstract
Résumé

We show that if a solution $y (x)$ of a sub-analytic differential equation admits an asymptotic expansion $\sum_{i = 1}^{\infty} c_{i} x^{μ_{i}}$ , $μ_{i} \in R^{+}$ , then the exponents $μ_{i}$ belong to a finitely generated semi-group of $R^{+}$ . We deduce a similar result for the components of non-oscillating trajectories of real analytic vector fields in dimension n.

Nous montrons que si une solution $y (x)$ d'une équation différentielle sous-analytique admet un développement asymptotique de la forme $\sum_{i = 1}^{\infty} c_{i} x^{μ_{i}}$ , $μ_{i} \in R^{+}$ , alors les exposants $μ_{i}$ appartiennent à un semi-groupe finiment engendré de $R^{+}$ . Nous en déduisons un résultat analogue pour les composantes des trajectoires non oscillantes de champs de vecteurs analytiques réels en dimension n.

Received: 2005-09-26
Accepted: 2005-11-08
Published online: 2005-12-06

DOI: 10.1016/j.crma.2005.11.005

Author's affiliations:

Mickaël Matusinski ¹; Jean-Philippe Rolin ¹

¹ I.M.B., université de Bourgogne, 9, avenue Savary, B.P. 47870, 21078 Dijon cedex, France

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     title = {Generalised power series solutions of sub-analytic differential equations},
     journal = {Comptes Rendus. Math\'ematique},
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     number = {2},
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     language = {en},
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Mickaël Matusinski; Jean-Philippe Rolin. Generalised power series solutions of sub-analytic differential equations. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 99-102. doi : 10.1016/j.crma.2005.11.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.11.005/

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