Doubly-resonant saddle-nodes in ( 3 ,0) and the fixed singularity at infinity in Painlevé equations: analytic classification
Annales de l'Institut Fourier, Volume 68 (2018) no. 4, pp. 1715-1830.

In this work, we consider germs of analytic singular vector fields in 3 with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlevé equations (P j ) j=I,,V for generic values of the parameters. Under suitable assumptions, we prove a theorem of analytic normalization over sectorial domains, analogous to the classical one due to Hukuhara–Kimura–Matuda for saddle-nodes in 2 . We also prove that these sectorial normalizing maps are in fact the Gevrey-1 sums of the formal normalizing map, the existence of which has been proved in a previous paper. Finally we provide an analytic classification under the action of fibered diffeomorphisms, based on the study of the so-called Stokes diffeomorphisms obtained by comparing consecutive sectorial normalizing maps à la Martinet–Ramis / Stolovitch for 1-resonant vector fields.

Dans ce travail, nous considérons des germes de champs de vecteurs singuliers dans ( 3 ,0) ayant une singularité isolée doublement résonante de type noeud-col à l’origine. Ces champs de vecteurs proviennent de systèmes différentiels irréguliers en dimension deux, avec deux valeurs propres opposées non-nulles, et apparaissent par exemple dans l’étude des singularités irrégulières à l’infini des équations de Painlevé (P j ) j=I,,V pour des valeurs génériques des paramètres. Sous des conditions adéquates, nous démontrons un théorème de normalisation analytique sur des domaines sectoriels, analogue à un résultat de Hukuhara, Kimura et Matuda pour les noeud-cols dans 2 . Nous prouvons également que ces normalisations sectorielles sont en fait les sommes 1-Gevrey de la normalisation formelle, dont l’existence a été prouvée dans un précédent papier. Nous terminons en fournissant une classification analytique sous l’action de difféomorphismes fibrés, basée sur l’étude des difféomorphismes de Stokes obtenus en comparant les normalisations sectorielles consécutives à la Martinet–Ramis / Stolovitch pour des champs de vecteurs 1-résonants.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3197
Classification: 34M30,  34M35,  34M40,  34M55
Keywords: Painlevé equations, singular vector field, irregular singularity, resonant singularity, analytic classification, Stokes diffeomorphisms.
Bittmann, Amaury 1

1 Université de Strasbourg IRMA 7, rue René Descartes 67084 Strasbourg Cedex (France)
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Bittmann, Amaury. Doubly-resonant saddle-nodes in $(\protect \mathbb{C}^3,0)$ and the fixed singularity at infinity in Painlevé equations: analytic classification. Annales de l'Institut Fourier, Volume 68 (2018) no. 4, pp. 1715-1830. doi : 10.5802/aif.3197. https://aif.centre-mersenne.org/articles/10.5802/aif.3197/

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