Doubly-resonant saddle-nodes in $(\mathbb{C}^{3},0)$ and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization.

Amaury Bittmann (1)

1 IRMA - Institut de Recherche Mathématique Avancée

In this work, following [Bit15], we consider analytic singular vector fields in $(\mathbb{C}^{3},0)$ 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 nor-malization over sectorial domains, analogous to the classical one due to Hukuhara-Kimura-Matuda [HKM61] for saddle-nodes in $(\mathbb{C}^{2},0)$. We also prove that the normalizing map is essentially unique and weakly Gevrey-1 summable.

Mots clés: resonant singularity; Painlevé equations; singular vector field; irregular singularity; Gevrey-1 summability; normal form

Domaines: Systèmes dynamiques [math.DS]

Amaury Bittmann. Doubly-resonant saddle-nodes in $(\mathbb{C}^{3},0)$ and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization.. 2016. ⟨hal-01314950v3⟩