Doubly-resonant saddle-nodes in (\protect \mathbb{C}^3,0) and the fixed singularity at infinity in Painlevé equations: analytic classification

Abstract

In this work, we consider germs of analytic singular vector fields in ℂ 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 ℂ 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.