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

TL;DR

This paper classifies certain doubly-resonant saddle-node singularities in complex three-dimensional vector fields, especially those arising in Painlevé equations, by establishing analytic normalization and describing their Stokes phenomena.

Contribution

It extends the analytic normalization theory to doubly-resonant saddle-nodes in (C^3,0) and classifies them analytically using Stokes diffeomorphisms, relating to Painlevé equations.

Findings

Proves sectorial normalization similar to classical saddle-node results.

Shows the normalizing maps are Gevrey-1 sums of formal maps.

Provides an analytic classification via Stokes diffeomorphisms.

Abstract

In this work, we consider germs of analytic singular vector elds in (C^3,0) with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector elds come from irregular two-dimensional dierential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at innity in Painlev{\'e} 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^2,0). We also prove that these 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 classication under the action of bered dieomorphisms, based on the study of the so-called Stokes dieomorphisms obtained by comparing consecutive sectorial normalizing maps {\`a} la Martinet-Ramis / Stolovitch for 1-resonant vector fields.