2017/09/28 by Amaury Bittmann, Bittmann, Amaury
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1709.10333
openalex publication_date 2017/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we consider germs of analytic singular vector elds in (C3,0) 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 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 (C2,0). 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 à la Martinet-Ramis / Stolovitch for 1-resonant vector fields.