Doubly-resonant saddle-nodes in (ℂ³,0) and the fixed singularity at infinity in the Painlevé equations (part II): sectorial normalization

2016/05/17 by Amaury Bittmann, Bittmann, Amaury
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Meromorphic and Entire Functions #Nonlinear Waves and Solitons #math.DS

paper · pdf · doi:10.48550/arxiv.1605.05052

openalex publication_date 2016/05/17 · openalex created_date 2016/06/24 · arxiv created 2016/11/18 · arxiv updated 2016/11/21 · openalex updated_date 2026/07/28

Abstract

In this work, following [Bit15], we consider analytic singular vector fields in (ℂ³,0) 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 (ℂ²,0). We also prove that the normalizing map is essentially unique and weakly Gevrey-1 summable.

Related

• Doubly-resonant saddle-nodes in (ℂ³,0) and the fixed singularity at infinity in the Painlevé equations (part III): local analytic classification
• Doubly-resonant saddle-nodes in (C³,0) and the fixed singularity at infinity in Painlevé equations: analytic classification
• Doubly-resonant saddle-nodes in C³ and the fixed singularity at infinity in the Painlevé equations: formal classification
• Analytic normal forms for planar resonant saddle vector fields
• Stokes phenomenon and confluence in non-autonomous Hamiltonian systems