2015/05/23 by Amaury Bittmann, Bittmann, Amaury
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · doi:10.48550/arxiv.1505.06300
openalex publication_date 2015/05/23 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
In this work we consider formal singular vector fields in C3with an isolated and doubly-resonant singularity of saddle-node typeat the origin. Such vector fields come from irregular two-dimensionalsystems with two opposite non-zero eigenvalues, and appear for instancewhen studying the irregular singularity at infinity in Painlevé equations(P_j)_j∈(I,II,III,IV,V), for generic values of the parameters.Under generic assumptions we give a complete formal classificationfor the action of formal diffeomorphisms (by changes of coordinates)fixing the origin and fibered in the independent variable. Wealso identify all formal isotropies (self-conjugacies) of the normalforms. In the particular case where the flow preserves a transversesymplectic structure, e.g. for Painlevé equations, we provethat the normalizing map can be chosen to preserve the transversesymplectic form.