Doubly-resonant saddle-nodes in $C^3$ and the fixed singularity at infinity in the Painlevé equations: formal classification.

Amaury Bittmann (1)

1 IRMA - Institut de Recherche Mathématique Avancée

In this work we consider formal 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 systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlevé equations $(P_{j})_{j\in(I,II,III,IV,V)}$, for generic values of the parameters. Under generic assumptions we give a complete formal classification for the action of formal diffeomorphisms (by changes of coordinates) fixing the origin and fibered in the independent variable. We also identify all formal isotropies (self-conjugacies) of the normal forms. In the particular case where the flow preserves a transverse symplectic structure, e.g. for Painlevé equations, we prove that the normalizing map can be chosen to preserve the transverse symplectic form.

Mots clés: Irregular singularity; Normal form; Resonant singularity; Singular vector field; Painlevé equations

Domaines: Systèmes dynamiques [math.DS]

Amaury Bittmann. Doubly-resonant saddle-nodes in $C^3$ and the fixed singularity at infinity in the Painlevé equations: formal classification.. Qualitative Theory of Dynamical Systems, 2016, ⟨10.1007/s12346-016-0214-z⟩. ⟨hal-01145314v3⟩