Doubly-Resonant Saddle-Nodes in C3 and the Fixed Singularity at Infinity in the Painlevé Equations: Formal Classification
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Amaury Bittmann [1]
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[1] University of Strasbourg
University of Strasbourg
Arrondissement de Strasbourg-Ville, Francia
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- Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 16, Nº 3, 2017, págs. 491-529
- Idioma: inglés
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Resumen
In this work we consider formal singular vector fields in C3 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 Pj , j ∈ {I, I I, III, I V, 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 x. 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.
