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Christiane Rousseau; Colin Christopher
Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle
(Déploiement de difféomorphismes résonants)
Annales de l'institut Fourier, 57 no. 1 (2007), p. 301-360, doi: 10.5802/aif.2260
Article PDF | Reviews MR 2316241 | Zbl 1127.37039 | 1 citation in Cedram
Class. Math.: 34M35, 37F75, 32S65
Keywords: Unfolding of a resonant diffeomorphism, modulus of analytic classification, unfolding of a resonant saddle, unfolding of Écalle modulus, unfolding of Martinet-Ramis modulus, unfolding of holonomy map, parametric resurgence phenomenon, transcritical bifurcation.

Résumé - Abstract

We consider germs of one-parameter generic families of resonant analytic diffeomorphims and we give a complete modulus of analytic classification by means of the unfolding of the Écalle modulus. We describe the parametric resurgence phenomenon. We apply this to give a complete modulus of orbital analytic classification for the unfolding of a generic resonant saddle of a 2-dimensional vector field by means of the unfolding of its holonomy map. Here again the modulus is an unfolding of the Martinet-Ramis modulus of the resonant saddle. When the saddle passes through the resonance we observe a “transcritical bifurcation”: the dynamics in the neighborhood of the saddle is governed by different parts of the unfolding of the modulus on each side of the bifurcation. We then include the time dependence and give a complete modulus of analytic conjugacy for the unfolding of a generic resonant saddle.

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