Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle
Annales de l'Institut Fourier, Volume 57 (2007) no. 1, p. 301-360
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.
On considère des germes de familles génériques à un paramètre déployant un germe de difféomorphisme résonant et on montre que le déploiement du module d’Ecalle donne un module complet de classification analytique. On décrit le phénomène de résurgence paramétrique. On applique les résultats précédents à la construction d’un module complet de classification analytique orbitale pour le déploiement d’un point de selle résonant générique au moyen du déploiement de son difféomorphisme d’holonomie. Ce module est le déploiement du module de Martinet-Ramis pour un point de selle résonant. Quand le point de selle passe par la résonance on observe une “bifurcation transcritique” : la dynamique du point de selle est contrôlée par des parties différentes du déploiement du module de chaque côté de la bifurcation. On regarde aussi la dépendance du temps et on donne un module complet de conjugaison analytique pour le déploiement d’un point de selle résonant générique.
DOI : https://doi.org/10.5802/aif.2260
Classification:  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.
@article{AIF_2007__57_1_301_0,
     author = {Rousseau, Christiane and Christopher, Colin},
     title = {Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {1},
     year = {2007},
     pages = {301-360},
     doi = {10.5802/aif.2260},
     zbl = {1127.37039},
     mrnumber = {2316241},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_1_301_0}
}
Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle. Annales de l'Institut Fourier, Volume 57 (2007) no. 1, pp. 301-360. doi : 10.5802/aif.2260. https://aif.centre-mersenne.org/item/AIF_2007__57_1_301_0/

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