Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle

Christiane Rousseau[1]; Colin Christopher[2]

  • [1] Université de Montréal, CP 6128, Succ. Centre-ville H3C 3J7 Montréal Qc (Canada)
  • [2] University of Plymouth School of Mathematics and Statistics Devon PL4 8AA (United Kingdom)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 1, page 301-360
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Rousseau, Christiane, and Christopher, Colin. "Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle." Annales de l’institut Fourier 57.1 (2007): 301-360. <http://eudml.org/doc/10223>.

@article{Rousseau2007,
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.},
affiliation = {Université de Montréal, CP 6128, Succ. Centre-ville H3C 3J7 Montréal Qc (Canada); University of Plymouth School of Mathematics and Statistics Devon PL4 8AA (United Kingdom)},
author = {Rousseau, Christiane, Christopher, Colin},
journal = {Annales de l’institut Fourier},
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; resonant diffeomorphisms; modulus of analytic classifications},
language = {eng},
number = {1},
pages = {301-360},
publisher = {Association des Annales de l’institut Fourier},
title = {Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle},
url = {http://eudml.org/doc/10223},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Rousseau, Christiane
AU - Christopher, Colin
TI - Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 1
SP - 301
EP - 360
AB - 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.
LA - eng
KW - 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; resonant diffeomorphisms; modulus of analytic classifications
UR - http://eudml.org/doc/10223
ER -

References

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