The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations

Waldo Arriagada-Silva[1]; Christiane Rousseau[2]

  • [1] Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, AB T2N 1N4, Canada; and Instituto de matemáticas, Universidad Austral de Chile, Casilla 567 - Valdivia, Chile.
  • [2] Département de Mathématiques et de Statistique, Université de Montréal C.P. 6128, Succ. Centre-ville Montréal, Québec H3C 3J7, Canada.

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: 3, page 541-580
  • ISSN: 0240-2963

Abstract

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In this paper we study equivalence classes of generic 1 -parameter germs of real analytic families 𝒬 ε unfolding codimension 1 germs of diffeomorphisms 𝒬 0 : ( ℝ , 0 ) → ( ℝ , 0 ) with a fixed point at the origin and multiplier - 1 , under (weak) analytic conjugacy. These germs are generic unfoldings of the flip bifurcation. Two such germs are analytically conjugate if and only if their second iterates, 𝒫 ε = 𝒬 ε ∘ 2 , are analytically conjugate. We give a complete modulus of analytic classification: this modulus is an unfolding of the Ecalle modulus of the resonant germ 𝒬 0 with special symmetry properties reflecting the real character of the germ 𝒬 ε . As an application, this provides a complete modulus of analytic classification under weak orbital equivalence for a germ of family of planar vector fields unfolding a weak focus of order 1 ( i . e . undergoing a generic Hopf bifurcation of codimension 1 ) through the modulus of analytic classification of the germ of family 𝒫 ε = 𝒬 ε ∘ 2 , where 𝒫 ε is the Poincaré monodromy of the family of vector fields.

How to cite

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Arriagada-Silva, Waldo, and Rousseau, Christiane. "The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations." Annales de la faculté des sciences de Toulouse Mathématiques 20.3 (2011): 541-580. <http://eudml.org/doc/219760>.

@article{Arriagada2011,
abstract = {In this paper we study equivalence classes of generic $1$-parameter germs of real analytic families $\{\mathcal\{Q\}\}_\{\varepsilon \}$ unfolding codimension $1$ germs of diffeomorphisms $\{\mathcal\{Q\}\}_0: (\{\mathbb\{R\}\},0)\rightarrow (\{\mathbb\{R\}\},0)$ with a fixed point at the origin and multiplier $-1,$ under (weak) analytic conjugacy. These germs are generic unfoldings of the flip bifurcation. Two such germs are analytically conjugate if and only if their second iterates, $\{\mathcal\{P\}\}_\{\varepsilon \}=\{\mathcal\{Q\}\}_\{\varepsilon \}^\{\circ 2\},$ are analytically conjugate. We give a complete modulus of analytic classification: this modulus is an unfolding of the Ecalle modulus of the resonant germ $\{\mathcal\{Q\}\}_0$ with special symmetry properties reflecting the real character of the germ $\{\mathcal\{Q\}\}_\{\varepsilon \} .$ As an application, this provides a complete modulus of analytic classification under weak orbital equivalence for a germ of family of planar vector fields unfolding a weak focus of order $1$$(i.e.$ undergoing a generic Hopf bifurcation of codimension $1)$ through the modulus of analytic classification of the germ of family $\{\mathcal\{P\}\}_\{\varepsilon \}=\{\mathcal\{Q\}\}_\{\varepsilon \}^\{\circ 2\},$ where $\{\mathcal\{P\}\}_\{\varepsilon \}$ is the Poincaré monodromy of the family of vector fields.},
affiliation = {Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, AB T2N 1N4, Canada; and Instituto de matemáticas, Universidad Austral de Chile, Casilla 567 - Valdivia, Chile.; Département de Mathématiques et de Statistique, Université de Montréal C.P. 6128, Succ. Centre-ville Montréal, Québec H3C 3J7, Canada.},
author = {Arriagada-Silva, Waldo, Rousseau, Christiane},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {singularity theory of differentiable mappings; unfolding of the codimension-one flip; Hopf bifurcation; analytic classification},
language = {eng},
month = {7},
number = {3},
pages = {541-580},
publisher = {Université Paul Sabatier, Toulouse},
title = {The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations},
url = {http://eudml.org/doc/219760},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Arriagada-Silva, Waldo
AU - Rousseau, Christiane
TI - The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/7//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - 3
SP - 541
EP - 580
AB - In this paper we study equivalence classes of generic $1$-parameter germs of real analytic families ${\mathcal{Q}}_{\varepsilon }$ unfolding codimension $1$ germs of diffeomorphisms ${\mathcal{Q}}_0: ({\mathbb{R}},0)\rightarrow ({\mathbb{R}},0)$ with a fixed point at the origin and multiplier $-1,$ under (weak) analytic conjugacy. These germs are generic unfoldings of the flip bifurcation. Two such germs are analytically conjugate if and only if their second iterates, ${\mathcal{P}}_{\varepsilon }={\mathcal{Q}}_{\varepsilon }^{\circ 2},$ are analytically conjugate. We give a complete modulus of analytic classification: this modulus is an unfolding of the Ecalle modulus of the resonant germ ${\mathcal{Q}}_0$ with special symmetry properties reflecting the real character of the germ ${\mathcal{Q}}_{\varepsilon } .$ As an application, this provides a complete modulus of analytic classification under weak orbital equivalence for a germ of family of planar vector fields unfolding a weak focus of order $1$$(i.e.$ undergoing a generic Hopf bifurcation of codimension $1)$ through the modulus of analytic classification of the germ of family ${\mathcal{P}}_{\varepsilon }={\mathcal{Q}}_{\varepsilon }^{\circ 2},$ where ${\mathcal{P}}_{\varepsilon }$ is the Poincaré monodromy of the family of vector fields.
LA - eng
KW - singularity theory of differentiable mappings; unfolding of the codimension-one flip; Hopf bifurcation; analytic classification
UR - http://eudml.org/doc/219760
ER -

References

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