• DOI:10.5802/AFST.1317
• Corpus ID: 56464589

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

@article{ArriagadaSilva2011TheMO,
  title={The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations},
  author={Waldo Arriagada-Silva and Christiane Rousseau},
  journal={Annales de la Facult{\'e} des Sciences de Toulouse},
  year={2011},
  volume={20},
  pages={541-580}
}

• Waldo Arriagada-Silva, C. Rousseau
• Published 2011
• Mathematics
• Annales de la Faculté des Sciences de Toulouse

In this paper we study equivalence classes of generic 1-parameter germs of real analytic families Qε unfolding codimension 1 germs of diffeomorphisms Q0 : (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, Pε = Q ε , are analytically conjugate. We give a complete modulus of analytic classification: this modulus is an unfolding of the Ecalle modulus of the resonant germ Q0 with special symmetry properties reflecting the real character of the germ Qε. 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 Pε = Q ε , where Pε is the Poincaré first return map of the family of vector fields.