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}
}
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… 

The modulus of unfoldings of cusps in conformal geometry

The equivalence problem in analytic dynamics for $1$-resonance

When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in the neighborhood of a singular point? A way to answer is to use normal forms. But

Parametric rigidity of real families of conformal diffeomorphisms tangent to x→−x

  • W. Arriagada
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2018
We prove that one-parameter families of real germs of conformal diffeomorphisms tangent to the involution x ↦−x are rigid in the parameter. We establish a connection between the dynamics in the

Temporally Normalizable Generic Unfoldings of Order-1 Weak Foci

In this paper, we describe the obstructions preventing the germ of an order-1 elliptic family from being temporally normalizable in the analytic case. We describe two classes of symmetries on the

Temporally Normalizable Generic Unfoldings of Order-1 Weak Foci

In this paper, we describe the obstructions preventing the germ of an order-1 elliptic family from being temporally normalizable in the analytic case. We describe two classes of symmetries on the

Analytic obstructions to isochronicity in codimension 1*

This paper studies the analytical obstructions preventing the germ of a generic analytic family of elliptic ordinary differential equations from being isochronous and sets the formal obstructions in terms of commutators.

References

SHOWING 1-10 OF 24 REFERENCES

Characterization of the unfolding of a weak focus and modulus of analytic classification

The thesis gives a geometric description for the germ of the singular holomorphic foliation associated with the complexication of a germ of generic analytic family unfolding a real analytic vector

MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMS

In this paper we give a complete modulus of analytic classi- fication under weak equivalence for generic analytic 1-parameter unfold- ings of dieomorphisms with a generic parabolic point. The modulus

Analytical Moduli for Unfoldings of Saddle-Node Vector Fields

In this paper we consider germs of k-parameter generic families of analytic 2-dimensional vector fields unfolding a saddle-node of codimension k and we give a complete modulus of analytic

Characterization of the generic unfolding of a weak focus

The Mandelbrot Set, Theme and Variations

Introduction L.Tan Preface J. Hubbard 1. The Mandelbrot set is universal C. McMullen 2. Baby Mandelbrot sets are born in cauliflowers A. Douady, X. Buff, R. Devaney and P. Sentenac 3. Modulation dans

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

On considere des germes de familles generiques a un parametre deployant un germe de diffeomorphisme resonant et on montre que le deploiement du module d'Ecalle donne un module complet de

Lectures on quasiconformal mappings

The Ahlfors Lectures: Acknowledgments Differentiable quasiconformal mappings The general definition Extremal geometric properties Boundary correspondence The mapping theorem Teichmuller spaces

Classification analytique des 'equations di 'erentielles non lin'eaires r'esonantes du premier ordre

On etudie les germes d'equations differentielles analytiques ω=A(x,y)dx+B(n,y)dy=0 singulieres a l'origine de C 2 . Classes de conjugaison des diffeomorphismes resonnants. Classification analytique