Waldo Arriagada-Silva; Christiane Rousseau
The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations
Annales de la faculté des sciences de Toulouse Sér. 6, 20 no. 3 (2011), p. 541-580, doi: 10.5802/afst.1317
Article PDF | Reviews MR 2894838 | Zbl 1242.58021
Résumé - 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.
Bibliography
[2] Arriagada-Silva (W.).— Characterization of the unfolding of a weak focus and modulus of analytic classification. PhD thesis, Université de Montréal, (2010).
[3] Christopher (C.) and Rousseau (C.).— The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point. Preprint, (2008). Zbl 1134.37021
[4] Ecalle (J.).— Les fonctions résurgentes. Publications mathématiques d’Orsay, (1985). Zbl 0602.30029
[5] Fatou (P.).— Sur les équations fonctionnelles. Bull. Soc. Math. France, Paris, 47, 48: p. 161-271, p. 33-94, p. 208-314, (1919-1920). JFM 47.0921.02
[6] Freitag (E.).— Complex Analysis 2. Universitext, Springer-Verlag Berlin Heidelberg, (2011). MR 2810329 | Zbl pre05913316
[7] Giné (J.) and Grau (M.).— Characterization of isochronous foci for planar analytic differential systems. Proc. Roy. Soc. Edinburgh Sect. A, 135: p. 985-998, (2005). MR 2187221 | Zbl 1092.34014
[8] Glutsyuk (A.A.).— Congruence of singular points and nonlinear stokes phenomenon. Trans. Moscow Math. Soc., 62: p. 49-95, (2001). MR 1907251 | Zbl 1004.34081
[9] Mattei (J.F.) and Moussu (R.).— Holonomie et intégrales premières. Ann. Scient. Éc. Norm. Sup., 4e série, 13: p. 469-523, (1980). Numdam | MR 608290 | Zbl 0458.32005
[10] Mardešić (P.), Roussarie (R.), and Rousseau (C.).— Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms. Moscow Mathematical Journal, 4: p. 455-498, (2004). MR 2108445 | Zbl 1077.37035
[11] Martinet (J.).— Remarques sur la bifurcation noeud-col dans le domaine complexe. Astérisque, 150-151: p. 131-149, (1987). MR 923597 | Zbl 0655.58025
[12] Martinet (J.) and Ramis (J.P.).— Problèmes de modules pour des équations différentielles non linéaires du premier ordre. Publ. IHES, 55: p. 63-164, (1982). Numdam | MR 672182 | Zbl 0546.58038
[13] Martinet (J.) and Ramis (J.P.).— Classification analytique des équations différentielles non linéaires résonnantes du premier ordre. Ann. Scient. Éc. Norm. Sup., 4e série, 16: p. 571-621, (1983). Numdam | MR 740592 | Zbl 0534.34011
[14] Pérez-Marco (R.) and Yoccoz (J.-C.).— Germes de feuilletages holomorphes à holonomie prescrite. S.M.F., Astérisque, 222: p. 345-371, (1994). Zbl 0809.32008
[15] Rousseau (C.).— The moduli space of germs of generic families of analytic diffeomorphisms unfolding of a codimension one resonant diffeomorphism or resonant saddle. J. Differential Equations, 248: p. 1794-1825, (2010). MR 2593608 | Zbl 1204.37048
[16] Rousseau (C.) and Christopher (C.).— Modulus of analytical classification for the generic unfolding of a codimension one resonant diffeomorphism or resonant saddle. Annales de l’Institut Fourier, 57: p. 301-360, (2007). Cedram | MR 2316241 | Zbl 1127.37039
[17] Rousseau (C.) and Teyssier (L.).— Analytical moduli for unfoldings of saddle node vector fields. Moscow Mathematical Journal, 8: p. 547-614, (2008). MR 2483224 | Zbl 1165.37016
[18] Shishikura (M.).— Bifurcation of parabolic fixed points. “The Mandelbrot set, theme and variations", London Math. Society Lecture Notes, 274: p. 325-363, (2000). MR 1765097 | Zbl 1062.37043
[19] Voronin (S. M.).— Analytic classification of germs of conformal maps $({\mathbb{C}} ,0) \rightarrow ({\mathbb{C}} ,0)$ with identical linear part. Funktsional. Anal. i Prilozhen and Func. Anal. Appl., 15: p. 1-17 (Russian), p. 1-13 (English), (1981). MR 609790 | Zbl 0463.30010
