Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle
Annales de l'Institut Fourier, Volume 57 (2007) no. 1, pp. 301-360.

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.

On considère des germes de familles génériques à un paramètre déployant un germe de difféomorphisme résonant et on montre que le déploiement du module d’Ecalle donne un module complet de classification analytique. On décrit le phénomène de résurgence paramétrique. On applique les résultats précédents à la construction d’un module complet de classification analytique orbitale pour le déploiement d’un point de selle résonant générique au moyen du déploiement de son difféomorphisme d’holonomie. Ce module est le déploiement du module de Martinet-Ramis pour un point de selle résonant. Quand le point de selle passe par la résonance on observe une “bifurcation transcritique” : la dynamique du point de selle est contrôlée par des parties différentes du déploiement du module de chaque côté de la bifurcation. On regarde aussi la dépendance du temps et on donne un module complet de conjugaison analytique pour le déploiement d’un point de selle résonant générique.

DOI: 10.5802/aif.2260
Classification: 34M35, 37F75, 32S65
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.
Mot clés : déploiement de difféomorphisme résonant, module de classification analytique, déploiement d’un point de selle résonant, déploiement du module d’Ecalle, déploiement du module de Martinet-Ramis, déploiement de l’holonomie, phénomène de résurgence paramétrique, bifurcation transcritique
Rousseau, Christiane 1; Christopher, Colin 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)
@article{AIF_2007__57_1_301_0,
     author = {Rousseau, Christiane and Christopher, Colin},
     title = {Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle},
     journal = {Annales de l'Institut Fourier},
     pages = {301--360},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {1},
     year = {2007},
     doi = {10.5802/aif.2260},
     mrnumber = {2316241},
     zbl = {1127.37039},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2260/}
}
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
SP  - 301
EP  - 360
VL  - 57
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2260/
DO  - 10.5802/aif.2260
LA  - en
ID  - AIF_2007__57_1_301_0
ER  - 
%0 Journal Article
%A Rousseau, Christiane
%A Christopher, Colin
%T Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle
%J Annales de l'Institut Fourier
%D 2007
%P 301-360
%V 57
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2260/
%R 10.5802/aif.2260
%G en
%F AIF_2007__57_1_301_0
Rousseau, Christiane; Christopher, Colin. Modulus of analytic classification for the generic unfolding of a codimension 1 resonant diffeomorphism or resonant saddle. Annales de l'Institut Fourier, Volume 57 (2007) no. 1, pp. 301-360. doi : 10.5802/aif.2260. https://aif.centre-mersenne.org/articles/10.5802/aif.2260/

[1] Arnold, V. Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir Moscow, 1980 (English transl.: Geometric theory of ordinary differential equations, Springer-Verlag) | MR | Zbl

[2] Brjuno, A. D. Analytic form of differential equations, Trans. Moscow Math. Soc., Volume 25 (1971), pp. 131-288 | MR | Zbl

[3] Christopher, C.; Mardešić, P.; Rousseau, C. Normalizable, integrable and linearizable points in complex quadratic systems in 2 , J. Dynam. Control Systems, Volume 9 (2003), pp. 311-363 | DOI | MR | Zbl

[4] Christopher, C.; Mardešić, P.; Rousseau, C. Normalizability, synchronicity and relative exactness for vector fields in  2 , J. Dynam. Control Systems, Volume 10 (2004), pp. 501-525 | DOI | MR | Zbl

[5] Christopher, C.; Rousseau, C. Normalizable, integrable and linearizable saddle points in the Lotka-Volterra system, Qual. Theory Dynam. Syst., Volume 5 (2004), pp. 11-61 | DOI | MR | Zbl

[6] Douady, A. Does a Julia set depend continuously on the polynomial?, Complex dynamical systems. The mathematics behind the Mandelbrot and Julia sets, Robert L. Devaney Ed., Amer. Math. Soc. Short Course Lecture Notes, Proceedings of Symposia in Applied Mathematics, Volume 49, American Mathematical Society, 1994, pp. 91-138 | MR | Zbl

[7] Écalle, J. Les fonctions résurgentes, Publications mathématiques d’Orsay, 1985 | Zbl

[8] Glutsyuk, A. A. Confluence of singular points and nonlinear Stokes phenomenon, Trans. Moscow Math. Soc., Volume 62 (2001), pp. 49-95 | MR | Zbl

[9] Hörmander, L. An introduction to complex analysis in several variables, North-Holland, American Elsevier, 1973 | MR | Zbl

[10] Ilyashenko, Y. Divergence of series reducing an analytic differential equation to linear normal form at a singular point, Funct. Anal. Appl., Volume 13 (1979), pp. 227-229 | MR | Zbl

[11] Ilyashenko, Y. Nonlinear Stokes phenomena, Nonlinear Stokes phenomena, Y. Ilyashenko Ed., Advances in Soviet Mathematics, Volume 14, Amer. Math. Soc., Providence, RI, 1993, pp. 1-55 | MR | Zbl

[12] Ilyashenko, Y. S.; Pyartli, A. S. Materialization of Poincaré resonances and divergence of normalizing series, J. Sov. Math., Volume 31 (1985), pp. 3053-3092 | DOI | Zbl

[13] Lavaurs, P. Systèmes dynamiques holomorphes : explosion de points périodiques paraboliques, Université Paris-Sud (1989) (Ph. D. Thesis)

[14] Mardešić, P.; Roussarie, R.; Rousseau, C. Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms, Moscow Math. J., Volume 4 (2004), pp. 455-502 | MR | Zbl

[15] Martinet, J. Remarques sur la bifurcation nœud-col dans le domaine complexe, Astérisque, Volume 150-151 (1987), pp. 131-149 | Zbl

[16] Martinet, J.; Ramis, J.-P. Classification analytique des équations différentielles non linéaires résonantes du premier ordre, Ann. Sci. École Norm. Sup., Volume 16 (1983), pp. 571-621 | Numdam | MR | Zbl

[17] Mattei, J.-F.; Moussu, R. Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. 4e série, Volume 13 (1980), pp. 469-523 | Numdam | MR | Zbl

[18] Oudkerk, R. The parabolic implosion for f 0 ( z ) = z + z ν + 1 + O ( z ν + 2 ) , University of Warwick (1999) (Ph. D. Thesis)

[19] Pérez-Marco, R. Solution complète au problème de Siegel de linéarisation d’une application holomorphe au voisinage d’un point fixe (d’après J.-C. Yoccoz), Astérisque, Volume 206 (1992), pp. 273-310 | Numdam | MR | Zbl

[20] Pérez-Marco, R.; Yoccoz, J.-C. Germes de feuilletages holomorphes à holonomie prescrite, Astérisque, Volume 222 (1994), pp. 345-371 | MR | Zbl

[21] Rousseau, C. Normal forms, bifurcations and finiteness properties of vector fields, Normal forms, bifurcations and finiteness properties of vector fields, NATO Sci. Ser. II Math. Phys. Chem., Volume 137, Kluwer Acad. Publ., Dordrecht, 2004, pp. 431-470 | MR

[22] Rousseau, C. Modulus of orbital analytic classification for a family unfolding a saddle-node, Moscow Math. J., Volume 5 (2005), pp. 245-268 | MR | Zbl

[23] Shishikura, M. Bifurcations of parabolic fixed points, The Mandelbrot set, theme and variations, Tan Lei Ed., London Math. Soc. Lecture Note Ser., Volume 274, Cambridge Univ. Press, Cambridge, 2000, pp. 325-363 | MR | Zbl

[24] Teyssier, L. Analytical classification of singular saddle-node vector fields, J. Dynam. Control Systems, Volume 10 (2004), pp. 577-605 | DOI | MR | Zbl

[25] Teyssier, L. Équation homologique et cycles asymptotiques d’une singularité nœud-col, Bull. Sci. Math., Volume 128 (2004), pp. 167-187 | DOI | Zbl

[26] Voronin, S. M.; Grintchy, A. A. An analytic classification of saddle resonant singular points of holomorphic vector fields in the complex plane, J. Dynam. Control Syst., Volume 2 (1996), pp. 21-53 | DOI | MR | Zbl

[27] Yoccoz, J.-C. Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque, Volume 231 (1995), pp. 3-88 | MR

Cited by Sources: