no. 6
Gevrey normal form for unfoldings of nilpotent contact points of planar slow-fast systems
[Forme normale de Gevrey pour les déploiements de points de contact nilpotents de systèmes lent-rapide dans le plan]
De Maesschalck, Peter1 ; Doan, Thai Son2
1 Hasselt University Agoralaan, gebouw D, B-3590 Diepenbeek (Belgium)
2 Institute of Mathematic Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi (Viet Nam)
Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2597-2621.

On présente des formes normales pour les déploiements de points de contact nilpotents des systèmes lent-rapide dans le plan. Les formes normales sont utiles pour traiter des points de contact reguliers et singuliers (y compris les points tournant). Pour les points de contact reguliers, on obtient une forme normale de type Liénard, tandis que pour les points de contact singuliers, la forme normale est de type Liénard sauf une erreur exponentiellement petite. Nos techniques sont basées sur des estimations de Gevrey des séries formelles et la sommation de Gevrey. Il s’agit d’une extension des résultats connus, basée sur une version de Gevrey de la théorème de préparation de Levinson.

We present normal forms for unfoldings of nilpotent contact points of slow-fast systems in the plane. The normal forms are useful in the treatment of regular and singular contact points (including turning points). For regular contact points, we obtain a normal form of Liénard type, while for singular contact points, the normal form is of Liénard type up to exponentially small error. Our techniques are based on Gevrey estimates on formal power series and Gevrey summation. This extension of earlier results is based on a Gevrey version of Levinson’s preparation theorem.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3144
Classification : 34D14, 34A26, 34M30, 34M60
Keywords: singular perturbations, slow-fast vector field, normal forms, Gevrey asymptotics, Liénard system
Mot clés : perturbation singulière, champs de vecteurs lent-rapide, formes normales, asymptotique Gevrey, systèmes Liénard
De Maesschalck, Peter 1 ; Doan, Thai Son 2

1 Hasselt University Agoralaan, gebouw D, B-3590 Diepenbeek (Belgium)
2 Institute of Mathematic Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Ha Noi (Viet Nam)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2017__67_6_2597_0,
     author = {De Maesschalck, Peter and Doan, Thai Son},
     title = {Gevrey normal form for unfoldings of nilpotent contact points of planar slow-fast systems},
     journal = {Annales de l'Institut Fourier},
     pages = {2597--2621},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     doi = {10.5802/aif.3144},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3144/}
}
TY  - JOUR
AU  - De Maesschalck, Peter
AU  - Doan, Thai Son
TI  - Gevrey normal form for unfoldings of nilpotent contact points of planar slow-fast systems
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 2597
EP  - 2621
VL  - 67
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3144/
DO  - 10.5802/aif.3144
LA  - en
ID  - AIF_2017__67_6_2597_0
ER  - 
%0 Journal Article
%A De Maesschalck, Peter
%A Doan, Thai Son
%T Gevrey normal form for unfoldings of nilpotent contact points of planar slow-fast systems
%J Annales de l'Institut Fourier
%D 2017
%P 2597-2621
%V 67
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3144/
%R 10.5802/aif.3144
%G en
%F AIF_2017__67_6_2597_0
De Maesschalck, Peter; Doan, Thai Son. Gevrey normal form for unfoldings of nilpotent contact points of planar slow-fast systems. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2597-2621. doi : 10.5802/aif.3144. https://aif.centre-mersenne.org/articles/10.5802/aif.3144/

[1] Canalis-Durand, Mireille; Ramis, Jean-Pierre; Schäfke, Reinhard; Sibuya, Yasutaka Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., Volume 518 (2000), pp. 95-129 | DOI | MR | Zbl

[2] De Maesschalck, Peter Gevrey normal forms for nilpotent contact points of order two, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 2, pp. 677-688 | DOI | MR | Zbl

[3] De Maesschalck, Peter; Dumortier, Freddy; Roussarie, Robert Cyclicity of common slow-fast cycles, Indag. Math., Volume 22 (2011) no. 3-4, pp. 165-206 | DOI | MR | Zbl

[4] Fenichel, Neil Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equations, Volume 31 (1979) no. 1, pp. 53-98 | DOI | MR | Zbl

[5] Fruchard, Augustin; Schäfke, Reinhard Composite asymptotic expansions, Lecture Notes in Mathematics, 2066, Springer, 2013, x+161 pages | DOI | MR | Zbl

[6] Huzak, Renato Normal forms of Liénard type for analytic unfoldings of nilpotent singularities, Proc. Am. Math. Soc., Volume 145 (2017) no. 10, pp. 4325-4336 | DOI | Zbl

[7] Jardón-Kojakhmetov, Hildeberto Formal normal form of A k slow fast systems, C. R. Acad. Sci. Paris, Ser. I, Volume 353 (2015), pp. 795-800 | DOI | Zbl

[8] Levinson, Norman A canonical form for an analytic function of several variables at a critical point, Bull. Am. Math. Soc., Volume 66 (1960), pp. 68-69 | DOI | MR | Zbl

[9] Neĭshtadt, Anatolií I. The separation of motions in systems with rapidly rotating phase, Prikl. Mat. Mekh., Volume 48 (1984) no. 2, pp. 197-204 | DOI | MR | Zbl

[10] Parusiński, Adam; Rolin, Jean-Philippe A note on the Weierstrass preparation theorem in quasianalytic local rings, Can. Math. Bull., Volume 57 (2014) no. 3, pp. 614-620 | DOI | MR | Zbl

[11] Ramis, Jean-Pierre; Schäfke, Reinhard Gevrey separation of fast and slow variables, Nonlinearity, Volume 9 (1996) no. 2, pp. 353-384 | DOI | MR | Zbl

[12] Roberts, Anthony J. Normal form transforms separate slow and fast modes in stochastic dynamical systems, Phys. A, Volume 387 (2008) no. 1, pp. 12-38 | DOI | MR

[13] Szmolyan, Peter; Wechselberger, Martin Canards in ℝ 3 , J. Differ. Equations, Volume 177 (2001) no. 2, pp. 419-453 | DOI | MR | Zbl

Cité par Sources :