Comptes Rendus
Complex Analysis
Gevrey properties of real planar singularly perturbed systems
Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 195-198.

By applying geometric techniques to real analytic singularly perturbed vector fields on the plane, we develop a way to give a bound on the Gevrey type of the Taylor development of center manifolds at normally hyperbolic turning points, and show that the same technique is useful in the study of degenerate planar turning points and their corresponding canard manifolds. At the end of the Note, we motivate the interest in Gevrey asymptotics by briefly discussing its relation with bifurcation delay.

Suivant l'approche géométrique dans l'étude de problèmes de perturbations singulières dans le plan, nous développons une méthode pour majorer le type Gevrey des variétés centrales aux points normalement hyperboliques, et des variétés canards aux points tournants. A la fin de la note, nous motivons l'intérêt de l'asymptotique Gevrey en décrivant le rapport avec le retard à la bifurcation.

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DOI: 10.1016/j.crma.2004.12.020
Peter De Maesschalck 1

1 Limburgs Universitair Centrum, Universitaire Campus Gebouw D, Departement WNI, Campuslaan 1, B-3590 Diepenbeek, Belgique
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Peter De Maesschalck. Gevrey properties of real planar singularly perturbed systems. Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 195-198. doi : 10.1016/j.crma.2004.12.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.020/

[1] E. Benoît; A. Fruchard; R. Schäfke; G. Wallet Solutions surstables des équations différentielles complexes lentes-rapides à point tournant, Ann. Fac. Sci. Toulouse Math., Volume 6 (1998), p. 7

[2] P. De Maesschalck, Geometry and Gevrey asymptotics of two-dimensional turning points, Ph.D. thesis, 2003

[3] F. Dumortier; R. Roussarie Canard cycles and center manifolds, Mem. Amer. Math. Soc., Volume 121 (1996), p. 577

[4] A. Fruchard; R. Schäfke Overstability and resonance, Ann. Inst. Fourier, Volume 53 (2003) no. 1, pp. 227-264

[5] R. Schäfke On the Borel transform, C. R. Acad. Sci. Paris, Ser. I, Volume 323 (1996)

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