Resurgence of inner solutions for              perturbations of the McMillan map

Pau Martín; David Sauzin; Tere M. Seara

doi:10.3934/dcds.2011.31.165

Article Contents

2011, Volume 31, Issue 1: 165-207. Doi: 10.3934/dcds.2011.31.165

This issue Previous Article Time-dependent attractor for the Oscillon equation Next Article Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force

Resurgence of inner solutions for perturbations of the McMillan map

1.
Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Ed-C3, Jordi Girona, 1-3, 08034 Barcelona
2.
IMCCE, Observatoire de Paris, 77 Avenue Denfert-Rochereau, 75014 Paris
3.
Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona

Received: December 31, 2009

Revised: December 31, 2010

Published: December 2010

Abstract Related Papers Cited by

Abstract

A sequence of "inner equations" attached to certain perturbations of the McMillan map was considered in [5], their solutions were used in that article to measure an exponentially small separatrix splitting. We prove here all the results relative to these equations which are necessary to complete the proof of the main result of [5]. The present work relies on ideas from resurgence theory: we describe the formal solutions, study the analyticity of their Borel transforms and use Écalle's alien derivations to measure the discrepancy between different Borel-Laplace sums.

Keywords:

Mathematics Subject Classification: 37J10, 34C37, 34E10, 34M37.

Citation:

Related Papers

Cited by

References

[1]	B. Candelpergher, J.-C. Nosmas and F. Pham, "Approche de la Résurgence," (French) [Approach to resurgence] Actualités Mathématiques, [Current Mathematical Topics] Hermann, Paris, 1993.
[2]	J. Écalle, "Les Fonctions Résurgentes. Tome I," (French) [Resurgent functions. Vol. I] Les algébres de fonctions résurgentes. [The algebras of resurgent functions] With an English foreword. Publications Mathématiques d'Orsay 81 [Mathematical Publications of Orsay 81], 5, Université de Paris-Sud, Département de Mathématique, Orsay, 1981, 247 pp.
[3]	V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map, Ann. Inst. Fourier, Grenoble, 51 (2001), 513-567.
[4]	B. Malgrange, Resommation des séries divergentes, (French) [Summation of divergent series], Exp. Math., {13} (1995), 163-222.
[5]	P. Martín, T. M. Seara and D. Sauzin, Exponentially small splitting of separatrices in perturbations of the McMillan map, preprint, 2009.
[6]	C. Olivé, D. Sauzin and T. Seara, Resurgence in a Hamilton-Jacobi equation, Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002), Ann. Inst. Fourier, Grenoble, 53 (2003), 1185-1235.
[7]	D. Sauzin, Resurgent functions and splitting problems, RIMS Kokyuroku, 1493 (2005), 48-117.