Exponentiallysmall splitting of separatrices in the perturbed McMillan map

Pau Martín; David Sauzin; Tere M. Seara

doi:10.3934/dcds.2011.31.301

Article Contents

2011, Volume 31, Issue 2: 301-372. Doi: 10.3934/dcds.2011.31.301

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Exponentially small splitting of separatrices in the perturbed 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: January 31, 2011

Published: May 2011

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Abstract

The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coefficients involved in the expansion have a resurgent origin, as shown in [14].

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Mathematics Subject Classification: 37J10, 34C37, 37C29, 34E10, 34M37.

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References

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