Unfolding of resonant saddles and the Dulac time

Pavao Mardešić; David Marín; Jordi Villadelprat

doi:10.3934/dcds.2008.21.1221

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2008, Volume 21, Issue 4: 1221-1244. Doi: 10.3934/dcds.2008.21.1221

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Unfolding of resonant saddles and the Dulac time

1.
Institut de Mathématiques de Bourgogne, UFR Sciences et Techniques, Université de Bourgogne, UMR 5584 du CNRS, B.P. 47870, 21078 Dijon
2.
Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona
3.
Departament d’Enginyeria Informàtica i Matemàtiques, ETSE, Universitat Rovira i Virgili, 43007 Tarragona

Received: June 30, 2007

Revised: November 30, 2007

Published: September 2008

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Abstract

In this work we study unfoldings of planar vector fields in a neighbourhood of a resonant saddle. We give a $\mathcal C^k$ normal form for the unfolding with respect to the conjugacy relation. Using our normal form we determine an asymptotic development, uniform with respect to the parameters, of the Dulac time of a resonant saddle deformation. Conjugacy relation instead of weaker equivalence relation is necessary when studying the time function. The Dulac time of a resonant saddle can be seen as the basic building block of the total period function of an unfolding of a hyperbolic polycycle.

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Mathematics Subject Classification: Primary: 37G05; Secondary: 34C07, 37C10, 37G15.

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