November  2008, 21(4): 1221-1244. doi: 10.3934/dcds.2008.21.1221

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, France

2. 

Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

3. 

Departament d’Enginyeria Informàtica i Matemàtiques, ETSE, Universitat Rovira i Virgili, 43007 Tarragona, Spain

Received  July 2007 Revised  December 2007 Published  May 2008

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.
Citation: Pavao Mardešić, David Marín, Jordi Villadelprat. Unfolding of resonant saddles and the Dulac time. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1221-1244. doi: 10.3934/dcds.2008.21.1221
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