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Transactions of the American Mathematical Society

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The inverse integrating factor and the Poincaré map


Authors: Isaac A. García, Héctor Giacomini and Maite Grau
Journal: Trans. Amer. Math. Soc. 362 (2010), 3591-3612
MSC (2010): Primary 37G15, 37G20, 34C05
DOI: https://doi.org/10.1090/S0002-9947-10-05014-2
Published electronically: February 17, 2010
MathSciNet review: 2601601
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Abstract: This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincaré return map in terms of the inverse integrating factor. In particular, we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it. We also apply this result to study the homoclinic loop bifurcation. We only consider homoclinic loops whose critical point is a hyperbolic saddle and whose Poincaré return map is not the identity. A local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method.


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Additional Information

Isaac A. García
Affiliation: Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69. 25001 Lleida, Spain
Email: garcia@matematica.udl.cat

Héctor Giacomini
Affiliation: Laboratoire de Mathématiques et Physique Théorique, C.N.R.S. UMR 6083, Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont 37200 Tours, France
Email: Hector.Giacomini@lmpt.univ-tours.fr

Maite Grau
Affiliation: Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69. 25001 Lleida, Spain
Email: mtgrau@matematica.udl.cat

DOI: https://doi.org/10.1090/S0002-9947-10-05014-2
Keywords: Inverse integrating factor, Poincar\'e map, limit cycle, homoclinic loop
Received by editor(s): October 17, 2007
Received by editor(s) in revised form: May 8, 2008
Published electronically: February 17, 2010
Additional Notes: The authors were partially supported by a DGICYT grant number MTM2005-06098-C02-02.
Dedicated: Dedicated to Professor Javier Chavarriga
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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