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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The bifurcation set of the period function of the dehomogenized Loud's centers is bounded


Authors: F. Mañosas and J. Villadelprat
Journal: Proc. Amer. Math. Soc. 136 (2008), 1631-1642
MSC (2000): Primary 34C07, 34C23; Secondary 34C25
Published electronically: January 23, 2008
MathSciNet review: 2373592
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Abstract: This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely

\begin{displaymath} \left\{ \begin{array}{l} \dot x=-y+xy, \\ [1pt] \dot y=x+Dx^2+Fy^2. \end{array}\right. \end{displaymath}

In this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle $ K=(-7,2)\times(0,4).$ More concretely, we prove that if $ (D,F)\notin K$, then the period function of the center is monotonically increasing.


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Additional Information

F. Mañosas
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra, Barcelona, Spain

J. Villadelprat
Affiliation: Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09131-4
PII: S 0002-9939(08)09131-4
Received by editor(s): October 18, 2006
Published electronically: January 23, 2008
Additional Notes: The authors were partially supported by the CONACIT through the grant 2005-SGR-00550 and by the DGES through the grant MTM-2005-06098-C02-1.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2008 American Mathematical Society