%0 Journal Article
%T A new approach for the study of limit cycles
%+ Departamento de Matemática y Fı́sica Aplicadas [Concepcion] (DMFA)
%+ Departament de Matemàtiques [Barcelona] (UAB)
%+ Institut Denis Poisson (IDP)
%A García-Saldaña, Johanna D.
%A Gasull, A.
%A Giacomini, Hector
%Z J.D. G.-S. is supported by FONDECyT grant number 11171115.
%Z A.G. is supported by Ministerio de Ciencia, Innovacion y Universidades of the Spanish Government through grants MTM2016-77278-P (MINECO/AEI/FEDER, UE) and by grant 2017-SGR-1617 from AGAUR, Generalitat de Catalunya.
%Z 20 pages, 1 figure
%< avec comité de lecture
%@ 0022-0396
%J Journal of Differential Equations
%I Elsevier
%V 269
%8 2020-05-05
%D 2020
%Z 1910.08098
%Z MSC 2010 : 34C25, 34C07 (Primary) ; 34C37 (Secondary)
%Z Mathematics [math]Journal articles
%X We prove that star-like limit cycles of any planar polynomial system can also be seen either as solutions defined on a given interval of a new associated planar non-autonomous polynomial system or as heteroclinic solutions of a 3-dimensional polynomial system. We illustrate these points of view with several examples. One of the key ideas in our approach is to decompose the periodic solutions as the sum of two suitable functions. As a first application we use these new approaches to prove that all star-like reversible limit cycles are algebraic. As a second application we introduce a function whose zeroes control the periodic orbits that persist as limit cycles when we perturb a star-like reversible center. As far as we know this is the first time that this question is solved in full generality. Somehow, this function plays a similar role that an Abelian integral for studying perturbations of Hamiltonian systems.
%G English
%L hal-02879499
%U https://cnrs.hal.science/hal-02879499
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