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Bifurcation values for a familiy of planar vector fields of degree five
García Saldaña, Johanna Denise (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Gasull i Embid, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Giacomini, Hector (Université de Tours (France). Laboratoire de Mathématiques et Physique Théorique)

Data: 2015
Resum: We study the number of limit cycles and the bifurcation diagram in the Poincar´ sphere of a one-parameter family of planar differential equations of degree e ˙ five x = Xb (x) which has been already considered in previous papers. We prove that there is a value b∗ > 0 such that the limit cycle exists only when b ∈ (0, b∗ ) and that it is unique and hyperbolic by using a rational Dulac function. Moreover we provide an interval of length 27/1000 where b∗ lies. As far as we know the tools used to determine this interval are new and are based on the construction of algebraic curves without contact for the flow of the differential equation. These curves are obtained using analytic information about the separatrices of the infinite critical points of the vector field. To prove that the Bendixson-Dulac Theorem works we develop a method for studying whether one-parameter families of polynomials in two variables do not vanish based on the computation of the so called double discriminant.
Nota: Número d'acord de subvenció MCYT/MTM2008-03437
Nota: Número d'acord de subvenció CIRIT/2009/SGR-410
Drets: Tots els drets reservats.
Llengua: Anglès
Document: article ; recerca ; preprint
Matèria: Bifurcation ; Double discriminant ; Dulac function ; Phase portrait on the Poincaré sphere ; Polynomial planar system ; Uniqueness of limit cycles
Publicat a: Discrete and Continuous Dynamical Systems. Series A, Vol. 35 Núm. 2 (2015) , p. 669-701, ISSN 1553-5231

DOI: 10.3934/dcds.2015.35.669

33 p, 1.1 MB

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