The cyclicity of hyperbolic hemicycles
Marín Pérez, David (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Villadelprat Yagüe, Jordi (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2025
Abstract:	We consider families of planar polynomial vector fields of degree n and study the cyclicity of a type of unbounded polycycle Γ called hemicycle. Compactified to the Poincaré disc, Γ consists of an affine straight line together with half of the line at infinity and has two singular points, which are hyperbolic saddles located at infinity. We prove four main results. Theorem A deals with the cyclicity of Γ when perturbed without breaking the saddle connections. For the other results we consider the case n=2. More concretely they are addressed to the quadratic integrable systems belonging to the class Q and having two hemicycles, Γ and Γ, surrounding each one a center. Theorem B gives the cyclicity of Γ and Γ when perturbed inside the whole family of quadratic systems. In Theorem C we study the number of limit cycles bifurcating simultaneously from Γ and Γ when perturbed as well inside the whole family of quadratic systems. Finally, in Theorem D we show that for three specific cases there exists a simultaneous alien limit cycle bifurcation from Γ and Γ.
Note:	Altres ajuts: acords transformatius de la UAB
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Limit cycle ; Hemicycle ; Cyclicity ; Asymptotic expansion ; Dulac map
Published in:	Journal of differential equations, Vol. 433 (July 2025) , art. 113281, ISSN 1090-2732

DOI: 10.1016/j.jde.2025.113281

59 p, 1.3 MB

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