On the cyclicity of hyperbolic polycycles
Buzzi, Claudio (Universidade Estadual Paulista "Júlio de Mesquita Filho". Instituto de Biociências, Letras e Ciências Exatas)
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Santana, Paulo Henrique Reis (Universidade Estadual Paulista "Júlio de Mesquita Filho". Instituto de Biociências, Letras e Ciências Exatas)

Date:	2025
Abstract:	Let X be a planar smooth vector field with a polycycle Γ with n sides and all its corners, that are at most n singularities, being hyperbolic saddles. In this paper we study the cyclicity of Γ in terms of the hyperbolicity ratios of these saddles, giving explicit conditions that ensure that it is at least k, for any k⩽n. Our result extends old results and also provides a more accurate proof of the known ones because we rely on some recent powerful works that study in more detail the regularity with respect to initial conditions and parameters of the Dulac map of hyperbolic saddles for families of vector fields. We also prove that when X is polynomial there is a polynomial perturbation (in general with degree much higher that the one of X) that attains each of the obtained lower bounds for the cyclicities. Finally, we also study some related inverse problems and provide concrete examples of applications in the polynomial world.
Grants:	Agencia Estatal de Investigación PID2022-136613NB-I00 Agencia Estatal de Investigación CEX2020-001084-M Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
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 acceptada per publicar
Subject:	Polycycle ; Limit cycle ; Displacement map ; Cyclicity ; Heteroclinic and homoclinic orbits
Published in:	Journal of differential equations, Vol. 429 (June 2025) , p. 646-677, ISSN 1090-2732

DOI: 10.1016/j.jde.2025.02.061

Available from: 2027-06-30 Postprint

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles