Lower bounds for the local cyclicity of centers using high order developments and parallelization
Gouveia, Luiz Fernando (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2021
Abstract:	We are interested in small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point, that we locate at the origin. We develop a parallelization technique to study higher order developments, with respect to the parameters, of the return map near the origin. This technique is useful to study lower bounds for the local cyclicity of centers. We denote by M(n) the maximum number of limit cycles bifurcating from the origin via a degenerate Hopf bifurcation for a polynomial vector field of degree n. We get lower bounds for the local cyclicity of some known cubic centers and we prove that M(4)≥20, M(5)≥33, M(7)≥61, M(8)≥76, and M(9)≥88.
Grants:	Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617 Ministerio de Ciencia e Innovación MTM2016-77278-P European Commission 777911
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:	Small-amplitude limit cycle ; Polynomial vector field ; Center cyclicity ; Lyapunov constants ; Higher-order developments and parallelization
Published in:	Journal of differential equations, Vol. 271 (January 2021) , p. 447-479, ISSN 1090-2732

DOI: 10.1016/j.jde.2020.08.027

Postprint 30 p, 468.7 KB

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