Lower bounds for the local cyclicity for families of centers

Made available in DSpace on 2021-06-25T12:30:34Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-02-25 Catalan AGAUR Spanish Ministerio de Ciencia, Innovacion y Universidades -Agencia estatal de investigacion European Community Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21], to prove that there exists a family of cubic centers, labeled by C D-31(12) in [25], with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some crucial missing points in the arguments that we correct here. We take advantage of a better understanding of the bifurcation phenomenon in nongeneric cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorphic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields. (C) 2020 Elsevier Inc. All rights reserved. Univ Lleida, Dept Matemat, Avda Jaume II 69, Lleida 6925001, Catalonia, Spain Univ Autonoma Barcelona, Dept Matemat, Barcelona 08193, Catalonia, Spain Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Ctr Recerca Matemat, Campus Bellaterra, Barcelona 08193, Catalonia, Spain Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Catalan AGAUR: 2017SGR1617 Catalan AGAUR: 2017SGR127 Spanish Ministerio de Ciencia, Innovacion y Universidades -Agencia estatal de investigacion: MTM2017-84383-P Spanish Ministerio de Ciencia, Innovacion y Universidades -Agencia estatal de investigacion: PID2019-104658GB-I00 European Community: H2020-MSCA-RISE-2017-777911 CNPq: 200484/2015-0 FAPESP: 2020/04717-0