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Centers and Limit Cycles of Vector Fields Defined on Invariant Spheres

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Abstract

The aim of this paper is the study of the center-focus and cyclicity problems inside the class \(\mathfrak {X}\) of 3-dimensional vector fields that admit a first integral that leaves invariant any sphere centered at the origin. We classify the centers of linear, quadratic homogeneous and a family of quadratic vector fields \(\mathcal {F}\subset \mathfrak {X}\), restricted to one of these spheres. Moreover, we show the existence of at least 4 limit cycles in family \(\mathcal {F}\).

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Acknowledgements

This work has been realized thanks to the Catalonia AGAUR 2017 SGR 1617 grant; the Spanish Ministerio de Ciencia, Innovación y Universidades - Agencia Estatal de Investigación PID2019-104658GB-I00 grant; the Brazilian CNPq 304798/2019-3 and São Paulo Research Foundation (FAPESP) 2017/08779-8, 2019/00440-7, 2019/10269-3 grants; and the Coordenação de Aperfeiçoamento de aPedsds oal de Nível Superior - Brasil (CAPES), Finance Code 001, and the European Community H2020-MSCA-RISE-2017-777911 grant.

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Authors and Affiliations

  1. Instituto de Biociências Letras e Ciências Exatas, Universidade Estadual Paulista (UNESP), São José do Rio Preto - SP, 15054-000, Brazil

    Claudio A. Buzzi & Ana Livia Rodero

  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain

    Joan Torregrosa

  3. Centre de Recerca Matemàtica, Campus de Bellaterra, 08193, Bellaterra, Barcelona, Spain

    Joan Torregrosa

Authors
  1. Claudio A. Buzzi
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  2. Ana Livia Rodero
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  3. Joan Torregrosa
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Correspondence to Joan Torregrosa.

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Communicated by Jeff Moehlis.

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Buzzi, C.A., Rodero, A.L. & Torregrosa, J. Centers and Limit Cycles of Vector Fields Defined on Invariant Spheres. J Nonlinear Sci 31, 92 (2021). https://doi.org/10.1007/s00332-021-09751-z

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