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Limit Cycles Bifurcating from an Invisible Fold–Fold in Planar Piecewise Hamiltonian Systems

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Abstract

The aim of this article is twofold. Firstly, we study the existence of limit cycles in a family of piecewise smooth vector fields corresponding to an unfolding of an invisible fold–fold singularity. More precisely, given a positive integer k, we prove that this family has exactly k hyperbolic crossing limit cycles in a suitable neighborhood of this singularity. Secondly, we provide a complete study relating the existence and stability of these crossing limit cycles with the limit cycles of the family of smooth vector fields obtained by the regularization method. This relationship is obtained by studying the equivalence between the signs of the Lyapunov coefficients of the family of piecewise smooth vector fields and the ones of its regularization.

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Funding

The first, second, and fourth authors are partially supported by Fundação de Amparo à Pesquisa do Estado de Minas Gerais – FAPEMIG [grant number APQ–01158–17]. The third author was financed in part by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – CAPES [Finance Code 001].

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Authors and Affiliations

  1. Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, CEP 37.500–903, Itajubá, MG, Brazil

    Denis de Carvalho Braga, Alexander Fernandes da Fonseca & Luis Fernando Mello

  2. Instituto de Biociências, Letras e Ciências Exatas, UNESP, Rua Cristóvão Colombo, 2265, CEP 15054–000, São José do Rio Preto, SP, Brazil

    Luiz Fernando Gonçalves

Authors
  1. Denis de Carvalho Braga
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  2. Alexander Fernandes da Fonseca
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  3. Luiz Fernando Gonçalves
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  4. Luis Fernando Mello
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Correspondence to Luis Fernando Mello.

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de Carvalho Braga, D., Fernandes da Fonseca, A., Gonçalves, L.F. et al. Limit Cycles Bifurcating from an Invisible Fold–Fold in Planar Piecewise Hamiltonian Systems. J Dyn Control Syst 27, 179–204 (2021). https://doi.org/10.1007/s10883-020-09478-2

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  • DOI: https://doi.org/10.1007/s10883-020-09478-2

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