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Creation of Limit Cycles in Piecewise Smooth Vector Fields Tangent to Nested Tori

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Abstract

The main goal of this paper is to present the behavior generated by piecewise smooth vector fields tangent to foliations. We consider two smooth foliations \(\mathcal {F}_1\) and \(\mathcal {F}_2\) that are coupled to produce a foliation composed by nested topological tori. Moreover, a piecewise smooth vector field composed of periodic orbits and tangent to these tori is considered. We perturb this piecewise vector field (and, consequently, the foliations) and the birth of either finite or infinitely many limit cycles for the 3D piecewise smooth vector field is observed.

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References

  1. Andronov, A., Vitt, A., Khaikin, S.: Theory of Oscillators. Adiwes International Series in Physics. Pergamon Press (1966)

  2. Arnold, V.I.: Mathematical methods of classical mechanics, volume 60 of Graduate Texts in Mathematics. Springer, New York (1978)

  3. Belev, S.A., Tyurin, N.A.: Nontoric foliations by Lagrangian tori of toric Fano varieties. Math. Notes 87(1), 43–51 (2010)

    Article  MathSciNet  Google Scholar 

  4. Buzzi, C.A., de Carvalho, T., Teixeira, M.A.: Birth of limit cycles bifurcating from a nonsmooth center. J. Math. Pures Appl. 102(1), 36–47 (2014)

    Article  MathSciNet  Google Scholar 

  5. Camacho, C., Lins Neto, A.: Geometric Theory of Foliations, 1st edn. Birkhauser (1985)

  6. Carvalho, T., Cristiano, R., Gonçalves, L.F., Tonon, D.: Global analysis of the dynamics of a mathematical model to intermittent HIV treatment. Nonlinear Dyn. 101, 719–739 (2020)

    Article  Google Scholar 

  7. Carvalho, T., Novaes, D.D., Gonçalves, L.F.: Sliding Shilnikov connection in Filippov-type predator-prey model. Nonlinear Dyn. 100, 2973–2987 (2020)

    Article  Google Scholar 

  8. Carvalho, T., Teixeira, M.A.: On piecewise smooth vector fields tangent to nested tori. J. Differ. Equ. 261(7), 4008–4029 (2016)

    Article  MathSciNet  Google Scholar 

  9. Chicone, C.: Ordinary Differential Equations with Applications, volume 34 of Texts in Applied Mathematics. Springer, New York (1999)

  10. Chua, L., Komuro, M., Matsumoto, T.: The double scroll family. IEEE Trans. Circuits Syst. 33(11), 1072–1118 (1986)

    Article  Google Scholar 

  11. di Bernardo, M., Budd, C.J., Champneys, A.R. Kowalczyk, P.: Piecewise-smooth Dynamical Systems: Theory and Applications, 1st edn, vol 163. Springer, London (2008)

  12. di Bernardo, M., Johansson, K.H., Vasca, F.: Self-oscillations and sliding in relay feedback systems: symmetry and bifurcations. Int. J. Bifurc. Chaos 11(04), 1121–1140 (2001)

    Article  Google Scholar 

  13. Ekeland, I.: Discontinuites de champs Hamiltoniens et existence de solutions optimales en calcul des variations. Publications Mathématiques, Inst. Hautes Étud. Sci. 47, 5–32 (1977)

  14. Erdi, P., Tóth, J.: Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models. Nonlinear Science, Theory and Application. Manchester University Press (1989)

  15. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Mathematics and its Applications, 1st edn. Kluwer Academic Publishers (1988)

  16. Leine, R., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems, 1st edn. Springer, Berlin (2004)

    Book  Google Scholar 

  17. Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Physica D 241(22), 1826–1844 (2012)

    Article  MathSciNet  Google Scholar 

  18. Panazzolo, D., da Silva, P.R.: Regularization of discontinuous foliations: blowing up and sliding conditions via Fenichel theory. J. Differ. Equ. 263(12), 8362–8390 (2017)

    Article  MathSciNet  Google Scholar 

  19. Scofield, P.D.: Some deformations of the Hopf foliation are also Kähler. Proc. Am. Math. Soc. 119(1), 251–253 (1993)

    MathSciNet  MATH  Google Scholar 

  20. Simpson, D.J.W.: Bifurcations in Piecewise-Smooth Continuous Systems, volume 69 of World Scientific Series on Nonlinear Science, Series A. World Scientific Publishing (2010)

  21. Tsukada, K.: The canonical foliation of a compact generalized Hopf manifold. Differ. Geom. Appl. 11(1), 13–28 (1999)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Tiago Carvalho is partially supported by São Paulo Research Foundation (FAPESP grants 2019/10450-0 and 2019/10269-3) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq Grant 304809/2017-9). The second author was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior CAPES [Finance Code 001].

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  1. Departamento de Computação e Matemática; Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Avenida Bandeirantes 3900, Ribeirão Preto, SP, 14040-901, Brazil

    Tiago Carvalho

  2. Instituto de Biociências, Letras e Ciências Exatas, Universidade Estadual Paulista, Rua Cristóvão Colombo 2265, São José do Rio Preto, SP, 15054-000, Brazil

    Luiz Fernando Gonçalves

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  1. Tiago Carvalho
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  2. Luiz Fernando Gonçalves
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Correspondence to Luiz Fernando Gonçalves.

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Carvalho, T., Gonçalves, L.F. Creation of Limit Cycles in Piecewise Smooth Vector Fields Tangent to Nested Tori. Qual. Theory Dyn. Syst. 20, 55 (2021). https://doi.org/10.1007/s12346-021-00491-9

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