Abstract
In this paper we extend three results about polycycles (also known as graphs) of planar smooth vector field to planar non-smooth vector fields (also known as piecewise vector fields, or Filippov systems). The polycycles considered here may contain hyperbolic saddles, semi-hyperbolic saddles, saddle-nodes and tangential singularities of any degree. We determine when the polycycle is stable or unstable. We prove the bifurcation of at most one limit cycle in some conditions and at least one limit cycle for each singularity in other conditions.
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References
Andrade, K., Gomide, O., Novaes, D.: Qualitative Analysis of Polycycles in Filippov Systems, arXiv:1905.11950v2 (2019)
Andrade, K., Jeffrey, M., Martins, R., Teixeira, M.: On the Dulac’s problem for piecewise analytic vector fields. J. Diff. Equ. 4, 2259–2273 (2019)
Andronov, A., Vitt, A., Khaikin, S.: Theory of Oscillators, Translated from the Russian by F. Immirzi; translation edited and abridged by W. Fishwick. Pergamon Press, Oxford (1966)
Bonet, C., Jeffrey, M.R., Martín, P., Olm, J.M.: Ageing of an oscillator due to frequency switching. Commun. Nonlinear Sci. Numer. Simul. 102, 105950 (2021). https://doi.org/10.1016/j.cnsns.2021.105950
Buzzi, C., Carvalho, T., Euzébio, R.: On Poincaré-Bendixson theorem and non-trivial minimal sets in planar nonsmooth vector fields. Publicacions Matemàtiques 62, 113–131 (2018)
Cherkas, L.: The stability of singular cycles. Differ. Uravn. 4, 1012–1017 (1968)
Duff, G.: Limit-cycles and rotated vector fields. Ann. Math. 57, 15–31 (1953)
Dulac, H.: Sur les cycles limites. Bull. Soc. Math. France 62, 45–188 (1923)
Dumortier, F., Roussarie, R., Rousseau, C.: Elementary graphics of cyclicity 1 and 2. Nonlinearity 7, 1001–1043 (1994)
Dumortier, F., Morsalani, M., Rousseau, C.: Hilbert’s 16th problem for quadratic systems and cyclicity of elementary graphics. Nonlinearity 9, 1209–1261 (1996)
Filippov, A.: Differential Equations with Discontinuous Right-hand Sides, Translated from the Russian, Mathematics and its Applications (Soviet Series) 18. Kluwer Academic Publishers Group, Dordrecht (1988)
Gasull, A., Mañosa, V., Mañosas, F.: Stability of certain planar unbounded polycycles. J. Math. Anal. Appl. 269, 332–351 (2002)
Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York (1983)
Han, M., Wu, Y., Bi, P.: Bifurcation of limit cycles near polycycles with n vertices. Chaos, Solitons Fractals 22, 383–394 (2004)
Holmes, P.: Averaging and chaotic motions in forced oscillations. SIAM J. Appl. Math. 38, 65–80 (1980)
Jeffrey, M.: Modeling with Nonsmooth Dynamics. Springer Nature, Switzerland AG (2020)
Jeffrey, M.R., Seidman, T.I., Teixeira, M.A., Utkin, V.I.: Into higher dimensions for nonsmooth dynamical systems. Phys. D: Nonlinear Phenomena 434, 133222 (2022). https://doi.org/10.1016/j.physd.2022.133222
Kelley, A.: The stable, center-stable, center, center-unstable, unstable manifolds. J. Diff. Equ. 3, 546–570 (1967)
Marin, D., Villadelprat, J.: Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: Local setting. J. Diff. Equ. 269, 8425–8467 (2020)
Marin, D., Villadelprat, J.: Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: General setting. J. Diff. Equ. 275, 684–732 (2021)
Mourtada, A., Cyclicite finie des polycycles hyperboliques de champs de vecteurs du plan mise sous forme normale, Springer-Verlag, Berlin Heidelberg. Lecture Notes in Mathematics, vol. 1455,: Bifurcations of Planar Vector Fields, pp. 272–314. Luminy Meeting Proceedings, France (1990)
Mourtada, A.: Degenerate and non-trivial hyperbolic polycycles with two vertices. J. Diff. Equ. 113, 68–83 (1994)
Novaes, D., Rondon, G.: On limit cycles in regularized Filippov systems bifurcating from homoclinic-like connections to regular-tangential singularities. Physica D 442, 133526 (2022)
Novaes, D., Teixeira, M., Zeli, I.: The generic unfolding of a codimension-two connection to a two-fold singularity of planar Filippov systems. Nonlinearity 31, 2083–2104 (2018)
Perko, L.: Homoclinic loop and multiple limit cycle bifurcation surfaces. Trans. Amer. Math. Soc 344, 101–130 (1994)
Perko, L.: Differential Equations and Dynamical Systems Texts in Applied Mathematics. Springer-Verlag, New York (2001)
Sotomayor, J.: Curvas Definidas por Equações Diferenciais no Plano. Instituto de Matemática Pura e Aplicada, Rio de Janeiro (1981)
Ye, Y., Cai, S., Lo, C.: Theory of Limit Cycles, Providence. American Mathematical Society, R.I. (1986)
Acknowledgements
We thank to the reviewers their comments and suggestions which help us to improve the presentation of this paper. The author is supported by São Paulo Research Foundation (FAPESP), grants 2019/10269-3 and 2021/01799-9.
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Santana, P. Stability and Cyclicity of Polycycles in Non-smooth Planar Vector Fields. Qual. Theory Dyn. Syst. 22, 142 (2023). https://doi.org/10.1007/s12346-023-00838-4
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DOI: https://doi.org/10.1007/s12346-023-00838-4