Abstract
In planar piecewise differential systems it is known that when the discontinuity curve is a straight line and both differential systems are linear centers, these piecewise differential systems have no limit cycles but if they are separated by other types of discontinuity curves, such as parabolas, then they have limit cycles. All these results are in the plane and although the qualitative theory of planar piecewise differential systems has been the subject of many research, this is not the case for piecewise differential systems in higher dimensions. In this paper, we study the maximum number of limit cycles of discontinuous piecewise differential systems in \(\mathbb {R}^3\) separated by a paraboloid (elliptic or hyperbolic), and formed by what we call two linear differential centers. We prove that these systems can have at most one limit cycle and that this upper bound is reached. We also provide systems of these types without periodic solutions and with a continuum of periodic solutions.
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The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Andronov, A., Vitt, A.M., Khaikin, S.: Theory of Oscillations. Pergamon Press, Oxford (1966)
Benterki, R., Damene, L., Llibre, J.: The limit cycles of discontinuous piecewise linear differential systems formed by centers and separated by irreducible cubic curves II. Differ. Equ. Dyn, Syst (2021)
Brunella, M.: Instability of equilibria in dimension three. Annales de l’Institut Fourier 48(5), 1345–1357 (1998)
Buzzi, C., Romano, Y., Llibre, J.: Crossing limit cycles of planar discontinuous piecewise differential systems formed by isochronous centres. Dyn. Syst. 37(4), 710–728 (2022)
di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences Series, vol. 163. Springer-Verlag, London (2008)
Dulac, H.: Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre. Bull. Sci. math. Sér. 2(32), 230–252 (1908)
Jimenez, J.J., Llibre, J., Medrado, J.C.: Crossing limit cycles for a class of piecewise linear differential centers separated by a conic. Electron. J. Differ. Equ. 41, 36 (2020)
Jimenez, J.J., Llibre, J., Medrado, J.C.: Crossing limit cycles for piecewise linear differential centers separated by a reducible cubic curve. Electron. J. Qual. Theory Differ. Equ. 19, 48 (2020)
Llibre, J., Nuñez, E., Teruel, A.E.: Limit cycles for planar piecewise linear differential systems via first integrals. Qual. Theory Dyn. Syst. 3(1), 29–50 (2002)
Llibre, J., Teixeira, M.A.: Periodic orbits of continuous and discontinuous piecewise linear differential systems via first integrals. São Paulo J. Math. Sci. 12(1), 121–135 (2018)
Llibre, J., Teixeira, M.A.: Piecewise linear differential systems with only centers can create limit cycles? Nonlinear Dyn. 91, 249–255 (2018)
Llibre, J., Tonon, D., Velter, M.: Crossing periodic orbits via first integrals. Int. J. Bifurc. Chaos 30(11), 2050163 (2020)
Llibre, J., Valls, C.: Crossing limit cycles for discontinuous piecewise differential systems formed by linear Hamiltonian saddles or linear centers separated by a conic. Chaos Solit. Fractals 159, 112076 (2022)
Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Physica D 241, 1826–1844 (2012)
Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle. J. de Mathématiques 37, 375–422 (1881)
Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle. J. de Mathématiques 8, 251–296 (1882)
Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle, Oeuvres de Henri Poincaré, vol. I, pp. 3–84. Gauthier-Villars, Paris (1951)
Simpson, D.J.W.: Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol. 69. World Scientific, Singapore (2010)
Van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701–710 (1920)
Van der Pol, B.: On relaxation-oscillations. Lond. Edinb. Dublin Philos. Mag. J. Sci. 2(7), 978–992 (1926)
Acknowledgements
The first author is supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00. The second author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is partially supported through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020.
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Jimenez, J., Llibre, J. & Valls, C. Limit Cycles for Discontinuous Piecewise Differential Systems in \(\mathbb {R}^3\) Separated by a Paraboloid. Differ Equ Dyn Syst (2023). https://doi.org/10.1007/s12591-023-00668-5
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DOI: https://doi.org/10.1007/s12591-023-00668-5