# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022034
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## Limit cycles of continuous piecewise differential systems separated by a parabola and formed by a linear center and a quadratic center

 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

*Corresponding author: Jaume Llibre

This paper is dedicated to my good friend Professor Jibin Li for his 80th birthday

Received  December 2021 Revised  January 2022 Early access February 2022

Fund Project: The author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00 (FEDER), and the H2020 European Research Council grant MSCA-RISE-2017-777911

Due to their applications to many physical phenomena during these last decades the interest for studying the continuous or discontinuous piecewise differential systems has increased strongly. The limit cycles play a main role in the study of any planar differential system. Up to now the major part of papers which study the limit cycles of the planar piecewise differential systems have considered systems formed by two pieces separated by one straight line.

Here we consider planar continuous piecewise differential systems separated by a parabola.

We prove that the planar continuous piecewise differential systems separated by a parabola and formed by a linear center and a quadratic center have at most one limit cycle. Moreover there are systems in this class exhibiting one limit cycle. So in particular we have solved the extension of the 16th Hilbert problem to this class of differential systems.

Citation: Jaume Llibre. Limit cycles of continuous piecewise differential systems separated by a parabola and formed by a linear center and a quadratic center. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022034
##### References:
 [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford-New York-Toronto, Ont. 1966. [2] N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type, American Math. Soc., 1954 (1954), 19 pp. [3] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci., vol. 163, Springer-Verlag, London, 2008. [4] D. C. Braga and L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dynam., 73 (2013), 1283-1288.  doi: 10.1007/s11071-013-0862-3. [5] C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936.  doi: 10.3934/dcds.2013.33.3915. [6] V. Carmona, F. Fernández-Sánchez and D. D. Novaes, A new simple proof for's the Lum–Chua conjecture revisited, Nonlinear Anal. Hybrid Syst., 40 (2021), Paper No. 100992, 7 pp. doi: 10.1016/j.nahs.2020.100992. [7] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer-Verlag, New York, 2006. [8] E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073-2097.  doi: 10.1142/S0218127498001728. [9] E. Freire, E. Ponce and F. Torres, A general mechanism to generate three limit cycles in planar Filippov systems with two zones, Nonlinear Dynamics, 78 (2014), 251-263.  doi: 10.1007/s11071-014-1437-7. [10] F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632.  doi: 10.1088/0951-7715/14/6/311. [11] M. Han and J. Yang, The maximum number of zeros of functions with parameters and application to differential equations, J. Nonlinear Modeling and Analysis, 3 (2021), 13-34. [12] M. Han and W. Zhang, On Hopf bifurcation in non–smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002. [13] D. Hilbert, Mathematische probleme, Bull. Amer. Math. Soc. (N.S.), 37 (2000), 407-436.  doi: 10.1090/S0273-0979-00-00881-8. [14] S. M. Huan and X. S. Yang, On the number of limit cycles in general planar piecewise systems, Discrete Cont. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147. [15] Y. Ilyashenko, Centennial history of Hilbert's $16$th problem, Bull. Amer. Math. Soc. (N.S.), 39 (2002), 301-354.  doi: 10.1090/S0273-0979-02-00946-1. [16] W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, (Dutch), (1911), 1446–1457. [17] W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20 (1912), 1354-1365. [18] J. Li, Hilbert's $16$th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.  doi: 10.1142/S0218127403006352. [19] L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ., (2014), 14 pp. doi: 10.14232/ejqtde.2014.1.70. [20] J. Llibre, D. D. Novaes and M. A. Teixeira, Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1550144, 11 pp. doi: 10.1142/S0218127415501448. [21] J. Llibre, M. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in planar piecewise linear systems without symmetry, Nonlinear Anal. Real World Appl., 14 (2013), 2002-2012.  doi: 10.1016/j.nonrwa.2013.02.004. [22] J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. Contin. Discrete Impuls. Syst. Ser., 19 (2012), 325-335. [23] J. Llibre and M. A. Teixeira, Piecewise linear differential systems without equilibria produce limit cycles?, Nonlinear Dyn., 88 (2017), 157-164.  doi: 10.1007/s11071-016-3236-9. [24] J. Llibre and M. A. Teixeira, Piecewise linear differential systems with only centers can create limit cycles?, Nonlinear Dyn., 91 (2018), 249-255.  doi: 10.1007/s11071-017-3866-6. [25] R. Lum and L. O. Chua, Global properties of continuous piec ewise-linear vector fields. Part I: Simplest case in $\mathbb{R}^2$, Internat. J. Circuit Theory Appl., 19 (1991), 251-307.  doi: 10.1002/cta.4490190305. [26] R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part II: Simplest symmetric case in $\mathrm{R}^2$, Internat. J. Circuit Theory Appl., 20 (1992), 9-46.  doi: 10.1002/cta.4490200103. [27] O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Phys. D, 241 (2012), 1826-1844.  doi: 10.1016/j.physd.2012.08.002. [28] D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc., 338 (1993), 799-841.  doi: 10.1090/S0002-9947-1993-1106193-6. [29] D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Sci. Ser. Nonlinear Sci. Ser. A, vol. 70, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/7612. [30] N. I. Vulpe, Affine–invariant conditions for the topological discrimination of quadratic systems with a center, Differentsial' nye Uravneniya, 19 (1983), 371-379.

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##### References:
 [1] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford-New York-Toronto, Ont. 1966. [2] N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type, American Math. Soc., 1954 (1954), 19 pp. [3] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci., vol. 163, Springer-Verlag, London, 2008. [4] D. C. Braga and L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dynam., 73 (2013), 1283-1288.  doi: 10.1007/s11071-013-0862-3. [5] C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936.  doi: 10.3934/dcds.2013.33.3915. [6] V. Carmona, F. Fernández-Sánchez and D. D. Novaes, A new simple proof for's the Lum–Chua conjecture revisited, Nonlinear Anal. Hybrid Syst., 40 (2021), Paper No. 100992, 7 pp. doi: 10.1016/j.nahs.2020.100992. [7] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer-Verlag, New York, 2006. [8] E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073-2097.  doi: 10.1142/S0218127498001728. [9] E. Freire, E. Ponce and F. Torres, A general mechanism to generate three limit cycles in planar Filippov systems with two zones, Nonlinear Dynamics, 78 (2014), 251-263.  doi: 10.1007/s11071-014-1437-7. [10] F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632.  doi: 10.1088/0951-7715/14/6/311. [11] M. Han and J. Yang, The maximum number of zeros of functions with parameters and application to differential equations, J. Nonlinear Modeling and Analysis, 3 (2021), 13-34. [12] M. Han and W. Zhang, On Hopf bifurcation in non–smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002. [13] D. Hilbert, Mathematische probleme, Bull. Amer. Math. Soc. (N.S.), 37 (2000), 407-436.  doi: 10.1090/S0273-0979-00-00881-8. [14] S. M. Huan and X. S. Yang, On the number of limit cycles in general planar piecewise systems, Discrete Cont. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147. [15] Y. Ilyashenko, Centennial history of Hilbert's $16$th problem, Bull. Amer. Math. Soc. (N.S.), 39 (2002), 301-354.  doi: 10.1090/S0273-0979-02-00946-1. [16] W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, (Dutch), (1911), 1446–1457. [17] W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20 (1912), 1354-1365. [18] J. Li, Hilbert's $16$th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.  doi: 10.1142/S0218127403006352. [19] L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ., (2014), 14 pp. doi: 10.14232/ejqtde.2014.1.70. [20] J. Llibre, D. D. Novaes and M. A. Teixeira, Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1550144, 11 pp. doi: 10.1142/S0218127415501448. [21] J. Llibre, M. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in planar piecewise linear systems without symmetry, Nonlinear Anal. Real World Appl., 14 (2013), 2002-2012.  doi: 10.1016/j.nonrwa.2013.02.004. [22] J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. Contin. Discrete Impuls. Syst. Ser., 19 (2012), 325-335. [23] J. Llibre and M. A. Teixeira, Piecewise linear differential systems without equilibria produce limit cycles?, Nonlinear Dyn., 88 (2017), 157-164.  doi: 10.1007/s11071-016-3236-9. [24] J. Llibre and M. A. Teixeira, Piecewise linear differential systems with only centers can create limit cycles?, Nonlinear Dyn., 91 (2018), 249-255.  doi: 10.1007/s11071-017-3866-6. [25] R. Lum and L. O. Chua, Global properties of continuous piec ewise-linear vector fields. Part I: Simplest case in $\mathbb{R}^2$, Internat. J. Circuit Theory Appl., 19 (1991), 251-307.  doi: 10.1002/cta.4490190305. [26] R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part II: Simplest symmetric case in $\mathrm{R}^2$, Internat. J. Circuit Theory Appl., 20 (1992), 9-46.  doi: 10.1002/cta.4490200103. [27] O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Phys. D, 241 (2012), 1826-1844.  doi: 10.1016/j.physd.2012.08.002. [28] D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc., 338 (1993), 799-841.  doi: 10.1090/S0002-9947-1993-1106193-6. [29] D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Sci. Ser. Nonlinear Sci. Ser. A, vol. 70, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. doi: 10.1142/7612. [30] N. I. Vulpe, Affine–invariant conditions for the topological discrimination of quadratic systems with a center, Differentsial' nye Uravneniya, 19 (1983), 371-379.
The limit cycle of the planar continuous piecewise differential system separated by the parabola $y = x^2$ and formed by the linear center (26) and the quadratic center (27). This limit cycle is traveled in counterclockwise
The graphic of the function $f(x_2)$. The horitzontal straigh line is the $x_2$-axis
The graphic of the function $g(x_2)$. The horitzontal straigh line is the $x_2$-axis. In the case $A<0$, $2$ real roots eventually the minimum can be positive
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