Mathematica Bohemica, first online, pp. 1-13

On limit cycles of piecewise differential systems formed by arbitrary linear systems and a class of quadratic systems

Aziza Berbache

Received November 26, 2021.   Published online December 20, 2022.

Abstract:  We study the continuous and discontinuous planar piecewise differential systems separated by a straight line and formed by an arbitrary linear system and a class of quadratic center. We show that when these piecewise differential systems are continuous, they can have at most one limit cycle. However, when the piecewise differential systems are discontinuous, we show that they can have at most two limit cycles, and that there exist such systems with two limit cycles. Therefore, in particular, we have solved the extension of the 16th Hilbert problem to this class of differential systems.
Keywords:  discontinuous piecewise differential system; continuous piecewise differential system; first integral; non-algebraic limit cycle; linear system; quadratic center
Classification MSC:  34C05, 34C07, 37G15

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[1] J. C. Artés, J. Llibre, J. C. Medrado, M. A. Teixeira: Piecewise linear differential systems with two real saddles. Math. Comput. Simul. 95 (2014), 13-22. DOI 10.1016/j.matcom.2013.02.007 | MR 3127752 | Zbl 07312523
[2] A. Berbache: Two limit cycles for a class of discontinuous piecewise linear differential systems with two pieces. Sib. Èlectron. Mat. Izv. 17 (2020), 1488-1515. DOI 10.33048/semi.2020.17.104 | MR 4229126 | Zbl 1452.34001
[3] A. Berbache: Two explicit non-algebraic crossing limit cycles for a family of piecewise linear systems. Mem. Differ. Equ. Math. Phys. 83 (2021), 13-29. MR 4281028 | Zbl 1473.34033
[4] X. Chen, Z. Du: Limit cycles bifurcate from centers of discontinuous quadratic systems. Comput. Math. Appl. 59 (2010), 3836-3848. DOI 10.1016/j.camwa.2010.04.019 | MR 2651857 | Zbl 1198.34044
[5] D. de Carvalho Braga, L. F. Mello: Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane. Nonlinear Dyn. 73 (2013), 1283-1288. DOI 10.1007/s11071-013-0862-3 | MR 3083780 | Zbl 1281.34037
[6] M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk: Piecewise-Smooth Dynamical Systems: Theory and Applications. Appled Mathematical Sciences 163. Springer, London (2008). DOI 10.1007/978-1-84628-708-4 | MR 2368310 | Zbl 1146.37003
[7] A. F. Filippov: Differential Equations with Discontinuous Right-Hand Sides. Mathematics and Its Applications: Soviet Series 18. Kluwer Academic, Dordrecht (1988). DOI 10.1007/978-94-015-7793-9 | MR 1028776 | Zbl 0664.34001
[8] E. Freire, E. Ponce, F. Rodrigo, F. Torres: Bifurcation sets of continuous piecewise linear systems with two zones. Int. J. Bifurcation Chaos Appl. Sci. Eng. 8 (1998), 2073-2097. DOI 10.1142/S0218127498001728 | MR 1681463 | Zbl 0996.37065
[9] E. Freire, E. Ponce, F. Torres: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11 (2012), 181-211. DOI 10.1137/11083928X | MR 2902614 | Zbl 1242.34020
[10] D. Hilbert: Mathematical problems. Bull. Am. Math. Soc., New Ser. 37 (2000), 407-436. DOI 10.1090/S0273-0979-00-00881-8 | MR 1779412 | Zbl 0979.01028
[11] S.-M. Huan, X.-S. Yang: Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics. Nonlinear Anal., Theory Methods Appl., Ser. A 92 (2013), 82-95. DOI 10.1016/ | MR 3091110 | Zbl 1309.34042
[12] S.-M. Huan, X.-S. Yang: On the number of limit cycles in general planar piecewise linear systems of node-node types. J. Math. Anal. Appl. 411 (2014), 340-353. DOI 10.1016/j.jmaa.2013.08.064 | MR 3118489 | Zbl 1323.34022
[13] L. Li: Three crossing limit cycles in planar piecewise linear systems with saddle-focus type. Electron. J. Qual. Theory Differ. Equ. 2014 (2014), Article ID 70, 14 pages. DOI 10.14232/ejqtde.2014.1.70 | MR 3304196 | Zbl 1324.34025
[14] J. Llibre, A. C. Mereu: Limit cycles for discontinuous quadratic differential systems with two zones. J. Math. Anal. Appl. 413 (2014), 763-775. DOI 10.1016/j.jmaa.2013.12.031 | MR 3159803 | Zbl 1318.34049
[15] J. Llibre, M. Ordóñez, E. Ponce: On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry. Nonlinear Anal., Real World Appl. 14 (2013), 2002-2012. DOI 10.1016/j.nonrwa.2013.02.004 | MR 3043136 | Zbl 1293.34047
[16] J. Llibre, E. Ponce: Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 19 (2012), 325-335. MR 2963277 | Zbl 1268.34061
[17] R. Lum, L. O. Chua: Global properties of continuous piecewise linear vector fields. I: Simplest case in $R^2$. Int. J. Circuit Theory Appl. 19 (1991), 251-307. DOI 10.1002/cta.4490190305 | Zbl 0732.34029
[18] R. Lum, L. O. Chua: Global properties of continuous piecewise linear vector fields. II: Simplest symmetric case in $R^2$. Int. J. Circuit Theory Appl. 20 (1992), 9-46. DOI 10.1002/cta.4490200103 | Zbl 0732.34029
[19] Y. Xiong: Limit cycles bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters. J. Math. Anal. Appl. 421 (2015), 260-275. DOI 10.1016/j.jmaa.2014.07.013 | MR 3250477 | Zbl 1405.34026

Affiliations:   Aziza Berbache, University Mohamed El Bachir El Ibrahimi of Bordj Bou Arreridj, Faculty of Mathematics and Informatics, Department of Mathematics, El Anasser 34265, Algeria, e-mail:

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