SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE

Meriem Barkat; Rebiha Benterki; Louiza Baymout

SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE

Abstract

The study of Hilbert’s 16th problem for piecewise linear differential systems has received significant attention from many researchers. It was shown that the upper bound for the maximum number of limit cycles can vary according to the configuration of the discontinuous curve. The family of discontinuous piecewise differential systems formed by linear isochronous centers or four families of quadratic isochronous centers separated by a straight line have been studied, and the authors have found at most two limit cycles. In this paper, we study the same family but instead of a straight line, we consider an irregular line separation, and we prove that there are at most five crossing limit cycles intersecting the separation curve at two points.

Keywords

References

Andronov, A., Vitt, A. and Khaikin, S., (1966), Theory of Oscillations. Pergamon Press, Oxford.
Art´es, J. C., Llibre, J., Medrado, J. C. and Teixeira, M. A., (2013), Piecewise linear differential systems with two real saddles, Math. Comput. Simul., 95, pp. 13–22.
Baymout, L., Benterki, R. and Llibre, J., (2024), Limit Cycles of the Discontinuous Piecewise Differential Systems Separated by a Nonregular Line and Formed by a Linear Center and a Quadratic One, Int. J. Bifurc. Chaos., 34, 2450058.
Belousov, B. P., (1958), Periodically acting reaction and its mechanism, Collection of abstracts on radiation medicine, Moscow, pp. 145-147.
Benterki, R. and Llibre, J., (2020), The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential systems, Nonlinear Dyn., 102, pp. 2453–2466.
Braga, D. C. and Mello, L. F., (2013), Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dyn., 73, pp. 1283–1288.
Castillo, J., Llibre, J. and Verduzco, F., (2017), The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems, Nonlinear Dyn., 90, pp. 1829–1840.
Damene, L. and Benterki, R., (2021), Limit cycles of discontinuous piecewise linear differential systems formed by centers or Hamiltonian without equilibria separated by irreducible cubics, Moroccan Journal of Pure and Applied Analysis., 7(2), pp. 248-276.

di Bernardo, M., Budd, C. J., Champneys, A. R. and Kowalczyk, P., (2008), Piecewise-Smooth Dynamical Systems, Theory and Applications, Appl. Math. Sci. Series 163, Springer-Verlag, London.
Esteban, M., Llibre., J. and Valls, C., (2021), The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line, Chaos., 31, 043112.
Georgescu, P. and Hsieh, W. H., (2007), Global Dynamics of a Predator-Prey Model with Stage Structure for the Predator, SIAM Journal on Applied Mathematics., 67(5), pp. 1379-1395.
Hilbert, D., (1900), Probleme, Mathematische, Lecture, Second Internat. Congr. Math. (Paris), Nachr. Ges. Wiss. Göttingen Math. Phys. KL, pp. 253–297.
Hilbert, D., (2000), Problems in Mathematics, Bull. (New Series) Amer. Math. Soc., 37, pp. 407-436.
Li´enard, A. M., (1982), Etude des oscillations entrenues, Revue G´en´erale del Electricit´e., 23, pp. 901-912.
Liberzon, D. and Morse, A. S., (1999), Basic problem in stability and design of switched systems, IEEE Control Systems Magazine., 19. 59-70.
Llibre, J., Teixeira, M. A. and Torregrosa, J., (2013), Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation, Int. J. Bifurc. Chaos., 23, 1350066.
Llibre, J. and Teixeira, M. A., (2017), Piecewise linear differential systems without equilibria produce limit cycles?, Nonlinear Dyn., 88, pp. 157–164.
Llibre, J. and Teixeira, M. A., (2018), Piecewise linear differential systems with only centers can create limit cycles?, Nonlinear Dyn., 91, pp. 249-255.
Llibre, J, and Teixeira, M. A., (2018), Limit cycles in Filippov systems having a circle as switching manifold, preprint.
Llibre, J. and Zhang, X., (2018), Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center, J. Math. Anal. Appl., 467, pp. 537–549.
Llibre, J. and Zhang, X., (2019), Limit cycles for discontinuous planar piecewise linear differential systems separated by an algebraic curve, Int. J. Bifurc. Chaos., 29(2), 1950017.
Loud, W. S., (1964), Behaviour of the period of solutions of certain plane autonomous systems near centers, Contrib. Differ. Equations., 3, pp. 21-36.
Makarenkov, O. and Lamb, J. S. W., (2012), Dynamics and bifurcations of nonsmooth systems, a survey.Phys., D 241, pp. 1826-1844.
Mardesic, P., Rousseau, C. and Toni, B., (1995), Linearization of isochronous centers, J.Differential Equations., 121, pp. 67-108.
Poincaré, M., (1881), M´emoire sur les courbes d´efinies par une ´equations differentielle, I. J. Math, Pures Appl, S´er., 3(7), pp. 375-422.
Poincar´e, M., (1886), M´emoire sur les courbes d´efinies par une ´equations differentielle, IV. J. Math, Pures Appl, S´er., 4(2), pp. 155-217.
Simpson, D. J. W., (2010), Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science A. 69, World Scientific, Singapore.
Van Der Pol, B., (1920), A theory of the amplitude of free and forced triode vibrations. Radio Rev, (Later Wirel. World)., 1, pp. 701–710.
Van Der Pol, B., (1926), On relaxation-oscillations. Lond. Edinb. Dublin Ppil, Mag. J. Sci., 2, pp. 78–992.

Details

Primary Language

English

Subjects

Ordinary Differential Equations, Difference Equations and Dynamical Systems

Journal Section

Research Article

Authors

Meriem Barkat This is me
0000-0003-2394-0559
Algeria

Rebiha Benterki ^*
0000-0001-6745-2747
Algeria

Louiza Baymout This is me
0000-0002-2140-8086
Algeria

Publication Date

October 1, 2025

Submission Date

October 2, 2024

Acceptance Date

April 7, 2025

Published in Issue

Year 2025 Volume: 15 Number: 10

IZ

https://izlik.org/JA73JL72WW

Cite

RIS / Bibtex

APA

Barkat, M., Benterki, R., & Baymout, L. (2025). SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE. TWMS Journal of Applied and Engineering Mathematics, 15(10), 2489-2504. https://izlik.org/JA73JL72WW

AMA

1.Barkat M, Benterki R, Baymout L. SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE. JAEM. 2025;15(10):2489-2504. https://izlik.org/JA73JL72WW

Chicago

Barkat, Meriem, Rebiha Benterki, and Louiza Baymout. 2025. “SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE”. TWMS Journal of Applied and Engineering Mathematics 15 (10): 2489-2504. https://izlik.org/JA73JL72WW.

EndNote

Barkat M, Benterki R, Baymout L (October 1, 2025) SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE. TWMS Journal of Applied and Engineering Mathematics 15 10 2489–2504.

IEEE

[1]M. Barkat, R. Benterki, and L. Baymout, “SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE”, JAEM, vol. 15, no. 10, pp. 2489–2504, Oct. 2025, [Online]. Available: https://izlik.org/JA73JL72WW

ISNAD

Barkat, Meriem - Benterki, Rebiha - Baymout, Louiza. “SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE”. TWMS Journal of Applied and Engineering Mathematics 15/10 (October 1, 2025): 2489-2504. https://izlik.org/JA73JL72WW.

JAMA

1.Barkat M, Benterki R, Baymout L. SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE. JAEM. 2025;15:2489–2504.

MLA

Barkat, Meriem, et al. “SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE”. TWMS Journal of Applied and Engineering Mathematics, vol. 15, no. 10, Oct. 2025, pp. 2489-04, https://izlik.org/JA73JL72WW.

Vancouver

1.Meriem Barkat, Rebiha Benterki, Louiza Baymout. SYSTEMS BIFURCATION OF LIMIT CYCLES FOR A FAMILY OF DISCONTINUOUS PIECEWISE ISOCHRONOUS DIFFERENTIAL SYSTEMS SEPARATED BY IRREGULAR LINE. JAEM [Internet]. 2025 Oct. 1;15(10):2489-504. Available from: https://izlik.org/JA73JL72WW