Limit cycles in planar discontinuous piecewise linear Hamiltonian systems with three equal sectors

Ali Bakhshalizadeh; Jaume Llibre

doi:10.1063/5.0315024

journal article Apr 01, 2026

Limit cycles in planar discontinuous piecewise linear Hamiltonian systems with three equal sectors

Ali Bakhshalizadeh

Jaume Llibre

Chaos: An Interdisciplinary Journal of Nonlinear Science Vol. 36 No. 4 · AIP Publishing

View at Publisher Save 10.1063/5.0315024

Abstract

In these last decades, the interest on the discontinuous piecewise differential systems has increased strongly, mainly due to their big number of applications. In their study, the existence or not of limit cycles play a main role. In this paper, we study a class of planar discontinuous piecewise differential systems composed of three equal sectors of angle 2π/3, where each sector is governed by a distinct linear Hamiltonian system. The discontinuity set consists of three rays, which are the boundaries of these three sectors. We prove that such differential systems can exhibit at most three crossing limit cycles. Furthermore, we construct an explicit example illustrating the existence of a discontinuous piecewise differential system that attains this upper bound. So, we have solved the extended 16th Hilbert problem for these classes of discontinuous piecewise differential systems.

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Details

Published: Apr 01, 2026
Vol/Issue: 36(4)

Authors

A

Ali Bakhshalizadeh

Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University 1 , Tehran,

J

Jaume Llibre

Departament de Matematiques 2 , Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia,; Reial Acadèmia de Ciències i Arts de Barcelona 3 , La Rambla 115, 08002 Barcelona, Catalonia,

Funding

Agencia Estatal de Investigación Award: PID2022-136613NB-100

Cite This Article

Ali Bakhshalizadeh, Jaume Llibre (2026). Limit cycles in planar discontinuous piecewise linear Hamiltonian systems with three equal sectors. Chaos: An Interdisciplinary Journal of Nonlinear Science, 36(4). https://doi.org/10.1063/5.0315024

Limit cycles in planar discontinuous piecewise linear Hamiltonian systems with three equal sectors

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