Title: The limit cycles of a class of discontinuous piecewise differential systems

Authors: Louiza Baymout; Rebiha Benterki; Jaume Llibre

Addresses: Mathematical Analysis and Applications Laboratory, Department of Mathematics, University Mohamed El Bachir El Ibrahimi of Bordj Bou Arr´eridj, 34000, El Anasser, Algeria ' Mathematical Analysis and Applications Laboratory, Department of Mathematics, University Mohamed El Bachir El Ibrahimi of Bordj Bou Arr´eridj, 34000, El Anasser, Algeria ' Departament de Matematiques, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

Abstract: The determination of the maximum number of limit cycles and their possible positions in the plane is one of the most difficult problems in the qualitative theory of planar differential systems. This problem is related to the second part of the unsolved 16th Hilbert's problem. Due to their applications in modelling many natural phenomena, piecewise differential systems have recently attracted big attention. The upper bound number of limit cycles that a class of differential systems may exhibit is typically very difficult to determine. In this work we extend the second part of the 16th Hilbert's problem to the planar discontinuous piecewise differential systems separated by a straight line and formed by an arbitrary linear centre and an arbitrary cubic uniform isochronous centre. We provide for this class of piecewise differential systems an upper bound on its maximal number of limit cycles, and we prove that such an upper bound is reached.

Keywords: cubic uniform isochronous centre; linear centre; limit cycle; discontinuous piecewise differential system.

DOI: 10.1504/IJDSDE.2024.144873

International Journal of Dynamical Systems and Differential Equations, 2024 Vol.13 No.4, pp.339 - 368

Received: 02 Dec 2022
Accepted: 25 Oct 2023
Published online: 06 Mar 2025 *