Algebraic limit cycles of planar discontinuous piecewise linear differential systems with an angular switching boundary
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Xiong, Haichao (Sichuan University. Department of Mathematics)
Zhang, Weinian (Sichuan University. Department of Mathematics)

Date:	2025
Abstract:	Known results show that, with a θ-angular switching boundary for θ ∈(0, π], a planar piecewise linear differential system formed by two Hamiltonian linear sub-systems has no crossing algebraic limit cycles of type I, i. e. , those cycles crossing one of the two sides of the θ-angular switching boundary twice only, and at most two crossing algebraic limit cycles of type II, i. e. , those cycles crossing both sides of the θ-angular switching boundary once separately. In this paper, using the Chebyshev theory and Descartes' rule to overcome difficulties in applying Gröbner basis to solve polynomial systems, we study the number of crossing algebraic limit cycles for such a piecewise linear system having a Hamiltonian sub-system and a non-Hamiltonian sub-system. We prove that the maximum number of type I is one and the lower and upper bounds of the maximum number of type II are five and seven, respectively, and show the coexistence of type I and type II, which implies that a lower bound for the maximum number of all crossing algebraic limit cycles is six.
Grants:	Agencia Estatal de Investigación PID2022-136613NB-I00 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
Note:	Altres ajuts: Reial Acadèmia de Ciències i Arts de Barcelona
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Algebraic limit cycle ; Polynomial differential systems ; Polynomial first integral ; Rational first integral
Published in:	Journal of mathematical analysis and applications, Vol. 546, Issue 2 (June 2025) , art. 129224, ISSN 1096-0813

DOI: 10.1016/j.jmaa.2025.129224

Available from: 2027-06-30 Postprint 29 p, 593.6 KB

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles