On the number of limit cycles in piecewise planar quadratic differential systems
Braun, Francisco (Universidade Federal de São Carlos. Departamento de Matemática)
Pereira Costa da Cruz, Leonardo (Universidade de São Paulo. Instituto de Ciências Matemáticas e de Computação)
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2024
Abstract:	We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S1, S2, S3, and S4, as well as to non-smooth perturbations of non-smooth centers given by putting different Si's in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.
Grants:	Agencia Estatal de Investigación PID2019-104658GB-I00 Agencia Estatal de Investigación CEX2020-001084-M European Commission 777911 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
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:	Periodic solution ; Averaging method ; Non-smooth differential system
Published in:	Nonlinear Analysis: Real World Applications, Vol. 79 (October 2024) , art. 104124, ISSN 1468-1218

DOI: 10.1016/j.nonrwa.2024.104124

Available from: 2026-10-31 Postprint

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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