The planar discontinuous piecewise linear refracting systems have at most one limit cycle
Li, Shimin (Guangdong University of Finance and Economics. School of Mathematics and Statistics)
Liu, Changjian (Sun Yat-sen University. School of Mathematics (Zhuhai))
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Data:	2021
Resum:	In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles. The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus-focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.
Ajuts:	Ministerio de Economía y Competitividad MTM2013-40998-P Agencia Estatal de Investigación MTM2016-77278-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 Ministerio de Economía y Competitividad MDM-2014-0445
Drets:	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.
Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Piecewise linear systems ; Refracting systems ; Limit cycle
Publicat a:	Nonlinear Analysis: Hybrid Systems, Vol. 41 (August 2021) , art. 101045, ISSN 1751-570X

DOI: 10.1016/j.nahs.2021.101045

Postprint 17 p, 659.0 KB

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