Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems
Chen, Xingwu (Sichuan University (Xina). Department of Mathematics)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Zhang, Weinian (Sichuan University (Xina). Department of Mathematics)

Date:	2017
Abstract:	It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in \R^2 is 3. In contrast here we consider discontinuous differential systems in \R^2 defined in two half--planes separated by a straight line. In one half plane we have a general linear center at the origin of \R^2, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of \R^2. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function.
Grants:	Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 European Commission 318999 Ministerio de Economía y Competitividad MTM2013-40998-P Ministerio de Economía y Competitividad MTM2016-77278-P
Note:	Agraïments: The first author is supported by NSFC grant #11471228. The third author is supported by NSFC grants #11231001, #11221101.
Rights:	Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Cyclicity ; Discontinuous differential system ; Hopf bifurcation ; Limit cycles
Published in:	Discrete and continuous dynamical systems. Series B, Vol. 22 Núm. 10 (2017) , p. 3953-3965, ISSN 1553-524X

DOI: 10.3934/dcdsb.2017203

Postprint 15 p, 328.5 KB

The record appears in these collections:
Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles