Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
Huang, Jianfeng (Jinan University. Department of Mathematics)
Liang, Haihua (Guangdong Polytechnic Normal University. School of Mathematics and Systems Science)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2018
Abstract:	In this paper we study the limit cycles of the planar polynomial differential systems * x=ax-y P_n(x,y),\\ y=x ay Q_n(x,y), * where P_n and Q_n are homogeneous polynomials of degree n2, and a R. Consider the functions * &()=P_n() Q_n()\\ &()=Q_n()-P_n()\\ &_1()=a()-(),\\ &_2()=(n-1)(2a()-()) '(). *First we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of _1 and _2 with definite sign. This result improves previous knwon results. Furthermore, if _1(_1a-_2)0 for some _1,_20, we provide necessary and sufficient conditions for the non-existence, and the existence and uniqueness of the limit cycles of these differential systems. When one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin.
Grants:	Ministerio de Economía y Competitividad MTM2016-77278-P Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Homogeneous nonlinearities ; Limit cycles ; Non-existence and uniqueness ; Polynomial differential systems
Published in:	Journal of differential equations, Vol. 265, issue 9 (Nov. 2018) , p. 3888-3913, ISSN 1090-2732

DOI: 10.1016/j.jde.2018.05.019

Postprint 25 p, 473.1 KB

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