Home > Articles > Published articles > Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities |
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 |
Postprint 25 p, 473.1 KB |