Vector fields with homogeneous nonlinearities and many limit cycles
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Yu, Jiang (Shanghai Jiao Tong University. Department of Mathematics)
Zhang, Xiang (Shanghai Jiao Tong University. Department of Mathematics)

Date:	2015
Abstract:	Consider planar real polynomial differential equations of the form x'=Lx X_n(x), where x=(x,y) R^2, L is a 2×2 matrix and X_n is a homogeneous vector field of degree n > 1. Most known results about these equations, valid for infinitely many n, deal with the case where the origin is a focus or a node and give either non-existence of limit cycles or upper bounds of one or two limit cycles surrounding the origin. In this paper we improve some of these results and moreover we show that for n 3 odd there are equations of this form having at least (n 1)/2 limit cycles surrounding the origin. Our results include cases where the origin is a focus, a node, a saddle or a nilpotent singularity. We also discuss a mechanism for the bifurcation of limit cycles from infinity.
Grants:	Ministerio de Ciencia y Tecnología MTM 2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-410 European Commission 316338
Note:	Agraïments: The second author is partially supported by NNSF of China grant 11431008, NSF of Shanghai grant 15ZR1423700 and NSF of Jiangsu grant BK 20131285. The third author is partially supported by NNSF of China grant 11271252, by RFDP of Higher Education of China grant 20110073110054, by innovation program of Shanghai Municipal Education Commission grant 15ZZ012.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Limit cycles ; Nilpotent singularity ; Polynomial differential equations
Published in:	Journal of differential equations, Vol. 258 (2015) , p. 3286-3303, ISSN 1090-2732

DOI: 10.1016/j.jde.2015.01.009

Postprint 19 p, 397.8 KB

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