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Pàgina inicial > Articles > Articles publicats > Vector fields with homogeneous nonlinearities and many limit cycles |
Data: | 2015 |
Resum: | 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. |
Ajuts: | Ministerio de Ciencia y Tecnología MTM 2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-410 European Commission 316338 |
Nota: | 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. |
Drets: | Tots els drets reservats. |
Llengua: | Anglès |
Document: | Article ; recerca ; Versió acceptada per publicar |
Matèria: | Limit cycles ; Nilpotent singularity ; Polynomial differential equations |
Publicat a: | Journal of differential equations, Vol. 258 (2015) , p. 3286-3303, ISSN 1090-2732 |
Postprint 19 p, 397.8 KB |