Vector fields with homogeneous nonlinearities and many limit cycles

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:258

Article

Gasull, Armengol¹  Yu, Jiang²  Zhang, Xiang^2,3 

[1] Univ Autonoma Barcelona, Dept Matemat, E-08193 Barcelona, Catalonia, Spain

[2] Shanghai Jiao Tong Univ, Dept Math, Shanghai 200240, Peoples R China

[3] Shanghai Jiao Tong Univ, MOE LSC, Shanghai 200240, Peoples R China

关键词: Polynomial differential equations;    Node;    Focus;    Nilpotent singularity;    Limit cycle;    Homogeneous nonlinearities;   

DOI  :  10.1016/j.jde.2015.01.009

来源: Elsevier

【 摘 要 】

Consider planar real polynomial differential equations of the form (x) over dot = Lx + X-n(x), where x = (x, y) is an element of R-2, L is a 2 x 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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free