Vector fields with homogeneous nonlinearities and many limit cycles
Vector fields with homogeneous nonlinearities and many limit cycles
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DOI：
10.1016/j.jde.2015.01.009
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发表时间：
2015-05
影响因子：
2.4
通讯作者：
A. Gasull;Jiang Yu;Xiang Zhang
中科院分区：
文献类型：
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作者：
A. Gasull;Jiang Yu;Xiang Zhang
考虑平面真实的多项式微分方程x stec = Lx + Xn（x），其中x=（x，y）∈ R2，L是2× 2矩阵，Xn是n次齐次向量场.大多数已知的结果，这些方程，有效的无穷多个n，处理的情况下，原点是一个焦点或一个节点，并给出要么不存在极限环或上限的一个或两个极限环周围的起源。本文改进了其中的一些结果，并证明了当n≥ 3时，存在这种形式的方程至少有（n+ 1）/2个极限环围绕原点.我们的结果包括的情况下，原点是一个焦点，一个节点，一个鞍点或幂零奇点。我们还讨论了从无穷远分支极限环的一种机制。
Consider planar real polynomial differential equations of the form x˙= L x+ 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.