平面可微向量场在无穷远点的渐近稳定性
Asymptotic stability at infinity for differentiable vector fields of the plane
摘要：
设X：R2 \Dr ->R2是可微（但不一定是C1）向量场，其中r>0且Dr={z \in R2：|z| \le r}。如果对某个e>0和对所有p \in R2 \Dr，D_p X的特征值不属于（-e，0] \cup {z \in \C：\mathcal{R}（z）\ge 0}，则（a）对于所有p \in R2 \Dr，存在从p开始的X的唯一正半轨线;（b）\mathcal{I}（X），X在无穷远处的指数，是扩展实数线[- \infty，\infty]的一个明确定义的数;（c）存在一个常数矢量v \in R2，使得如果\mathcal{I}（X）小于零（分别为大于或等于零），则黎曼球面R2的无穷远点\cup是排斥者（分别为向量场X+ v的吸引子。
英文摘要：
Let X:R2\Dr->R2 be a differentiable (but not necessarily C1) vector field, where r>0 and Dr={z\in R2:|z|\le r}. If for some e>0 and for all p\in R2\Dr, no eigenvalue of D_p X belongs to (-e,0]\cup {z\in\C:\mathcal{R}(z)\ge 0}, then (a)For all p\in R2\Dr, there is a unique positive semi--trajectory of X starting at p; (b)\mathcal{I}(X), the index of X at infinity, is a well defined number of the extended real line [-\infty,\infty); (c) There exists a constant vector v\in R2 such that if \mathcal{I}(X) is less than zero (resp. greater or equal to zero), then the point at infinity \infty of the Riemann sphere R2\cup\set{\infty} is a repellor (resp. an attractor) of the vector field X+v.
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