Asymptotic stability at infinity for bidimensional Hurwitz vector fields

Abstract

Let $X:U-->R^2$ be a differentiable vector field. Set $Spc(X)={eigenvalues of DX(z) : z\in U}$. This $X$ is called Hurwitz if $Spc(X)\subset{z\in C:\Re(z)<0}$. Suppose that $X$ is Hurwitz and $U\subset R^2$ is the complement of a compact set. Then by adding to $X$ a constant $v$ one obtains that the infinity is either an attractor or a repellor for $X+v.$