Asymptotic stability at infinity for bidimensional Hurwitz vector fields

Let X : U -> R-2 be a differentiable vector field. Set Spc(X) = {eigenvalues of DX(z): z is an element of U}. This X is called Hurwitz if Spc(X) subset of {z is an element of C: R(z) < 0}. Suppose that X is Hurwitz and U subset of 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. That means: (i) there exists a unbounded sequence of closed curves, pairwise bounding an annulus the boundary of which is transversal to X + v, and (ii) there is a neighborhood of infinity with unbounded trajectories, free of singularities and periodic trajectories of X + v. This result is obtained after to proving the existence of <(X)over tilde> : R-2 -> R-2, a topological embedding such that (X) over tilde equals X in the complement of some compact subset of U. (C) 2013 Elsevier Inc. All rights reserved.