Asymptotic stability at infinity for bidimensional Hurwitz vector fields

Let X:U → R2 be a differentiable vector field. Set Spc(X) = {eigenvalues of DX(z): z∈ U}. This X is called Hurwitz if Spc(X)⊂{z∈C:ℜ(z)<0}. Suppose that X is Hurwitz and U⊂R2 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~:R2 → R2, a topological embedding such that X~ equals X in the complement of some compact subset of U. © 2013 Elsevier Inc.