Global asymptotic stability for differentiable vector fields of R2

(a) Let X: R-2 --> R-2 be a differentiable map (not necessarily C-1) and let Spec(X) be the set of (complex) eigenvalues of the derivative DXp when p varies in R-2. If, for some epsilon > 0, Spec(X) boolean AND [0, epsilon) = phi then X is injective. (b) Let X: R-2 --> R-2 be a differentiable vector field such that X(0) = 0 and Spec(X) subset of {z is an element of C : R(z) < 0}. Then, for all p is an element of R-2, there is a unique positive trajectory starting at p; moreover the omega-limit set of p is equal to {0}. (C) 2004 Elsevier Inc. All rights reserved.