Asymptotic stability at infinity for differentiable vector fields of the plane

Let X: R-2\(D) over bar (sigma) -> R-2 be a differentiable (but not necessarily C-1) vector field, where sigma > 0 and (D) over bar (sigma) = {z is an element of R-2 : parallel to z parallel to <= sigma. Denote by R(z) the real part of z is an element of C. If for some epsilon > 0 and for all p is an element of R-2\(D) over bar (sigma), no eigenvalue of DpX belongs to (-epsilon, 0]boolean OR{z is an element of C: R(z) >= 0), then: (a) for all p is an element of R-2\(D) over bar (sigma), there is a unique positive semi-trajectory of X starting at p; (b) it is associated to X, a well-defined number I(X) of the extended real line [-infinity, infinity) (called the index of X at infinity) such that for some constant vector v is an element of R-2 the following is satisfied: if I(X) is less than zero (respectively greater or equal to zero), then the point at infinity infinity of the Riemann sphere R-2 boolean OR {infinity} fool is a repellor (respectively an attractor) of the vector field X + v. (c) 2006 Elsevier Inc. All rights reserved.