Let X : R2 {set minus} over(D, -)σ → R2 be a differentiable (but not necessarily C1) vector field, where σ > 0 and over(D, -)σ = {z ∈ R2 : {norm of matrix} z {norm of matrix} ≤ σ}. Denote by R (z) the real part of z ∈ C. If for some ε{lunate} > 0 and for all p ∈ R2 {set minus} over(D, -)σ, no eigenvalue of Dp X belongs to (- ε{lunate}, 0] ∪ {z ∈ C : R (z) ≥ 0}, then: (a) for all p ∈ R2 {set minus} over(D, -)σ, 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 [- ∞, ∞) (called the index of X at infinity) such that for some constant vector v ∈ R2 the following is satisfied: if I (X) is less than zero (respectively greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R2 ∪ {∞} is a repellor (respectively an attractor) of the vector field X + v. © 2006 Elsevier Inc. All rights reserved.