Asymptotic Stability at Infinity for Differentiable Vector Fields of the Plane

Let X : R\Dσ → R 2 be a differentiable (but not necessarily C) vector field, where σ > 0 and Dσ = { z ∈ R : ‖z‖ ≤ σ } . If for some ǫ > 0 and for all p ∈ R\Dσ, no eigenvalue of DpX belongs to (−ǫ, 0] ∪ {z ∈ C : R(z) ≥ 0}, then a) For all p ∈ R\Dσ, there is a unique positive semi–trajectory of X starting at p; b) I(X), the index of X at infinity, is a well defined number of the extended real line [−∞,∞); c) There exists a constant vector v ∈ R such that if I(X) is less than zero (resp. greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R ∪ {∞} is a repellor (resp. an attractor) of the vector field X + v.