Asymptotic stability at infinity for differentiable vector fields of the plane

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 D_p 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.