Asymptotic Stability at Infinity for Differentiable Vector Fields of the Plane

@inproceedings{Pires2006AsymptoticSA,
title={Asymptotic Stability at Infinity for Differentiable Vector Fields of the Plane},
author={B Pires and Roland Rabanal},
year={2006}
}

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… CONTINUE READING