Asymptotic stability at infinity for differentiable vector fields of the plane
Let X:ℝ2\Dσ→ℝ2 be a differentiable (but not necessarily C1) vector field, where σ>0 and Dσ={z∈ℝ2: ∥z∥≤σ}. Denote by R(z) the real part of z∈ℂ. If, for some ϵ>0 and for all p∈ℝ2∖Dσ, no eigenvalue of DpX belongs to (−ϵ,0]∪{z∈ℂ: R(z)≥0}, then: (a) for all p∈ℝ2∖Dσ, there is a unique positive semi-trajectory of X starting at p; (b) there 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 than 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.
Autor(es):Gutierrez, Carlos
Pires, Benito
Rabanal, Roland
Año: 2006
Título de la revista: Journal of Differential Equations
Volumen: 231
Página inicial - Página final: 165-181
ISSN: 0022-0396
Url: https://www.sciencedirect.com/science/article/pii/S0022039606003020