An eigenvalue condition for the injectivity and asymptotic stability at infinity
Let U → ℝ2 be a differentiable vector field defined on the complement of a compact set. We study the intrinsic relation between the asymptotic behavior of the real eigenvalues of the differential DX z and the global injectivity of the local diffeomorphism given by X. This set U induces a neighborhood of ∞ in the Riemann Sphere ℝ2 ∪ {∞}. In this work we prove the existence of a sufficient condition which implies that the vector field X🙁U, ∞) → (ℝ2, 0), —which is differentiable in U\{∞} but not necessarily continuous at ∞,—has ∞ as an attracting or a repelling singularity. This improves the main result of Gutiérrez-Sarmiento: Asterisque, 287 (2003) 89–102.
Autor(es):Rabanal, Roland
Año: 2005
Título de la revista: Qualitative Theory of Dynamical Systems
Volumen: 6
Número: 2
Página inicial - Página final: 233-250
ISSN: 1575-5460
Url: http://link.springer.com/article/10.1007%2FBF02972675