Asymptotic stability at infinity for bidimensional Hurwitz vector fields
Let X:U→ℝ2 be a differentiable vector field. Set Spc(X)={eigenvalues of DX(z) : z∈U}. This X is called Hurwitz if Spc(X)⊂{z∈C : ℜ(z)<0}. Suppose that X is Hurwitz and U⊂ℝ2 is the complement of a compact set. Then by adding to X a constant v one obtains that the infinity is either an attractor or a repellor for X+v. That means: (i) there exists a unbounded sequence of closed curves, pairwise bounding an annulus the boundary of which is transversal to X+v, and (ii) there is a neighborhood of infinity with unbounded trajectories, free of singularities and periodic trajectories of X+v. This result is obtained after to proving the existence of Z:ℝ2→ℝ2, a topological embedding such that Z equals X in the complement of some compact subset of U.
Autor(es):Rabanal, Roland
Año: 2013
Título de la revista: Journal of Differential Equations
Volumen: 255
Página inicial - Página final: 1050-1066
ISSN: 0022-0396
Url: http://www.ams.org/mathscinet/pdf/3062761.pdf