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An eigenvalue condition for the injectivity and asymptotic stability at infinity

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LetU → 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 differentialDX z and the global injectivity of the local diffeomorphism given byX. This setU 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 fieldX:(U, ∞) → (ℝ2, 0), —which is differentiable inU/{∞} 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.

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Correspondence to Roland Rabanal.

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Partially supported by CAPES (Brazil) with grant number BEX-2256/05-3.

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Rabanal, R. An eigenvalue condition for the injectivity and asymptotic stability at infinity. Qual. Th. Dyn. Syst. 6, 233 (2005).

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Key Words

  • Planar vector fields
  • Asymptotic stability
  • Injectivity