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

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References

Publications referenced by this paper.
Showing 1-10 of 22 references

A remark on an eigenvalue condition for the global injectivity of maps of R2

  • C. Gutierrez, N. Van Chau
  • Notas do ICMC,
  • 2005
1 Excerpt

Global asymptotic stability for differentiable vector fields of R2

  • A. Fernandes, C. Gutierrez, R. Rabanal
  • J. Differential Equations,
  • 2004
1 Excerpt

On local diffeomorphisms of R that are injective

  • A. Fernandes, C. Gutierrez, R. Rabanal
  • Qual. Theory Dyn. Syst.,
  • 2004
1 Excerpt

Topics on singularities and bifurcations of vector fields. In Normal forms, bifurcations and finiteness problems in differential equations, volume

  • F. Dumortier, P. De Maesschalck
  • NATO Sci. Ser. II Math. Phys. Chem.,
  • 2004

Injectivity of C1 maps R2 → R2 at infinity and planar vector fields

  • C. Gutierrez, A. Sarmiento
  • Astérisque, (287):xviii,
  • 2003
3 Excerpts

De Maesschalck . Topics on singularities and bifurcations of vector fields

  • P.
  • Canad . J . Math .
  • 2002

On the injectivity of C1 maps of the real plane

  • M. Cobo, C. Gutierrez, J. Llibre
  • Canad. J. Math.,
  • 2002
1 Excerpt

Ordinary differential equations with applications, volume 34 of Texts in Applied Mathematics

  • C. Chicone
  • 1999

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