Journal of Dynamics and Differential Equations

, Volume 23, Issue 4, pp 885–902 | Cite as

Planar Vector Fields with a Given Set of Orbits

  • Jaume Llibre
  • Rafael Ramírez
  • Natalia Sadovskaia
Article
  • 69 Downloads

Abstract

We determine all the \({\mathcal{C}^1}\) planar vector fields with a given set of orbits of the form y y(x) = 0 satisfying convenient assumptions. The case when these orbits are branches of an algebraic curve is also study. We show that if a quadratic vector field admits a unique irreducible invariant algebraic curve \({g(x, y) = \sum_{j=0}^S a_j(x) y^{S-j}= 0}\) with S branches with respect to the variable y, then the degree of the polynomial g is at most 4S.

Keywords

Invariant curve Singular algebraic curve Branches Quadratic vector fields Orthogonal polynomial Orbits 

Mathematics Subject Classification (2000)

Primary 34C05 34A34 34C14 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Jaume Llibre
    • 1
  • Rafael Ramírez
    • 2
  • Natalia Sadovskaia
    • 3
  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain
  2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain
  3. 3.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain

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