Planar Vector Fields with a Given Set of Orbits

Jaume Llibre, Rafael Ramírez, Natalia Sadovskaia

Research output: Contribution to journalArticleResearchpeer-review


We determine all the C1 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)=Σsj=0 aj(x)yS-j = 0 with S branches with respect to the variable y, then the degree of the polynomial g is at most 4S. © 2011 Springer Science+Business Media, LLC.
Original languageEnglish
Pages (from-to)885-902
JournalJournal of Dynamics and Differential Equations
Issue number4
Publication statusPublished - 1 Dec 2011


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


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