Advertisement

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
  • 82 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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz M., Stegun I.: Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables. Dover, New York (1965)Google Scholar
  2. 2.
    Cerveau D., Lins Neto A.: Holomorphic foliations in \({\mathbb{C} \times \mathbb{P}^2}\) having an invariant algebraic curve. Ann. Math. 140, 289–294 (1994)CrossRefGoogle Scholar
  3. 3.
    Chavarriga J., Grau M.: A family of non–Darboux integrablequadratic polynomial differential systems with algebraic solutions of arbitrary high degree. Appl. Math. Lett. 16, 833–837 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chavarriga J., Llibre J.: Invariant algebraic curves and rational first integrals for planar polynomial vector fields. J. Differ. Equ. 169, 1–16 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Christopher C., Llibre J., Pantazi C., Zhang X.: Darboux integrability and invariant algebraic curves for planar polynomial systems. J. Phys. A 35, 2457–2476 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fulton W.: Algebraic Curves. An Introduction to Algebraic Geometry. W.A. Benjamin Inc, New York (1969)Google Scholar
  7. 7.
    Gasull A., Li S., Llibre J.: Chordal quadratic systems. Rocky Mt. J. Math. 16, 751–782 (1986)zbMATHCrossRefGoogle Scholar
  8. 8.
    Li W., Llibre J., Zhang X.: Planar vector fields with generalized rational first integrals. Bull. Sci. Math. 125, 341–361 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Llibre J., Ramírez R., Sadovskaia N.: On the 16th Hilbert problem for algebraic limit cycles. J. Differ. Equ. 248, 1401–1409 (2010)zbMATHCrossRefGoogle Scholar
  10. 10.
    Llibre J., Ramírez R., Sadovskaia N.: On the 16th Hilbert problem for limit cycles on nonsingular algebric curves. J. Differ. Equ. 250, 983–999 (2011)zbMATHCrossRefGoogle Scholar
  11. 11.
    Ramirez R., Sadovskaia N.: Differential equations on the plane with given solutions. Collect. Math. 47, 145–177 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ramírez, R., Sadovskaia, N.: Inverse approach in the study of ordinary differential equations, pp. 1–49. Preprint, Universitat Rovira i Virgili, Tarragona (2008)Google Scholar
  13. 13.
    Sadovskaia, N.: Problemas inversos en la teoría de las ecuaciones diferenciales ordinarias. Ph.D., Universitat Politécnica de Catalunya, Barcelona (2002) (in Spanish).Google Scholar
  14. 14.
    Tsygventsev A.: Algebraic invariant curves of plane polynomial differential systems. J. Phys. A 34, 663–672 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zhang, X.: The 16th Hilbert problem on algebraic limit cycles. Preprint (2010)Google Scholar

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

Personalised recommendations