Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere

Article
Received:

Abstract

LexX be a homogeneous polynomial vector field of degreen≥3 on S2 having finitely many invariant circles. Then, for such a vector fieldX we find upper bounds for the number of invariant circles, invariant great circles, invariant circles intersecting at a same point and parallel circles with the same director vector. We give examples of homogeneous polynomial vector fields of degree 3 on S2 having finitely many invariant circles which are not great circles, which are limit cycles, but are not great circles and invariant great circles that are limit cycles. Moreover, for the casen=3 we determine the maximum number of parallel invariant circles with the same director vector.

Key words and phrases

invariant circle invariant great circle polynomial vector field 
This is a preview of subscription content, log in to check access.

References

  1. [1]
    Andronov A.A., Leontovich E.A., Gordon I.I., Maier A.G.,Qualitative theory of second-order dynamic systems, John Wiley & Sons, 1973.Google Scholar
  2. [2]
    Artés J.C., Grünbaum B., Llibre J.,On the number of invariant straight lines for polynomial differential systems, Pacific J. Math.,184 (1998), 207–230MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Camacho M.I.,Geometric properties of homogeneous vector fields of degree two in ℝ3, Trans. Amer. Math. Soc.,268 (1981), 79–101.CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    Chavarriga J.,Integrable systems in the plane with a center type linear part, Applicationes Mathematicae,22 (1994), 285–309.MathSciNetMATHGoogle Scholar
  5. [5]
    Christopher C., Llibre J., Pereira J.V.,Multiplicity of invariant algebraic curves and Darboux integrability, to appear in Pacific J. of Math., 2005.Google Scholar
  6. [6]
    Christopher C., Llibre J.,Integrability via invariant algebraic curves for planar polynomial differential system, Ann. of Diff. Eqs.,16 (2000), 5–19.MathSciNetMATHGoogle Scholar
  7. [7]
    Gutierrez C, Llibre J.,Darbouxian integrability for polynomial vector fields on the 2-dimensional sphere, Extracta Mathematicae,17 (2002), 289–301.MathSciNetMATHGoogle Scholar
  8. [8]
    Llibre J., Pessoa C.,Homogeneous polynomial vector fields of degree 2on the 2-dimensional sphere, to appear in Discrete and Contin. Dyn. Syst. Ser. A, 2006.Google Scholar
  9. [9]
    Llibre J., Rodríguez G.,Invariant hyperplanes and darboux integrability for d-dimensional polynomial differential systems, Bull. Sci. Math.,124 (2000), 599–619.CrossRefMathSciNetMATHGoogle Scholar
  10. [10]
    Llibre J., Zhang X.,Darboux integrability of real polynomial vector fields on regular algebraic hypersurfaces, Rend. Cir. Mat. Palermo,51 (2002), 109–126.CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.2949-1 Department de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain

Personalised recommendations