Acta Mathematica Sinica, English Series

, Volume 34, Issue 5, pp 801–811 | Cite as

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

Article
  • 16 Downloads

Abstract

In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x2 + y2 + 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincaré disc.

Keywords

Quadratic system complex ellipse invariant algebraic curves phase portrait Poincaré disc 

MR(2010) Subject Classification

34C05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Artés, J. C., Llibre, J.: Quadratic Hamiltonian vector fields. J. Differential Equations, 107, 80–95 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Artés, J. C., Llibre, J., Vulpe, N.: Complete geometric invariant study of two classes of quadratic systems. Electronic J. Differential Equations, 2012(09), 1–35 (2012)MathSciNetMATHGoogle Scholar
  3. [3]
    Bautin, N. N.: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Mat. Sbornik 30, 181–196 (1952); Amer. Math. Soc. Transl., 100, 1–19 (1954)MathSciNetMATHGoogle Scholar
  4. [4]
    Coppel, W. A.: A survey of quadratic systems. J. Differential Equations, 2, 293–304 (1966)MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Date, T.: Classification and analysis of two-dimensional homogeneous quadratic differential equations systems. J. Differential Equations, 32, 311–334 (1979)MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Dulac, H.: Détermination et integration d’une certaine classe d’équations différentielle ayant par point singulier un centre. Bull. Sci. Math. Sér. (2), 32, 230–252 (1908)MATHGoogle Scholar
  7. [7]
    Dumortier, F., Llibre, J., Artés, J. C.: Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, 2006MATHGoogle Scholar
  8. [8]
    Kalin, Yu. F., Vulpe, N. I.: Affine-invariant conditions for the topological discrimination of quadratic Hamiltonian differential systems. Differential Equations, 34(3), 297–301 (1998)MathSciNetMATHGoogle Scholar
  9. [9]
    Kapteyn, W.: On the midpoints of integral curves of differential equations of the first degree (Dutch). Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 19, 1446–1457 (1911)Google Scholar
  10. [10]
    Kapteyn, W.: New investigations on the midpoints of integrals of differential equations of the first degree (Dutch). Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 20, 1354–1365 (1912), 21, 27–33 (1013)Google Scholar
  11. [11]
    Korol, N. A.: The integral curves of a certain differential equation (in Russian). Minsk. Gos. Ped. Inst. Minsk, 47–51 (1973)Google Scholar
  12. [12]
    Li, W., Llibre, J., Nicolau, N., et al.: On the differentiability of first integrals of two dimensional flows. Proc. Amer. Math. Soc., 130, 2079–2088 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Llibre, J., Schlomiuk, D.: On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete and Continuous Dynamical Systems Series A, 35, 1091–1102 (2015)MathSciNetMATHGoogle Scholar
  14. [14]
    Lunkevich, V. A., Sibirskii, K. S.: Integrals of a general quadratic differential system in cases of a center. Differential Equations, 18, 563–568 (1982)MathSciNetMATHGoogle Scholar
  15. [15]
    Lyagina, L. S.: The integral curves of the equation y’ = (ax 2 +bxy +cy 2)/(dx 2 +exy +fy 2) (in Russian). Usp. Mat. Nauk, 6-2(42), 171–183 (1951)MathSciNetGoogle Scholar
  16. [16]
    Markus, L.: Quadratic differential equations and non-associative algebras. Annals of Mathematics Studies, 45, Princeton University Press, Princeton, 1960, 185–213MathSciNetMATHGoogle Scholar
  17. [17]
    Markus, L.: Global structure of ordinary differential equations in the plane. Trans. Amer. Math Soc., 76, 127–148 (1954)MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Neumann, D. A.: Classification of continuous flows on 2-manifolds. Proc. Amer. Math. Soc., 48, 73–81 (1975)MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Newton, T. A.: Two dimensional homogeneous quadratic differential systems. SIAM Review, 20, 120–138 (1978)MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Peixoto, M. M.: Dynamical systems. Proceedings of a Symposium held at the University of Bahia, Acad. Press, New York, 1973, 389–420CrossRefGoogle Scholar
  21. [21]
    Qin, Y. X.: On the algebraic limit cycles of second degree of the differential equation dy/dx = Σ0≤i+j≤2 a ij x i y j0≤i+j≤2 b ij x i y j. Acta Math. Sin., 8, 23–35 (1958)Google Scholar
  22. [22]
    Schlomiuk, D.: Algebraic particular integrals, integrability and the problem of the center. Trans. Amer. Math. Soc., 338, 799–841 (1993)MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    Sibirskii, K. S., Vulpe, N. I.: Geometric classification of quadratic differential systems. Differential Equations, 13, 548–556 (1977)MathSciNetMATHGoogle Scholar
  24. [24]
    Vdovina, E. V.: Classification of singular points of the equation y’ = (a 0 x 2 +a 1 xy +a 2 y 2)/(b 0 x 2 +bnxy + b 2 y 2) by Forster’s method (in Russian). Differential Equations, 20, 1809–1813 (1984)Google Scholar
  25. [25]
    Ye, W. Y., Ye, Y.: On the conditions of a center and general integrals of quadratic differential systems. Acta Math. Sin., Engl. Ser., 17, 229–236 (2001)MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    Żoła̧dek, H.: Quadratic systems with center and their perturbations. J. Differential Equations, 109, 223–273 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, Barcelona, CataloniaSpain
  2. 2.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal

Personalised recommendations