Qualitative Theory of Dynamical Systems

, Volume 13, Issue 2, pp 305–351 | Cite as

Global Configurations of Singularities for Quadratic Differential Systems with Total Finite Multiplicity Three and at Most Two Real Singularities

  • Joan C. Artés
  • Jaume LlibreEmail author
  • Dana Schlomiuk
  • Nicolae Vulpe


In this work we consider the problem of classifying all configurations of singularities, finite and infinite, of quadratic differential systems, with respect to the geometric equivalence relation defined in Artés et al. (Rocky Mount J Math, 2014). This relation is deeper than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci of different orders. Such distinctions are however important in the production of limit cycles close to the foci in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows us to incorporate all these important geometric features which can be expressed in purely algebraic terms. This equivalence relation is also deeper than the qualitative equivalence relation introduced in [15]. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in [3] where the classification was done for systems with total multiplicity \(m_f\) of finite singularities less than or equal to one. That work was continued in [4] where the geometric classification was done for the case \(m_f=2\). The case \(m_f=3\) has been split in two separate papers because of its length. The subclass of three real distinct singular points was done in [5] and we complete this case here. In this article we obtain geometric classification of singularities, finite and infinite, for the remaining three subclasses of quadratic differential systems with \(m_f=3\) namely: (i) systems with a triple singularity (19 configurations); (ii) systems with one double and one simple real singularities (62 configurations) and (iii) systems with one real and two complex singularities (75 configurations). We also give here the global bifurcation diagrams of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for these subclasses of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of invariant polynomials. This provides an algorithm for computing the geometric configuration of singularities for any quadratic system in this class.


Quadratic vector fields Infinite and finite singularities Affine invariant polynomials Poincaré compactification Configuration of singularities Geometric equivalence relation 

Mathematics Subject Classification

Primary 58K45 34C05 34A34 


  1. 1.
    Artés, J.C., Llibre, J., Schlomiuk, D.: The geometry of quadratic differential systems with a weak focus of second order. Int. J. Bifurc. Chaos 16, 3127–3194 (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: From topological to geometric equivalence in the classification of singularities at infinity for quadratic vector fields. Rocky Mount. J. Math. (2014, To appear)Google Scholar
  3. 3.
    Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: Configurations of singularities for quadratic differential systems with total finite multiplicity lower than 2. Bul. Acad. Stiinte Repub. Mold. Mat. 1, 72–124 (2013)Google Scholar
  4. 4.
    Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: Algorithm for determining the global geometric configurations of singularities of total finite multiplicity 2 for quadratic differential systems. Electron. J. Differ. Equ. 2014(159), 1–79 (2014)Google Scholar
  5. 5.
    Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, D.: Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities. J. Fixed Point Theory Appl. 14, 555–618 (2014)Google Scholar
  6. 6.
    Artés, J.C., Llibre, J., Vulpe, N.I.: Singular points of quadratic systems: a complete classification in the coefficient space \(\mathbb{R}^{12}\). Int. J. Bifurc. Chaos 18, 313–362 (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Baltag, V.: Algebraic equations with invariant coefficients in qualitative study of the polynomial homogeneous differential systems. Bul. Acad. Stiinte Repub. Mold. Mat. 2, 31–46 (2003)MathSciNetGoogle Scholar
  8. 8.
    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. Sb. (N.S) 2 30(7), 181–196 (1952, Russian), Transl. Am. Math. Soc. Ser. (1) 100(1954), 397–413Google Scholar
  9. 9.
    Calin, Iu: On rational bases of \(GL(2,\mathbb{R})\)-comitants of planar polynomial systems of differential equations. Bul. Acad. Stiinte Repub. Mold. Mat. 2, 69–86 (2003)MathSciNetGoogle Scholar
  10. 10.
    Chavarriga, J., Sabatini, M.: A survey of isochronous centers. Qual. Theory Dyn. Syst. 1(1), 1–70 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Coll, B.: Qualitative study of some classes of vector fields in the plane, pp. 5–34. Ph. D., Universitat Autónoma de Barcelona (1987)Google Scholar
  12. 12.
    Conti, R.: Uniformly isochronous centers of polynomial systems in \(\mathbb{R}^2\). Lect. Notes Pure Appl. Math. 152, 21–31 (1994)MathSciNetGoogle Scholar
  13. 13.
    Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Universitext. Springer, Berlin (2008)Google Scholar
  14. 14.
    Gonzalez, E.A.: Velasco, generic properties of polynomial vector fields at infinity. Trans. Am. Math. Soc. 143, 201–222 (1969)CrossRefzbMATHGoogle Scholar
  15. 15.
    Jiang, Q., Llibre, J.: Qualitative classification of singular points. Qual. Theory Dyn. Syst. 6, 87–167 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Llibre, J., Schlomiuk, D.: Geometry of quadratic differential systems with a weak focus of third order. Can. J. Math. 56, 310–343 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Loud, W.S.: Behavior of the period of solutions of certain plane autonomous systems near centers. Contrib. Differ. Equ. 3, 21–36 (1964)MathSciNetGoogle Scholar
  18. 18.
    Nikolaev, I., Vulpe, N.: Topological classification of quadratic systems at infinity. J. Lond. Math. Soc. 2, 473–488 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pal, J., Schlomiuk, D.: Summing up the dynamics of quadratic Hamiltonian systems with a center. Can. J. Math. 56, 583–599 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Popa, M.N.: Applications of Algebraic Methods to Differential Systems. Piteşti Univers., The Flower Power Edit, Romania (2004)Google Scholar
  21. 21.
    Schlomiuk, D.: Algebraic and geometric aspects of the theory of polynomial vector fields. In: Schlomiuk, D. (ed.) Bifurcations and Periodic Orbits of Vector Fields, NATO ASI Series, Series C, vol. 408, pp. 429–467. Kluwer Academic Publishers, The Netherlands (1993)Google Scholar
  22. 22.
    Schlomiuk, D., Pal, J.: On the geometry in the neighborhood of infinity of quadratic differential phase portraits with a weak focus. Qual. Theory Dyn. Syst. 2, 1–43 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schlomiuk, D., Vulpe, N.I.: Geometry of quadratic differential systems in the neighborhood of infinity. J. Differ. Equ. 215, 357–400 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schlomiuk, D., Vulpe, N.I.: The full study of planar quadratic differential systems possessing a line of singularities at infinity. J. Dynam. Differ. Equ. 20, 737–775 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sibirskii, K.S.: Introduction to the algebraic theory of invariants of differential equations. Translated from the Russian. Nonlinear Science: Theory and Applications. Manchester University Press, Manchester (1988)Google Scholar
  26. 26.
    Vulpe, N.: Characterization of the finite weak singularities of quadratic systems via invariant theory. Nonlinear Anal. 74, 6553–6582 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Vulpe, N.I.: Polynomial bases of comitants of differential systems and their applications in qualitative theory. “Ştiinţa”, Kishinev (1986, in Russian)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Joan C. Artés
    • 1
  • Jaume Llibre
    • 1
    Email author
  • Dana Schlomiuk
    • 2
  • Nicolae Vulpe
    • 3
  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain
  2. 2.Département de Mathématiques et de StatistiquesUniversité de MontréalMontréalCanada
  3. 3.Institute of Mathematics and Computer ScienceAcademy of Science of MoldovaChişinăuMoldova

Personalised recommendations