Skip to main content

Characterization and bifurcation diagram of the family of quadratic differential systems with an invariant ellipse in terms of invariant polynomials


Consider the class QS of all non-degenerate planar quadratic systems and its subclass QSE of all its systems possessing an invariant ellipse. This is an interesting family because on one side it is defined by an algebraic geometric property and on the other, it is a family where limit cycles occur. Note that each quadratic differential system can be identified with a point of \({{\mathbb {R}}}^{12}\) through its coefficients. In this paper we provide necessary and sufficient conditions for a system in QS to have at least one invariant ellipse. We give the global “bifurcation” diagram of the family QS which indicates where an ellipse is present or absent and in case it is present, the diagram indicates if the ellipse is or it is not a limit cycle. The diagram is expressed in terms of affine invariant polynomials and it is done in the 12-dimensional space of parameters. This diagram is also an algorithm for determining for any quadratic system if it possesses an invariant ellipse and whether or not this ellipse is a limit cycle.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3


  1. Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: Geometric configurations of singularities of planar polynomial differential systems [A global classification in the quadratic case], Accepted for publication by Springer Nature Switzerland AG (January, 2019)

  2. Arnold, V.: Métodes mathématiques de la mécanique classique. Éditions MIR, Moscou (1976)

    Google Scholar 

  3. Baltag, V.: Algebraic equations with invariant coefficients in qualitative study of the polynomial homogeneous differential systems. Bull. Acad. Sci. Moldova. Math. 2, 31–46 (2003)

    MATH  Google Scholar 

  4. Bularas, D., Calin, I., Timochouk, L., Vulpe, N.: T-comitants of quadratic systems: a study via the translation invariants, Delft University of Technology, Faculty of Technical Mathematics and Informatics, Report no. 96-90, (1996).

  5. Cairó, L., Giacomini, H., Llibre, J.: Liouvillian first integrals for the planar Lotka–Volterra systems. Rend. Circ. Mat. Palermo 2(52), 389–418 (2003)

    MathSciNet  Article  Google Scholar 

  6. Cairó, L., Llibre, J.: Darbouxian first integrals and invariants for real quadratic systems having an invariant conic. J. Phys. A: Math. Gen. 35, 589–608 (2002)

    MathSciNet  Article  Google Scholar 

  7. Cairó, L., Feix, M.R., Llibre, J.: Integrability and algebraic solutions for planar polynomial differential systems with emphasis on the quadratic systems. Resenhas 4, 127–161 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Calin, I.: On rational bases of \(GL(2,\mathbb{R})\)-comitants of planar polynomial systems of differential equations. Bull. Acad. Sci. Moldova. Math. 2, 69–86 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Cao, F., Jiang, J.: The classification on the global phase portraits of two-dimensional Lotka–Volterra system. J. Dyn. Differ. Equ. 20(4), 797–830 (2008)

    MathSciNet  Article  Google Scholar 

  10. Chavarriga, J., Giacomini, H., Llibre, J.: Uniqueness of algebraic limit cycles for quadratic systems. J. Math. Anal. Appl. 261, 85–99 (2001)

    MathSciNet  Article  Google Scholar 

  11. Christopher, C.: Quadratic systems having a parabola as an integral curve. Proc. R. Soc. Edinb. 112A, 113–134 (1989)

    MathSciNet  Article  Google Scholar 

  12. Christopher, C.: Invariant algebraic curves and conditions for a center. Proc. R. Soc. Edinb. 124A, 1209–1229 (1994)

    Article  Google Scholar 

  13. Darboux, G.: Mémoire sur les équations différentielles du premier ordre et du premier degré. Bulletin de Sciences Mathématiques, 2me série, 2(1), 60–96; 123–144; 151–200 (1878)

  14. Druzhkova, T.A.: The algebraic integrals of a certain differential equation. Differ. Equ. 4, 736–739 (1968)

    MathSciNet  MATH  Google Scholar 

  15. Grace, J.H., Young, A.: The Algebra of Invariants. Stechert, New York (1941)

    MATH  Google Scholar 

  16. Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902). (Reprinted in Bull. Am. Math. 37 (2000), 407–436)

  17. Jouanolou, J.P.: Equations de Pfaff algébriques. Lecture Notes in Math, vol. 708. Springer, New York (1979)

    Book  Google Scholar 

  18. Kooij, R.E., Christopher, C.J.: Algebraic invariant curves and the integrability of polynomial systems. Appl. Math. Lett. 6(4), 5153 (1993)

    MathSciNet  Article  Google Scholar 

  19. Lawrence, J.D.: A Catalog of Special Planar Curves. Dover Publication (1972)

  20. Llibre, J., Swirszcz, G.: Classification of quadratic systems admitting the existence of an algebraic limit cycle. Bull. Sci. Math. 131, 405–421 (2007)

    MathSciNet  Article  Google Scholar 

  21. Llibre, J., Ramirez, R., Ramirez, V., Sadovskaia, N.: The 16th Hilbert problem restricted to circular algebraic limit cycles. J. Differ. Equ. 248, 1401–1409 (2010)

    Article  Google Scholar 

  22. Ollagnier, J.M.: Liouvillian integration of the Lotka–Volterra system. Qual. Theory Dyn. Syst. 2, 307–358 (2001)

    MathSciNet  Article  Google Scholar 

  23. Oliveira, R.D.S., Rezende, A.C., Vulpe, N.: Family of quadratic differential systems with invariant hyperbolas: a complete classification in the space \(\mathbb{R}^{12}\): Electron. J. Differ. Equ. 162 1–50 (2016)

  24. Oliveira, R.D.S., Rezende, A.C., Schlomiuk, D., Vulpe, N.: Geometric and algebraic classification of quadratic differential systems with invariant hyperbolas. Electron. J. Differ. Equ. 295 1–122 (2017)

  25. Olver, P.J.: Classical Invariant Theory, London Mathematical Society, Student Texts, vol. 44. Cambridge University Press (1999)

  26. Poincaré, H.: Sur les courbes définies par les équations differentielles, Hournal de Mathématiques Pures et Apliquées, 4eme série, 1 (1885), 167–244. (Oeuvre (1880–1890), Gauthier–Villar, Paris)

  27. Poincaré, H.: Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré. I. Rend. Circ. Mat. Palermo 5, 169–191 (1891)

    MATH  Google Scholar 

  28. Poincaré, H.: Sur l’intégration algébrique des équations différentielles. C. R. Acad. Sci. Paris 112, 761–764 (1891)

    MATH  Google Scholar 

  29. Popa, M.N.: Applications of Algebraic Methods to Differential Systems. Piteşti Univers, The Flower Power Edit, Romania (2004)

  30. Qin, Y.: On the algebraic limit cycles of second degree of the differential equation \(dy/dx =\Sigma _{0\le i+j \le 2} a_{ij} x^i y^j / \Sigma _{0\le i+j \le 2} b_{ij} x^i y^j \). Chin. Math. Acta 8, 608 (1996)

    Google Scholar 

  31. Reyn, J.W.: Phase portraits of a quadratic system of differential equations occurring frequently in applications. Nieuw Arch. Wisk. (4) 5(2), 107–151 (1987)

    MathSciNet  MATH  Google Scholar 

  32. Schlomiuk, D.: Algebraic particular integrals, integrability and the problem of the center. Trans. Am. Math. Soc. 338, 799–841 (1993)

    MathSciNet  Article  Google Scholar 

  33. Schlomiuk, D.: Algebraic and geometric aspects of the theory of polynomial vector fields. In: Bifurcations and Periodic Orbits of Vector Fields, Dana Schlomiuk Editor, NATO Advanced Science Institute Series C Mathematics Physics Science, vol. 408, pp. 429–467. Kluwer Academic Publishers, Dordrecht (1993)

  34. Schlomiuk, D.: Elementary first integrals and algebraic invariant curves of differential equations. Expo. Math. 11, 433–454 (1993)

    MathSciNet  MATH  Google Scholar 

  35. Schlomiuk, D.: Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields. Publ. Mat. 58, 461–496 (2014)

    MathSciNet  Article  Google Scholar 

  36. Schlomiuk, D., Guckenheimer, J., Rand, R.: Integrability of plane quadratic vector fields. Expo. Math. 8(1), 3–25 (1990)

    MathSciNet  MATH  Google Scholar 

  37. Schlomiuk, D., Vulpe, N.: Geometry of quadratic differential systems in the neighbourhood of the line at infinity. J. Differ. Equ. 215, 357–400 (2005)

    Article  Google Scholar 

  38. Schlomiuk, D., Vulpe, N.: Planar quadratic differential systems with invariant straight lines of total multiplicity four. Nonlinear Anal. 68(4), 681–715 (2008)

    MathSciNet  Article  Google Scholar 

  39. Schlomiuk, D., Vulpe, N.: Global classification of the planar Lotka–Volterra differential systems according to their configurations of invariant straight lines. J. Fixed Point Theory Appl. 8(1), 177–245 (2010)

    MathSciNet  Article  Google Scholar 

  40. Schlomiuk, D., Vulpe, N.: Global topological classification of Lotka–Volterra quadratic differential systems. Electron. J. Differ. Equ. 2012(64), 69 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Schlomiuk, D., Vulpe, N.: Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity. Rocky Mt. J. Math. 38(6), 1–60 (2008)

    MathSciNet  Article  Google Scholar 

  42. Schlomiuk, D., Vulpe, N.: Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four. Bul. Acad. Ştiinţe Repub. Mold. Mat. 1, 27–83 (2008)

    MathSciNet  MATH  Google Scholar 

  43. Schlomiuk, D., Vulpe, N.: Planar quadratic differential systems with invariant straight lines of at least five total multiplicity. Qual. Theory Dyn. Syst. 5, 135–194 (2004)

    MathSciNet  Article  Google Scholar 

  44. Schlomiuk, D., Vulpe, N.: The full study of planar quadratic differential systems possessing a line of singularities at infinity. J. Dyn. Differ. Equ. 20, 737–775 (2008)

    MathSciNet  Article  Google Scholar 

  45. 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. (This is a translation of the Russian original [“Shtiintsa”, Kishinev, 1982; MR0716501].)

  46. Vulpe, N.: Characterization of the finite weak singularities of quadratic systems via invariant theory. Nonlinear Anal. 74(4), 6553–6582 (2011)

    MathSciNet  Article  Google Scholar 

  47. Vulpe, N.I.: Polynomial bases of comitants of differential systems and their applications in qualitative theory. (Russian) “Shtiintsa”, Kishinev, (1986)

  48. Zhang, X.: The 16th Hilbert problem on algebraic limit cycles. J. Differ. Equ. 251, 1778–1789 (2011)

    MathSciNet  Article  Google Scholar 

Download references


The first author is partially supported by FAPESP Grants “Projeto Temático” 2014/00304-2 and 2017/20854-5. The work of the third and the fourth authors was partially supported by the NSERC Grant RN000355.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Regilene Oliveira.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Oliveira, R., Rezende, A.C., Schlomiuk, D. et al. Characterization and bifurcation diagram of the family of quadratic differential systems with an invariant ellipse in terms of invariant polynomials. Rev Mat Complut 35, 361–413 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Quadratic vector fields
  • Affine invariant polynomials
  • Invariant algebraic curve
  • Invariant ellipse
  • Limit cycle

Mathematics Subject Classification

  • 58K30
  • 34A26
  • 34C05
  • 34C40