The Resonant Center Problem for a 2:-3 Resonant Cubic Lotka–Volterra System

  • Jaume Giné
  • Colin Christopher
  • Mateja Prešern
  • Valery G. Romanovski
  • Natalie L. Shcheglova
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)


Using tools of computer algebra we derive the conditions for the cubic Lotka–Volterra system \(\dot x = x( 2 - a_{20} x^2 - a_{11} xy - a_{02} y^2)\), \(\dot y = y(-3 + b_{20} x^2 + b_{11} xy + b_{02} y^2)\) to be linearizable and to admit a first integral of the form Φ(x,y) = x 3 y 2 + ⋯ in a neighborhood of the origin, in which case the origin is called a 2: − 3 resonant center.


resonant center problem polynomial systems of differential equations first integral 

1991 Mathematics Subject classification

Primary 34C14 Secondary 34A26 37C27 34C25 


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  1. 1.
    Bibikov, Y.N.: Local theory of nonlinear analytic ordinary differential equations. Lecture Notes in Mathematics, vol. 702. Springer, Heidelberg (1979)zbMATHGoogle Scholar
  2. 2.
    Bruno, A.D.: A Local Method of Nonlinear Analysis for Differential Equations. Nauka, Moscow (1979) (in Russian); Local Methods in Nonlinear Differential Equations. Springer, Berlin (1989) (translated from Russian)Google Scholar
  3. 3.
    Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: On the integrability of two-dimensional flows. J. Differential Equations 157, 163–182 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: Darboux integrability and the inverse integrating factor. J. Differential Equations 194, 116–139 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chen, X., Giné, J., Romanovski, V.G., Shafer, D.S.: The 1: − q resonant center problem for certain cubic Lotka–Volterra systems. Appl. Math. Comput. (to appear)Google Scholar
  6. 6.
    Christopher, C., Li, C.: Limit Cycles of Differential Equations. Birkhäuser, Basel (2007)zbMATHGoogle Scholar
  7. 7.
    Christopher, C., Rousseau, C.: Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in ℂ2. Publ. Mat. 45, 95–123 (2001)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Christopher, C., Mardešic, P., Rousseau, C.: Normalizable, integrable and linearizable saddle points for complex quadratic systems in ℂ2. J. Dyn. Control Syst. 9, 311–363 (2003)zbMATHCrossRefGoogle Scholar
  9. 9.
    Decker, W., Pfister, G., Schönemann, H.A.: Singular 2.0 library for computing the primary decomposition and radical of ideals primdec.lib (2001)Google Scholar
  10. 10.
    Dulac, H.: Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre. Bull. Sci. Math. 32, 230–252 (1908)Google Scholar
  11. 11.
    Fronville, A., Sadovski, A.P., Żołądek, H.: Solution of the 1: − 2 resonant center problem in the quadratic case. Fund. Math. 157, 191–207 (1998)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Giné, J.: On the number of algebraically independent Poincaré-Liapunov constants. Appl. Math. Comput. 188, 1870–1877 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Giné, J., Mallol, J.: Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials. Nonlinear Anal. 71, e132–e137 (2009)Google Scholar
  14. 14.
    Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomials. J. Symbolic Comput. 6, 146–167 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gravel, S., Thibault, P.: Integrability and linearizability of the Lotka–Volterra System with a saddle point with rational hyperbolicity ratio. J. Differential Equations 184, 20–47 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Greuel, G.M., Pfister, G., Schönemann, H.: Singular 3.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2005),
  17. 17.
    Hu, Z., Romanovski, V.G., Shafer, D.S.: 1: − 3 resonant centers on ℂ2 with homogeneous cubic nonlinearities. Comput. Math. Appl. 56, 1927–1940 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Liapunov, M.A.: Problème général de la stabilité du mouvement. Ann. of Math. Stud. 17. Pricenton University Press, Princeton (1947)Google Scholar
  19. 19.
    Liu, C., Chen, G., Li, C.: Integrability and linearizability of the Lotka–Volterra systems. J. Differential Equations 198, 301–320 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Poincaré, H.: Mémoire sur les courbes définies par les équations différentielles. Journal de Mathématiques 37, 375–422 (1881); 8, 251–296 (1882), Oeuvres de Henri Poincaré, vol. I, pp. 3–84. Gauthier–Villars, Paris (1951)Google Scholar
  21. 21.
    Romanovski, V.G., Shafer, D.S.: On the center problem for p: − q resonant polynomial vector fields, Bull. Belg. Math. Soc. Simon Stevin 15, 871–887 (2008)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Romanovski, V.G., Shafer, D.S.: The Center and Cyclicity Problems: A Computational Algebra Approach. Birkhäuser, Boston (2009)zbMATHGoogle Scholar
  23. 23.
    Wang, P.S., Guy, M.J.T., Davenport, J.H.: P-adic reconstruction of rational numbers. SIGSAM Bull. 16, 2–3 (1982)zbMATHCrossRefGoogle Scholar
  24. 24.
    Żołądek, H.: The problem of center for resonant singular points of polynomial vector fields. J. Differential Equations 137, 94–118 (1997)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jaume Giné
    • 1
  • Colin Christopher
    • 2
  • Mateja Prešern
    • 3
  • Valery G. Romanovski
    • 4
    • 5
  • Natalie L. Shcheglova
    • 6
  1. 1.Departament de MatemàticaUniversitat de LleidaLleidaSpain
  2. 2.School of Computing and MathematicsPlymouth UniversityPlymouthUK
  3. 3.Department of Mathematics and StatisticsUniversity of StrathclydeGlasgowUnited Kingdom
  4. 4.CAMTP - Center for Applied Mathematics and Theoretical PhysicsUniversity of MariborMariborSlovenia
  5. 5.Faculty of Natural Science and MathematicsUniversity of MariborMariborSlovenia
  6. 6.Faculty of Mechanics and MathematicsBelarusian State UniversityMinskBelarus

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