Nonlinear Dynamics

, Volume 91, Issue 1, pp 487–496 | Cite as

Nondegenerate centers and limit cycles of cubic Kolmogorov systems

Original Paper
First Online:
Received:
Accepted:
  • 118 Downloads

Abstract

We characterize the center conditions for a cubic Kolmogorov differential system. We also study the number of small limit cycles that can bifurcate from the singular point of focus type for such systems.

Keywords

Center problem Kolmogorov systems Analytic integrability Limit cycles 

Mathematics Subject Classification

37C15 34A34 34C05 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The first and second authors are partially supported by a MINECO/ FEDER Grant No. MTM2014-56272-C2-2 and by the Consejería de Educación y Ciencia de la Junta de Andalucía (Projects P12-FQM-1658, FQM-276). The third author is partially supported by a MINECO/FEDER Grant No. MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) Grant No. 2014SGR 1204. The authors thank Alejandro Rodríguez Luis for his help in finding numerically five limit cycles.

References

  1. 1.
    Bomze, I.: Lotka–Volterra equations and replicator dynamics: a two dimensional classification. Biol. Cybern. 48, 201–211 (1983)CrossRefMATHGoogle Scholar
  2. 2.
    Chavarriga, J., Giné, J.: Integrability of cubic systems with degenerate infinity. Differ. Equ. Dyn. Syst. 6(4), 425–438 (1998)MathSciNetMATHGoogle Scholar
  3. 3.
    Chavarriga, J., Giné, J., García, I.A.: Isochronous centers of cubic systems with degenerate infinity. Differ. Equ. Dyn. Syst. 7, 221–238 (1999)MathSciNetMATHGoogle Scholar
  4. 4.
    Chen, X., Huang, W., Romanovski, V.G., Zhang, W.: Linearizability and local bifurcation of critical periods in a cubic Kolmogorov system. J. Comput. Appl. Math. 245, 86–96 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cozma, D.: Darboux integrability and rational reversibility in cubic systems with two invariant straight lines. Electron. J. Differ. Equ. 23, 1–19 (2013)MathSciNetMATHGoogle Scholar
  6. 6.
    Decker, W., Laplagne, S., Pfister, G., Schonemann, H.A.: SINGULAR 3-1 library for computing the prime decomposition and radical of ideals, primdec.lib, (2010)Google Scholar
  7. 7.
    Doedel, E., Wang, X., Fairgrieve, T.: Auto94: software for continuation and bifurcation problems in ODE. Applied Mathematics Report. User’s Manual, California Institute of Technology (1995)Google Scholar
  8. 8.
    Du, C., Huang, W.: Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model. Nonlinear Dyn. 72(1–2), 197–206 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dukarić, M., Giné, J.: Integrability of Lotka–Volterra planar complex cubic systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 26(1), 1650002, 16 (2016)MathSciNetMATHGoogle Scholar
  10. 10.
    Dulac, H.: Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un cente. Bull. Sci. Math. Sér. 2(32), 230–252 (1908)MATHGoogle Scholar
  11. 11.
    Feng, L.: Integrability and bifurcations of limit cycles in a cubic Kolmogorov system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 23(4), 1350061, 6 (2013)MathSciNetMATHGoogle Scholar
  12. 12.
    Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decompositions of polynomials. J. Symb. Comput. 6, 146–167 (1988)CrossRefMATHGoogle Scholar
  13. 13.
    Giné, J.: On some open problems in planar differential systems and Hilbert’s 16th problem. Chaos Solitons Fractals 31(5), 1118–1134 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Giné, J., Romanovski, V.G.: Linearizability conditions for Lotka–Volterra planar complex cubic systems. J. Phys. A 42(22), 225206, 15 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Giné, J., Santallusia, X.: Essential variables in the integrability problem of planar vector fields. Phys. Lett. A 375(3), 291–297 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Giné, J., Valls, C.: Integrability conditions of a resonant saddle perturbed with homogeneous quintic nonlinearities. Nonlinear Dyn. 81(4), 2021–2030 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Greuel, G.M., Pfister, G., Schönemann, H.A.: SINGULAR 3.0 A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserlautern. http://www.singular.uni-kl.de (2005)
  18. 18.
    Kapteyn, W.: On the midpoints of integral curves of differential equations of the first order abd the first degree. Nederl. Acad. Wetensch. Verslagen Afd. Natuurkunde Koninkl. Nederland 19, 1446–1457 (1911)Google Scholar
  19. 19.
    Kapteyn, W.: New investigations Oon the midpoints of integral curves of differential equations of the first order abd the first degree. Nederl. Acad. Wetensch. Verslagen Afd. Natuurkunde Koninkl. Nederland 20, 1354–1365 (1912); 21, 27–33 (1912)Google Scholar
  20. 20.
    Kolmogorov, A.: Sulla teoria di Volterra della lotta per l’esistenza. Giornale dell’ Istituto Italiano degli Attuari 7, 74–80 (1936)MATHGoogle Scholar
  21. 21.
    Llibre, J., Salhi, T.: On the dynamics of a class of Kolmogov systems. Appl. Math. Comput. 225, 242–245 (2013)MathSciNetMATHGoogle Scholar
  22. 22.
    Lloyd, N.G., Christopher, C.J., Devlin, J., Pearson, J.M., Yasmin, N.: Quadratic-like cubic systems. Differ. Equ. Dyn. Syst. 5, 329–345 (1997)MathSciNetMATHGoogle Scholar
  23. 23.
    Lloyd, N.G., Pearson, J.M., Sáez, E., Szántó, I.: Limit cycles of a cubic Kolmogorov system. Appl. Math. Lett. 9(1), 15–18 (1996)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lloyd, N.G., Pearson, J.M., Sáez, E., Szántó, I.: A cubic Kolmogorov system with six limit cycles. Comput. Math. Appl. 44(3–4), 445–55 (2002)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lotka, A.J.: Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. USA 6, 410–415 (1920)CrossRefGoogle Scholar
  26. 26.
    Poincaré, H.: Sur l’intégration des équations différentielles du premier order et du premier degré I and II. Rendiconti del circolo matematico di Palermo 5, 161–191 (1891); 11, 193–239 (1897)Google Scholar
  27. 27.
    Romanovski, V.G., Shafer, D.S.: The Center and Cyclicity Problems: A Computational Algebra Approach. Birkhäuser, Boston (2009)MATHGoogle Scholar
  28. 28.
    Romanovski, V.G., Prešern, M.: An approach to solving systems of polynomials via modular arithmetics with applications. J. Comput. Appl. Math. 236(2), 196–208 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Sáez, E., Szántó, I.: One-parameter family of cubic Kolmogorov system with an isochronous center. Collect. Math. 48(3), 297–301 (1997)MathSciNetMATHGoogle Scholar
  30. 30.
    Sáez, E., Szántó, I., González-Olivares, E.: Cubic Kolmogorov system with invariant straight lines. Proceedings of the Third World Congress of Nonlinear Analysts, Part 7 (Catania, 2000). Nonlinear Anal. 47(7), 4521–4525 (2001)Google Scholar
  31. 31.
    Suba, A., Cozma, D.: Solution of the problem of the centre for cubic differential system with three invariant straight lines in generic position. Qual. Theory Dyn. Syst. 6(1), 45–58 (2005)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Volterra, V.: Lecons sur la Théorie Mathématique de la Lutte pour la vie. Gauthier Villars, Paris (1931)MATHGoogle Scholar
  33. 33.
    Waldvogel, J.: The period in the Lotka–Volterra system is monotonic. J. Math. Anal. Appl. 114, 178–184 (1986)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wang, P.S., Guy, M.J.T., Davenport, J.H.: P-adic reconstruction of rational numbers. SIGSAM Bull. 16(2), 2–3 (1982)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Facultad de CienciasUniversidad de HuelvaHuelvaSpain
  2. 2.Departament de Matemàtica, Inspires Research CentreUniversitat de LleidaLleidaSpain

Personalised recommendations