Skip to main content
SpringerLink
Log in

Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

A class of degree four differential systems that have an invariant conic x 2 + Cy 2 = 1, C ∈ ℝ, is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. N. Bautin: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus. Mat. Sbornik 30(72) (1952), 181–196; English translation in AMS 1(5) (1962), 396–413.

    MathSciNet  Google Scholar 

  2. I. Bendixson: Sur les courbes définies par des équations différentielles. Acta Math. 24 (1900), 1–88. (In French.)

    Article  MATH  MathSciNet  Google Scholar 

  3. W. A. Coppel: Some quadratics systems with at most one limit cycle. Dynam. report. Expositions Dynam. Systems (N.S.) 2 (1989), 61–88.

    MathSciNet  Google Scholar 

  4. D. Cozma, A. Suba: The solution of the problem of center for cubic differential systems with four invariant straight lines. An. Stiint. Univ. Al. I. Cuza Iasi, Ser. Nuova, Mat. 44(Suppl.) (1999), 517–530.

    MathSciNet  Google Scholar 

  5. D. Cozma, A. Suba: Solution of the problem of center for a cubic differential systems with three invariant straight lines. Qual. Theory Dyn. Syst. 2 (2001), 129–143.

    MATH  MathSciNet  Google Scholar 

  6. J. Chavarriga, E. Sáez, I. Szántó, and M. Grau: Coexistence of limit cycles and invariant algebraic curves on a Kukles systems. Nonlinear Anal., Theory Methods Appl. 59 (2004), 673–693.

    Article  MATH  Google Scholar 

  7. C. Lansun, W. Mingshu: Relative position and number of limit cycles of a quadratic differential system. Acta Math. Sin. 22 (1979), 751–758. (In Chinese.)

    MATH  Google Scholar 

  8. L. A. Cherkas, L. I. Zhilevich: The limit cycles of some differential equations. Differ. Uravn. 8 (1972), 924–929. (In Russian.)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. D. Guoren, W. Songlin: Closed orbits and straight line invariants in E 3 systems. Acta Mathematica Sci. 9 (1989), 251–261. (In Chinese.)

    Google Scholar 

  11. H. Dulac: Sur les cycles limites. S. M. F. Bull. 51 (1923), 45–188. (In French.)

    MathSciNet  MATH  Google Scholar 

  12. D. Hilbert: Mathematical problems. American Bull (2) 8 (1902), 437–479.

    MathSciNet  Google Scholar 

  13. E. D. James, N. G. Lloyd: A cubic system with eigth small-amplitude limit cycles. IMA J. Appl. Math. 47 (1991), 163–171.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Kooij: Limit cycles in polynomial systems. PhD. thesis. University of Technology, Delft, 1993.

    Google Scholar 

  15. J. Li: Hilbert’s 16th problem for n = 3: H(3) ≥ 11. Kexue Tongbao 31 (1984), 718.

    Google Scholar 

  16. A. Liénard: Etude des oscillations entreteneues. Re. générale de l’électricité 23 (1928), 901–912. (In French.)

    Google Scholar 

  17. Z. H. Liu, E. Sáez, and I. Szántó: A cubic systems with an invariant triangle surrounding at last one limit cycle. Taiwanese J. Math. 7 (2003), 275–281.

    MATH  MathSciNet  Google Scholar 

  18. N. G. Lloyd, J. M. Pearson, E. Sáez, and I. Szántó: Limit cycles of a cubic Kolmogorov system. Appl. Math. Lett. 9 (1996), 15–18.

    Article  MATH  Google Scholar 

  19. N. G. Lloyd, J. M. Pearson, E. Sáez, and I. Szántó: A cubic Kolmogorov system with six limit cycles. Computers Math. Appl. 44 (2002), 445–455.

    Article  MATH  Google Scholar 

  20. N. G. Lloyd, J. M. Pearson: Five limit cycles for a simple cubic system. Publications Mathematiques 41 (1997), 199–208.

    MATH  MathSciNet  Google Scholar 

  21. N. G. Lloyd, T. R. Blows, and M. C. Kalenge: Some cubic systems with several limit cycles. Nonlinearity (1988), 653–669.

  22. S. Ning, S. Ma, K. H. Kwek, and Z. Zheng: A cubic systems with eight small-amplitude limit cycles. Appl. Math. Lett. 7 (1994), 23–27.

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Poincaré: Mémorie sur les courbes définies par lés équations differentialles I–VI, Oeuvre I. Gauthier-Villar, Paris, 1880–1890. (In French.)

    Google Scholar 

  24. L. S. Pontryagin: On dynamical systems close to Hamiltonian ones. Zh. Eksper. Teoret. Fiz. 4 (1934), 883–885. (In Russian.)

    Google Scholar 

  25. J. W. Reyn: A bibliography of the qualitative theory of quadratic systems of differential equation in the plane. Report TU Delft 92-17, second edition. 1992.

  26. E. Sáez, I. Szántó, and E. González-Olivares: Cubic Kolmogorov system with four limit cycles and three invariant straight lines. Nonlinear Anal., Theory Methods Appl. 47 (2001), 4521–4525.

    Article  MATH  Google Scholar 

  27. S. Songling: A concrete example of the existence of four limit cycles for planar quadratics systems. Sci. Sin. XXIII (1980), 153–158.

    Google Scholar 

  28. S. Songling: System of equation (E 3) has five limit cycles. Acta Math. Sin. 18 (1975). (In Chinese.)

  29. S. Guangjian, S. Jifang: The n-degree differential system with 1/2(n − 1)(n + 1) straight line solutions has no limit cycles. Proc. Conf. Ordinary Differential Equations and Control Theory, Wuhan 1987 (1987), 216–220. (In Chinese.)

  30. B. van der Pol: On relaxation-oscillations. Philos. Magazine 7 (1926), 978–992.

    Google Scholar 

  31. W. Dongming: A class of cubic differential systems with 6-tuple focus. J. Differ. Equations 87 (1990), 305–315.

    Article  MATH  Google Scholar 

  32. Y. Xinan: A survey of cubic systems. Ann. Differ. Equations 7 (1991), 323–363.

    MATH  Google Scholar 

  33. Y. Yanqian, Y. Weiyin: Cubic Kolmogorov differential system with two limit cycles surrounding the same focus. Ann. Differ. Equations 1 (1985), 201–207.

    MATH  Google Scholar 

  34. P. Yu and M. Han: Twelve limit cycles in a cubic order planar system with Z 2-symmetry. Communications on pure and applied analysis 3 (2004), 515–526.

    Article  MATH  MathSciNet  Google Scholar 

  35. H. Zoladek: Eleven small limit cycles in a cubic vector field. Nonlinearity 8 (1995), 843–860.

    Article  MATH  MathSciNet  Google Scholar 

  36. W. Stephen: A System for Doing Mathematics by Computer. Wolfram Research Mathematica, 1988.

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile

    Eduardo Sáez & Iván Szántó

  2. Instituto de Matemáticas y Física, Universidad de Valparaíso, Casilla 5030, Valparaíso, Chile

    Eduardo Stange

Authors
  1. Eduardo Sáez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eduardo Stange
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Iván Szántó
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sáez, E., Stange, E. & Szántó, I. Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four. Czech Math J 57, 105–114 (2007). https://doi.org/10.1007/s10587-007-0047-7

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-007-0047-7

Keywords

Search

Navigation