Skip to main content
SpringerLink
Log in

Linear estimate of the number of zeros of Abelian integrals for quadratic centers having almost all their orbits formed by cubics

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

We study the number of zeros of Abelian integrals for the quadratic centers having almost all their orbits formed by cubics, when we perturb such systems inside the class of all polynomial systems of degreen

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. Arnold, V. I., Loss of stability of self-oscillation close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl., 1977, 11: 1–10.

    Article  Google Scholar 

  2. Han, M., Bifurcations of invariant for tori and subharmonic solutions for period perturbed system, Science in China, Ser. A, 1994, 37(11): 1325–1336.

    MATH  Google Scholar 

  3. Khovansky, A. G., Real analytic manifolds with finiteness properties and complex Abelian integrals, Funct. Anal. Appl., 1984, 18: 118–128.

    Google Scholar 

  4. Varchenko, A. N., Estimate of the number of zeros of an Abelian integrals depending on a parameter and limit cycles, Funct. Anal. Appl., 1984, 18: 98–108.

    Article  MATH  Google Scholar 

  5. Gavrilov, L., Nonoscillation of elliptic integrals related to cubic polynomials with symmetry of order three, Bull. London. Math. Soc., 1998, 30: 267–273.

    Article  MATH  MathSciNet  Google Scholar 

  6. Gavrilov, L., Abelian integrals related to Morse polynomials and perturbations of plane Hamiltonian vector fields, Ann. Inst. Fourier, 1999, 49(2): 611–652.

    MATH  MathSciNet  Google Scholar 

  7. Horozov, E., Iliev, I. D., Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians, Nonlinearity, 1998, 11: 1521–1537.

    Article  MATH  MathSciNet  Google Scholar 

  8. Iliev, I. D., On the limit cycles obtainable from polynomial perturbations of the Bogdanov-Takens Hamiltonian, Israel J. Math., 2000, 115: 269–284.

    Article  MATH  MathSciNet  Google Scholar 

  9. Li, B., Zhang, Z., A note on a result of G.S.Petrov about the weakened bert problem, J. Math. Anal. Appl., 1995, 190: 489–516.

    Article  MATH  MathSciNet  Google Scholar 

  10. Novikov, D., Yakovenko, S., Simple exponential estimate for the number of real zeros of complete Abelian integrals, Ann. Inst. Fourier, 1995, 45: 897–927.

    MATH  MathSciNet  Google Scholar 

  11. Petrov, G. S., Complex zeros of an elliptic integral, Funct. Anal. Appl., 1989, 23: 88–89.

    Article  Google Scholar 

  12. Petrov, G. S., Non-oscillation of elliptic integrals, Funct. Anal. Appl., 1990, 24: 45–50.

    MathSciNet  Google Scholar 

  13. Rousseau, C., Zoladeck, H., Zeros of complete elliptic integrals for 1:2 resonance, J. Differential Equations, 1991, 94: 42–54.

    Article  Google Scholar 

  14. Zhao, Y., Zhang, Z., Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians, J. Differential Equations, 1999, 155: 73–88.

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhao, Y., Zhang, Z., Bifurcations of limit cycles from cubic Hamiltonian system system with a center and a saddle-loop, Publicacions Mateatiques, 2000, 44: 205–235.

    MATH  MathSciNet  Google Scholar 

  16. Zhao, Y., Zhang, Z., Abelian integrals for cubic Hamiltonian vector fields, Annali di Matematica Pura Applicata, 1999, CLXXVI: 251–272.

    Article  MathSciNet  Google Scholar 

  17. Zhao, Y., Liang, Z., Lu, G., The cyclicity of the period annulus of the quadratic Hamiltonian system with non-Morsean point, J. Differential Equations, 2000, 162: 199–223.

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhao, Y., Zhu, S., Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment, Bulletin des Sciences Mathematiques, 2001, 125(2): 109–138.

    Article  MATH  MathSciNet  Google Scholar 

  19. Han, M., Ye, Y., Zhu, D., Cyclicity of homoclinic loops and degenerate cubic Hamiltonian, Science in China, Ser. A, 1999, 42(6): 605–617.

    Article  MATH  MathSciNet  Google Scholar 

  20. Han, M., Cyclicity of planar homoclinic loops and quadratic integrable systems, Science in China, Ser. A, 1997, 40(12): 1247–1258.

    Article  MATH  Google Scholar 

  21. Chicone, C., Jacobs, M., Bifurcation of limit cycles from quadratic isochrones, J. Differential Equations, 1991, 91: 268–326.

    Article  MATH  MathSciNet  Google Scholar 

  22. Hainzl, J., Uniqueness of limit cycles and quotients of Abelian integrals, Nonlinear Analysis, Theory, Methods & Applications, 1997, 30: 4789–4798.

    Article  MATH  MathSciNet  Google Scholar 

  23. Iliev, I. D., Perturbations of quadratic centers, Bull. Sci. Math., 1998, 122: 107–161.

    Article  MATH  MathSciNet  Google Scholar 

  24. Shafer, D. S., Zegeling, A., Bifurcation of limit cycles from quadratic centers, J. Differential Equations, 1995, 122: 48–70.

    Article  MATH  MathSciNet  Google Scholar 

  25. Li, C., Li, W., Llibre, J., Zhang, Z., Bifurcation of limit cycles from quadratic isochronous centers, preprint.

  26. Li, C., Li, W., Llibre, J., Zhang, Z., Linear estimation for the number of zeros of Abelian integrals for some quadratic isochronous centers, Nonlinearity, 2000, 13(5): 1775–1880.

    Article  MATH  MathSciNet  Google Scholar 

  27. Gasull, A., Li, W., Llibre, J., Zhang, Z., Chebyshev property of complete elliptic integrals and its application to Abelian integrals, to appear in Pacific Journal of Mathematics.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhongshan University, 510275, Guangzhou, China

    Yulin Zhao, Weigu Li, Chengzhi Li & Zhifen Zhang

  2. School of Mathematical Sciences, Peking University, 100871, Beijing, China

    Yulin Zhao, Weigu Li, Chengzhi Li & Zhifen Zhang

Authors
  1. Yulin Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weigu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chengzhi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhifen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yulin Zhao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, Y., Li, W., Li, C. et al. Linear estimate of the number of zeros of Abelian integrals for quadratic centers having almost all their orbits formed by cubics. Sci. China Ser. A-Math. 45, 964–974 (2002). https://doi.org/10.1007/BF02879979

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02879979

Keywords

Search

Navigation