Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

High order Melnikov functions and the problem of uniformity in global bifurcation

  • 86 Accesses

  • 8 Citations

Summary

In studying bifurcation of Hamiltonian system with small perturbation, it can not always be obtained the uniqueness of limit cycle uniformly for0 < ε ≪ 1 from monotonicity of I2(h)/I1(h), in case the first order Melnikov function M1(h)=μI1(h) + vI2(h) is degenerate at the end points of the interval of its definition. In this paper two examples have been constructed for showing the above conclusion.

References

  1. [1]

    J. A. Sanders,Melnikov's method and averaging, Celestial Mechanics,28 (1982), pp. 171–181.

  2. [2]

    S. N. Chow -J. K. Hale -J. Mallet-Parret,An example of bifurcation to homoclinic orbits, J. Diff. Equat.,37 (1980), pp. 351–373.

  3. [3]

    R. I. Bogdanov,Bifurcation of limit cycle of a family of plane vector field, Sel. Math. Sov.,4 (1981), pp. 373–387.

  4. [4]

    R. Roussarie,On the number of limit cycles which appear by perturbation of separatric loop of planar vector fields, Bol. Soc. Brasil. Mat.,17 (1986), pp. 67–101.

  5. [5]

    Feng Beiyie -Xiang Min,Stability of saddle-loop and conditions of bifurcation out limit cycles, Acta Mathematica, Sinica,28, 1 (1985), pp. 53–70.

  6. [6]

    A. A.Andronov - E. A.Leontovich -I. I.Gordon - A. G.Maier,Theory of Bifurcations of Dynamics System on a Plane, Wiley (1973).

  7. [7]

    F. Dumortier -R. Roussarie -J. Sotomayor,Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynamical Systems,7 (1986), pp. 375–413.

  8. [8]

    S. N.Chow - J. K.Hale,Methods of Bifurcation Theory, Springer-Verlag (1982).

  9. [9]

    R. Cushman -J. A. Sanders,A codimension two bifurcation with a third order Picard-Fuchs equations, J. Diff. Equat.,59 (1985), pp. 243–256.

  10. [10]

    J. Carr -S. N. Chow -J. K. Hale,Abelian integrals and bifurcation theory, J. Diff. Equat.,59 (1985), pp. 413–437.

  11. [11]

    D. Stoffer,On the approach of Holmes and Sanders to the Melnikov procedure in the method of averaging, Z.A.M.P.,34, 6 (1983), pp. 948–952.

  12. [12]

    C. Holmes -P. J. Holmes,Second order averaging and bifurcations to subharmonics in Buffing's equation, J. Sound Vibr.,78, 2 (1981), pp. 161–174.

  13. [13]

    J. P. Ottoy,Perturbation method for small periodically forces systems. I -For purely non-linear dynamical systems, Int. J. Control,40, 3 (1984), pp. 555–569.

  14. [14]

    B. Drachman -S. A. van Gils -Zhifen Zhang,Abelian integrals of quadratic vector fields, J. Reine Angew. Math.,382 (1987), pp. 165–180.

Download references

Author information

Additional information

This work was partially supported by Natural Science Foundation of China.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Zhi-Fen, Z., Bao-Yi, L. High order Melnikov functions and the problem of uniformity in global bifurcation. Annali di Matematica 161, 181–212 (1992). https://doi.org/10.1007/BF01759638

Download citation

Keywords

  • Periodic Orbit
  • Hamiltonian System
  • Homoclinic Orbit
  • Closed Orbit
  • Lower Order Term