Skip to main content
SpringerLink
Log in

Equivalence of the Melnikov Function Method and the Averaging Method

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

There is a folklore about the equivalence between the Melnikov method and the averaging method for studying the number of limit cycles, which are bifurcated from the period annulus of planar analytic differential systems. But there is not a published proof. In this short paper, we prove that for any positive integer k, the kth Melnikov function and the kth averaging function, modulo both Melnikov and averaging functions of order less than k, produce the same number of limit cycles of planar analytic (or \(C^\infty \)) near-Hamiltonian 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.

Similar content being viewed by others

References

  1. Arnold, V.I.: Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl. 11, 85–92 (1977)

    Article  MATH  Google Scholar 

  2. Buica, A., Llibre, J.: Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math. 128, 7–22 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Christopher, C., Li, C.: Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel (2007)

    Google Scholar 

  4. Françoise, J.P.: Successive derivatives of a first return map, application to the study of quadratic vector fields. Ergod. Theory Dyn. Syst. 16(1), 87–96 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Giné, J., Grau, M., Llibre, J.: Averaging theory at any order for computing periodic orbits. Phys. D 250, 58–65 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    Book  MATH  Google Scholar 

  7. Hale, J.K.: Ordinary Differential Equations. Robert E. Krieger Publishing Co, Malabar (1980)

    MATH  Google Scholar 

  8. Han, M.: Asymptotic expansions of Melnikov functions and limit cycle bifurcations. Int. J. Bifurc. Chaos 22(12), 1250296 (2012). 30 pp

    Article  MathSciNet  MATH  Google Scholar 

  9. Han, M.: Bifurcation Theory of Limit Cycles. Science Press, Beijing (2013)

    Google Scholar 

  10. Han, M.: Bifurcations of invariant tori and subharmonic solutions for periodic perturbed systems. Sci. China Ser. A 37(11), 1325–1336 (1994)

    MathSciNet  MATH  Google Scholar 

  11. Han, M., Jiang, K., Green Jr, D.: Bifurcations of periodic orbits, subharmonic solutions and invariant tori of high-dimensional systems. Nonlinear Anal. 36, 319–329 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Iliev, I.D., Perko, L.M.: Higher order bifurcations of limit cycles. J. Differ Equ. 154(2), 339–363 (1999)

  13. Jebrane, A., Żoła̧dek, H.: A note on higher order Melnikov functions. Qual. Theory Dyn. Syst. 6(2), 273–287 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurc. Chaos 13(1), 47–106 (2003)

    Article  MATH  Google Scholar 

  15. Llibre, J., Mereu, A.C., Novaes, D.D.: Averaging theory for discontinuous piecewise differential systems. J. Differ. Equ. 258, 4007–4032 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Llibre, J., Zhang, X.: Hopf bifurcation in higher dimensional differential systems via the averaging method. Pac. J. Math. 240(2), 321–341 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Roussarie, R.: Melnikov functions and Bautin ideal. Qual. Theory Dyn. Syst. 2(1), 67–78 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems, second ed. In: Applied Mathematical Sciences, vol. 59. Springer, New York (2007)

Download references

Acknowledgments

We are grateful to the referee for his/her helpful suggestions which helped to improve the presentation of the paper. The first two authors acknowledge the support of the study by the National Natural Science Foundation of China (No. 1271261 and No. 11431008) and the Slovenian Research Agency respectively. The third author is partially supported by NNSF of China Grant Number 11271252, by RFDP of Higher Education of China Grant Number 20110073110054, and by innovation program of Shanghai Municipal Education Commission Grant 15ZZ012. All the three authors are supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme, FP7-PEOPLE-2012-IRSES-316338.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Maoan Han

  2. Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, 2000, Maribor, Slovenia

    Valery G. Romanovski

  3. Faculty of Natural Science and Mathematics, University of Maribor, 2000, Maribor, Slovenia

    Valery G. Romanovski

  4. Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Xiang Zhang

Authors
  1. Maoan Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valery G. Romanovski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maoan Han.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, M., Romanovski, V.G. & Zhang, X. Equivalence of the Melnikov Function Method and the Averaging Method. Qual. Theory Dyn. Syst. 15, 471–479 (2016). https://doi.org/10.1007/s12346-015-0179-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-015-0179-3

Keywords

Search

Navigation