Skip to main content
SpringerLink
Log in

Studies of the Petrov Module for a Family of Generalized Liénard Integrable Systems

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

Abstract

In this article we use the Lambert function in order to study a family of integrable generalized Liénard equations \(X_f\) which display a center. We first prove a conjugation lemma inside a continuum of nested periodic orbits. Then we deduce an explicit operator of Gelfand–Leray associated with the Hamiltonian of equation \(X_f\). Afterwards, we provide a generating family for the associated Petrov module. Finally, by using the Lambert function, we study the monotonicity of the Abelian integral of this generating family’s elements.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the lambert w function. Adv. Comput. Math. 5(1), 329–359 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Desroches, M., Franoise, J.-P., Mégret, L.: Canard-induced loss of stability across a homoclinic bifurcation. ARIMA Rev. Afr. Rech. Inform. Math. Appl. 20, 47–62 (2015)

    MathSciNet  Google Scholar 

  3. Doedel, E.J., Champneys, A.R., Dercole, F., Fairgrieve, T.F., Kuznetsov, Y.A., Oldeman, B., Paffenroth, R.C., Sandstede, B., Wang, X.J., Zhang, C.H.: Auto-07p: Continuation and bifurcation software for ordinary differential equations. http://indy.cs.concordia.ca (2007)

  4. Doedel, E.J., Keller, H.B., Kernévez, J.-P.: Numerical analysis and control of bifurcation problems part i: Bifurcation in finite dimensions. Int. J. Bifurc. Chaos 1(3), 493–520 (1991)

    Article  MATH  Google Scholar 

  5. Doedel, E.J., Keller, H.B., Kernévez, J.-P.: Numerical analysis and control of bifurcation problems part ii: Bifurcation in finite dimensions. Int. J. Bifurc. Chaos 1(4), 745–772 (1991)

    Article  MATH  Google Scholar 

  6. Dumortier, F., Li, C.: Perturbations from an elliptic hamiltonian of degree four. i. Saddle loop and two saddle cycle. J. Differ. Equ. 176, 114–157 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dumortier, F., Li, C.: Perturbations from an elliptic hamiltonian of degree four. ii. Cuspidal loop. J. Differ. Equ. 175, 209–243 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dumortier, F., Li, C., Zhang, Z.: Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J. Differ. Equ. 139, 146–193 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dumortier, F., Panazzolo, D., Roussarie, R.: More limit cycles than expected in liénard equations. Proc. Am. Math. Soc. 135(6), 1895–1904 (2007)

    Article  MATH  Google Scholar 

  10. Figueras, J.L., Tucker, W., Villadelprat, J.: Computer-assisted techniques for the verification of the chebyshev property of abelian integrals. J. Differ. Equ. 254, 3647–3663 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Françoise, J.-P., Pugh, C.C.: Keeping track of limit cycles. J. Differ. Equ. 65(2), 139–157 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. Françoise, J.-P., Xiao, D.: Perturbation theory of a symmetric center within liénard equations. J. Differ. Equ. 259(6), 2408–2429 (2015)

    Article  MATH  Google Scholar 

  13. Gavrilov, L.: Petrov modules and zeros of abelian integrals. Bull. Sci. Math. 122, 571–584 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gavrilov, L., Iliev, I.D.: Perturbations of quadratic hamiltonian two-saddle cycles. Ann. Inst. H. Poincar Anal. Non Linaire 32(2), 307–324 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grau, M., Manosas, F., Villadelprat, J.: A chebyshev criterion for abelian integrals. Trans. Am. Math. Soc. 363, 109–129 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Karamata, J.: Sur quelques problmes poss par ramanujam. J. Indian Math. Soc. 24, 343–365 (1960)

    MathSciNet  Google Scholar 

  17. Li, C., Zhang, Z.: A criterion for determining the monotonicity of the ratio of two abelian integrals. J. Differ. Equ. 127, 407–424 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lynch, S.: Limit cycles of generalized linard equations. Appl. Math. Lett. 8(6), 15–17 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. De Maesschalck, P., Desroches, M.: Numerical continuation techniques for planar slow-fast systems. SIAM J. Appl. Dyn. Syst. 12(3), 1159–1180 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. De Maesschalck, P., Dumortier, F.: Bifurcations of multiple relaxation oscillations in polynomial linard equations. Proc. Am. Math. Soc. 139(6), 2073–2085 (2011)

    Article  MATH  Google Scholar 

  21. De Maesschalck, P., Wechselberger, M.: Neural excitability and singular bifurcations. J. Math. Neurosci. 5(1), 16 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Peng, L.P.: Unfolding of a quadratic integrable system with a homoclinic loop. Acta Math. Sin. 18, 737–754 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Petrov, G.: Complex zeros of an elliptic integral. Funktsional. Anal. i Prilozhen 21(3), 87–88 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Smale, S.: Mathematical problems for the next century. Math. Intell. 20(2), 7–15 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Jacques-Louis Lions Laboratory, Pierre and Marie Curie University, 4 Place Jussieu, 75005, Paris, France

    Lucile Mégret

  2. Team MYCENAE, INRIA, 2 Rue Simone IFF, 75012, Paris, France

    Lucile Mégret

Authors
  1. Lucile Mégret
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lucile Mégret.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mégret, L. Studies of the Petrov Module for a Family of Generalized Liénard Integrable Systems. Qual. Theory Dyn. Syst. 17, 519–539 (2018). https://doi.org/10.1007/s12346-017-0250-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-017-0250-3

Keywords

Mathematics Subject Classification

Search

Navigation