Skip to main content
SpringerLink
Log in

A symplectic map and its application to the persistence of lower dimensional invariant tori

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

Abstract

We give a nonlinear symplectic coordinator transformation, which can move the normal frequencies of the lower dimensional torus up to (k,w) where ω is the frequency vector of the torus. That means the normal frequencies with a difference (k,w) may be regarded as the same. As an application, we derive a persistence result on lower dimensional tori of nearly integrable Hamiltonian systems when the second Melnikov’s condition is partially violated.

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. Bourgain, J., On Melnikov’s persistency problem, Math. Res. Lett., 1997, 4: 445–458.

    MATH  MathSciNet  Google Scholar 

  2. Craig, W., Wayne, C.E., Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure and App. Math., 1993, 46: 1409–1498

    Article  MATH  MathSciNet  Google Scholar 

  3. Eliasson, L. H., Perturbations of stable invariant tori for Hamiltonian systems, Ann. Sc. Norm. Super Pisa, 1988, 15: 115–147.

    MATH  MathSciNet  Google Scholar 

  4. Kuksin, S. B., Perturbation theory for quasiperiodic solutions of infinite dimensional Hamiltonian systems, and its applications to the Korteweg de Vries equations, Matem. Sbornik, 1988, 136(178): 3.

    Google Scholar 

  5. Kuksin, S. B., Nearly integrable infinite dimensional Hamitonian systems, Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1993, 1556.

    Google Scholar 

  6. Melnikov, V. K., On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function, Sov. Math. Dokl., 1965, 6: 1592–1596.

    Google Scholar 

  7. Melnilov, V. K., A family of conditionally periodic solutions of a Hamiltonian systems, Sov. Math. Dokl., 1968, 9: 882–886.

    Google Scholar 

  8. Pöschel, J., On elliptic lower dimensional tori in Hamiltonian systems, Math. Z., 1989, 202(4): 559–608.

    Article  MATH  MathSciNet  Google Scholar 

  9. Whitney, H., Analytical extensions of differentiable functions defined in closed sets, Trans. A. M. S., 1934, 36: 63–89.

    Article  MathSciNet  Google Scholar 

  10. You, J., Perturbations of lower dimensional tori for Hamiltonian systems, J. Differential Equations, 1999, 152: 1–29.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Southeast University, 210018, Nanjing, China

    Junxiang Xu

  2. Department of Mathematics, Nanjing University, 210093, Nanjing, China

    Jiangong You

Authors
  1. Junxiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiangong You
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junxiang Xu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, J., You, J. A symplectic map and its application to the persistence of lower dimensional invariant tori. Sci. China Ser. A-Math. 45, 598–603 (2002). https://doi.org/10.1360/02ys9064

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1360/02ys9064

Keywords

Search

Navigation