SpringerLink
Log in

Invariant Curves of Almost Periodic Twist Map**s

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

In this paper we are concerned with the existence of invariant curves of planar twist map**s which are almost periodic in a spatial variable. As an application of this result to differential equations we will prove the existence of almost periodic solutions and the boundedness of all solutions for superlinear Duffing’s equation with an almost periodic external force.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dieckerhoff, R., Zehnder, E.: Boundedness of solutions via the twist theorem. Ann. Sci. Norm. Sop. Pisa. Cl. Sci. 14, 79–95 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer, Berlin (1999)

    Book  Google Scholar 

  3. Herman, M.R.: Surles courbes invariantes par les difféomorphismes de l’anneau I. Astérisque 103–104 (1983)

  4. Herman, M.R.: Surles courbes invariantes par les difféomorphismes de l’anneau II. Astérisque 144 (1986)

  5. Huang, P., Li, X., Liu, B.: Invariant curves of smooth quasi-periodic map**s. Discrt. Contin. Dyn. Syst. Ser. A 38, 131–154 (2018)

  6. Huang, P., Li, X., Liu, B.: Quasi-periodic solutions for an asymmetric oscillation. Nonlinearity 29, 3006–3030 (2016)

    Article  MathSciNet  Google Scholar 

  7. Littlewood, J.E.: Some Problems in Real and Complex Analysis. Heath, Lexington Mass (1968)

  8. Levi, M.: Quasi-periodic motions in superquadratic time-periodic potentials. Commun. Math. Phys. 143, 43–83 (1986)

    Article  Google Scholar 

  9. Levi, M., Moser, J.: A Lagrangian proof of the invariant curve theorem for twist map**s, Smooth Ergodic Theory and its Applications, (Seattle, WA, 1999) (Proc. Symp. Pure Math. vol 69) (Providence, RI: American Mathematical Society) (2001) 733–746

  10. Liu, B., You, J.: Quasi-periodic solutions of Duffing’s equations. Nonlinear Anal. 33, 645–655 (1998)

    Article  MathSciNet  Google Scholar 

  11. Liu, B.: Invariant curves of quasi-periodic reversible map**. Nonlinearity 18, 685–701 (2005)

    Article  MathSciNet  Google Scholar 

  12. Morris, G.R.: A case of boundedness in Littlewood’s problem on oscillatory differential equations. Bull. Austral. Math. Soc. 14, 71–93 (1976)

    Article  MathSciNet  Google Scholar 

  13. Moser, J.: On invariant curves of area-perserving map**s of an annulus, Nachr. Akad. Wiss. Göttingen Math. -Phys. vol II (1962) 1–20

  14. Moser, J.: Stable and random motions in dynamical systems, Ann. of Math. Studies 77, Princeton, NJ (1973)

  15. Moser, J.: A stability theorem for minimal foliations on a torus. Ergod. Theory Dyn. Syst. 8, 251–81 (1988)

    Article  MathSciNet  Google Scholar 

  16. Ortega, R.: Boundedness in a piecewise linear oscillator and a variant of the small twist theorem. Proc. Lond. Math. Soc. 79, 381–413 (1999)

    Article  MathSciNet  Google Scholar 

  17. Ortega, R.: Invariant curves of map**s with averaged small twist. Adv. Nonlinear Stud. 1, 14–39 (2001)

    Article  MathSciNet  Google Scholar 

  18. Pöschel, J.: Small divisors with spatial structure in infinite dimensional Hamiltonian systems. Commun. Math. Phys. 127, 351–393 (1990)

    Article  MathSciNet  Google Scholar 

  19. Rüssmann, H.: Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1970) 67–105

  20. Siegel, C., Moser, J.: Lectures on Celestial Mechanics. Springer, Berlin (1997)

    MATH  Google Scholar 

  21. Zharnitsky, V.: Invariant curve theorem for quasi-periodic twist map**s and stability of motion in the Fermi–Ulam problem. Nonlinearity 13, 1123–36 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, People’s Republic of China

    Peng Huang

  2. School of Mathematics Sciences, Bei**g Normal University, Bei**g, 100875, People’s Republic of China

    **ong Li

  3. School of Mathematical Sciences, Peking University, Bei**g, 100871, People’s Republic of China

    Bin Liu

Authors
  1. Peng Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. **ong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to **s&author=Peng%20Huang%20et%20al&contentID=10.1007%2Fs10884-021-10033-1&copyright=The%20Author%28s%29%2C%20under%20exclusive%20licence%20to%20Springer%20Science%2BBusiness%20Media%2C%20LLC%2C%20part%20of%20Springer%20Nature&publication=1040-7294&publicationDate=2021-06-25&publisherName=SpringerNature&orderBeanReset=true">Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, P., Li, X. & Liu, B. Invariant Curves of Almost Periodic Twist Map**s. J Dyn Diff Equat 34, 1997–2033 (2022). https://doi.org/10.1007/s10884-021-10033-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-021-10033-1

Keywords

Search

Navigation