Skip to main content
SpringerLink
Log in

Invariant tori in nonlinear oscillations

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

Abstract

The boundedness of all the solutions for semilinear Duffing equationx″ + ω2 x + φ(x) =p(t), ω ∈ ℝ+ℕ is proved, wherep (t) is a smooth 2π-periodic function and the perturbation ⌽(x) is bounded.

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Diecherhoff, R., Zehnder, E., Boundedness of solutions via the Twist Theorem,Ann. Scuola Norm. Sup. Pisa C1. Sci. (1), 1987, 14: 79.

    Google Scholar 

  3. Levi, M., Quasiperiodic motions in superquadratic time-periodic potentials,Comm. Math. Phys., 1991, 143: 13.

    Article  Google Scholar 

  4. You, J., Boundedness for solutions of superlinear Duffing’s equations via twist curves theorems,Science in China, 1992, 35: 399.

    MATH  Google Scholar 

  5. Yuan, X., Invariant tori of Duffing-type equations,J. Differential Eqs., 1998, 142: 231.

    Article  MATH  Google Scholar 

  6. Ortega, R., Asymmetric oscillators and twist mappings,J. London Math. Soc., 1996, 53: 325.

    MATH  MathSciNet  Google Scholar 

  7. Liu, B., Boundedness in nonlinear oscillations at resonance,J. Differential Eqs., 1999, 153: 142.

    Article  MATH  Google Scholar 

  8. Kunze, M., Kupper, T., You, J., On the application of KAM theory to discontinous dynamical systems,J. Differential Eqs, 1997, 139: 1.

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu, B., Boundedness of solutions for semilinear Duffing equations,J. Differential Eqs, 1998, 145: 119.

    Article  MATH  Google Scholar 

  10. Moser, J., On invariant curves of area- preserving mappings of an annulus,Nachr. Akad. wiss, Gottingen Math. -Phys. kl II, 1962, 1–20.

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, 100871, Beijing, China

    Bin Liu & Yiqian Wang

Authors
  1. Bin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yiqian Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, B., Wang, Y. Invariant tori in nonlinear oscillations. Sci. China Ser. A-Math. 42, 1047–1058 (1999). https://doi.org/10.1007/BF02889506

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02889506

Keywords

Search

Navigation