Skip to main content

Twist Mappings, Invariant Curves and Periodic Differential Equations

  • Chapter

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 43))

Abstract

Let us consider the periodic differential system in the plane, whereXsatisfies. In this course we will study the quasi-periodic solutions of this equation and we will show that these solutions play an important role in the study of the dynamics when the equation has a hamiltonian structure.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V.M. Alekseev, Quasirandom dynamical systems II. One-dimensional nonlinear oscillations in a field with periodic perturbationMath. USSR Sb. 6(1968), 505–560.

    Article  Google Scholar 

  2. J.M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonanceNonlinearity 9(1996), 1099–1111.

    Article  MathSciNet  MATH  Google Scholar 

  3. J.M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillatorJ. Diff. Equations 143(1998), 201–220.

    Article  MathSciNet  MATH  Google Scholar 

  4. M.R. HermanSur les courbes invariantes par les difféomorphismes de l’anneau I, Asterisque 103–104(1983).

    Google Scholar 

  5. M.R. HermanSur les courbes invariantes par les difféomorphismes de l’anneau II, Asterisque 144(1986).

    Google Scholar 

  6. M.R. HermanDémonstration du théorème des courbes invariantes par les difféomorphismes de l’anneau,unpublished manuscript.

    Google Scholar 

  7. H. Jacobowitz, Periodic solutions ofx” + f (t, x) = 0via the PoincaréBirkhoff theorem,J. Differential Equations 20(1976), 37–52.

    Google Scholar 

  8. M. Kunze, Remarks on boundedness of semilinear oscillators, in this volume.

    Google Scholar 

  9. M. Kunze, T. Köpper and J. You, On the application of KAM theory to discontinuous dynamical systemsJ. Diff. Equations 139(1997), 1–21.

    Article  MATH  Google Scholar 

  10. A.C. Lazer and J.P. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysisSIAM Review 32(1990), 537–578.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Levi, Quasiperiodic motions in superquadratic time-periodic potentialsComm. Math. Phys.143 (1991), 43–83.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Liu, Boundedness of solutions for semilinear Duffing equationsJ. Differential Equations145 (1998), 119–144.

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Liu, Boundedness in asymmetric oscillationsJ. Math. Anal. Apps.231 (1999), 355–373.

    Article  MATH  Google Scholar 

  14. G.R. Morris, A case of boundedness in Littlewood’s problem on os-cillatory differential equationsBull. Austral. Math. Soc.14 (1976), 71–93.

    Article  MathSciNet  MATH  Google Scholar 

  15. J.K. Moser, On invariant curves of area-preserving mappings of an annulusNachr. Akad. Wiss. Gottingen Math. Phys. II(1962), 1–20.

    Google Scholar 

  16. R. Ortega, Asymmetric oscillators and twist mappingsJ. London Math. Soc.53 (1996), 325–342.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theoremProc. London Math. Soc.79 (1999), 381–413.

    Article  MathSciNet  MATH  Google Scholar 

  18. E. Picard, Sur l’application des méthodes d’approximations successives á l’étude de certaines équations différentielles ordinairesJounal deLiouville9 (1893), 217–271; reprinted in Oeuvres de Emile Picard, vol. 2, Editions du CNRS (Paris, 1979).

    Google Scholar 

  19. H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzier-barer Abbildungen eines KreisringesNachr. Akad. Wiss. Gottingen Math. Phys. Kl II(1970), 67–105.

    Google Scholar 

  20. G. Seifert, Resonance in an undamped second-order nonlinear equation with periodic forcingQuart. Appl. Math.48 (1990), 527–530.

    MathSciNet  MATH  Google Scholar 

  21. C.L. Siegel and J. MoserLectures on Celestial MechanicsSpringer-Verlag, Berlin, 1971.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Science+Business Media New York

About this chapter

Cite this chapter

Ortega, R. (2001). Twist Mappings, Invariant Curves and Periodic Differential Equations. In: Grossinho, M.R., Ramos, M., Rebelo, C., Sanchez, L. (eds) Nonlinear Analysis and its Applications to Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 43. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0191-5_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0191-5_5

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6654-9

  • Online ISBN: 978-1-4612-0191-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics