Twist Mappings, Invariant Curves and Periodic Differential Equations

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


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.


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  1. [1]
    V.M. Alekseev, Quasirandom dynamical systems II. One-dimensional nonlinear oscillations in a field with periodic perturbationMath. USSR Sb. 6(1968), 505–560.CrossRefGoogle Scholar
  2. [2]
    J.M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonanceNonlinearity 9(1996), 1099–1111.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    J.M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillatorJ. Diff. Equations 143(1998), 201–220.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    M.R. HermanSur les courbes invariantes par les difféomorphismes de l’anneau I, Asterisque 103–104(1983).Google Scholar
  5. [5]
    M.R. HermanSur les courbes invariantes par les difféomorphismes de l’anneau II, Asterisque 144(1986).Google Scholar
  6. [6]
    M.R. HermanDémonstration du théorème des courbes invariantes par les difféomorphismes de l’anneau,unpublished manuscript.Google Scholar
  7. [7]
    H. Jacobowitz, Periodic solutions ofx” + f (t, x) = 0via the PoincaréBirkhoff theorem,J. Differential Equations 20(1976), 37–52.Google Scholar
  8. [8]
    M. Kunze, Remarks on boundedness of semilinear oscillators, in this volume.Google Scholar
  9. [9]
    M. Kunze, T. Köpper and J. You, On the application of KAM theory to discontinuous dynamical systemsJ. Diff. Equations 139(1997), 1–21.MATHCrossRefGoogle Scholar
  10. [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.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    M. Levi, Quasiperiodic motions in superquadratic time-periodic potentialsComm. Math. Phys.143 (1991), 43–83.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    B. Liu, Boundedness of solutions for semilinear Duffing equationsJ. Differential Equations145 (1998), 119–144.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    B. Liu, Boundedness in asymmetric oscillationsJ. Math. Anal. Apps.231 (1999), 355–373.MATHCrossRefGoogle Scholar
  14. [14]
    G.R. Morris, A case of boundedness in Littlewood’s problem on os-cillatory differential equationsBull. Austral. Math. Soc.14 (1976), 71–93.MathSciNetMATHCrossRefGoogle Scholar
  15. [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. [16]
    R. Ortega, Asymmetric oscillators and twist mappingsJ. London Math. Soc.53 (1996), 325–342.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theoremProc. London Math. Soc.79 (1999), 381–413.MathSciNetMATHCrossRefGoogle Scholar
  18. [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. [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. [20]
    G. Seifert, Resonance in an undamped second-order nonlinear equation with periodic forcingQuart. Appl. Math.48 (1990), 527–530.MathSciNetMATHGoogle Scholar
  21. [21]
    C.L. Siegel and J. MoserLectures on Celestial MechanicsSpringer-Verlag, Berlin, 1971.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Rafael Ortega
    • 1
  1. 1.Departamento de Matemática AplicadaFacultad de Ciencias Universidad de GranadaGranadaSpain

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