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On Littlewood's boundedness problem for sublinear Duffing equations

Author: Bin Liu
Journal: Trans. Amer. Math. Soc. 353 (2001), 1567-1585
MSC (1991): Primary 34C15, 58F27
Published electronically: December 18, 2000
MathSciNet review: 1806727
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Abstract | References | Similar Articles | Additional Information


In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations

\begin{displaymath}x^{\prime\prime} + g(x) = e(t), \end{displaymath}

where the 1-periodic function $e(t)$ is a smooth function and $g(x)$satisfies sublinearity:

\begin{displaymath}{sign}(x)\cdot g(x)\to+\infty,\quad g(x)/x\to 0 \quad {as}\,\,\, \vert x\vert\to+\infty. \end{displaymath}

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  • 1. V.I.Arnold, Mathematical Methods of Classical Mechanics, Berlin, Heidelberg, New York: Springer 1978. MR 57:14033b
  • 2. R.Dieckerhoff and E.Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (1)14 (1987), 79-95. MR 89e:34066
  • 3. T.Küpper and J.You, Existence of quasiperiodic solutions and Littlewood's boundedness problem of Duffing equations with subquadratic potentials, Nonlinear Anal. 35 (1999), 549-559. MR 99i:34064
  • 4. S.Laederich and M.Levi, Invariant curves and time-dependent potentials, Ergod.Th. & Dynam. Sys. 11 (1991), 365-378. MR 92g:58110
  • 5. M.Levi, Quasi-periodic motions in superquadratic time-periodic potentials, Comm. Math. Phys., 143(1)(1991), 43-83. MR 93i:34080
  • 6. J.Littlewood, ``Some Problems in Real and Complex Analysis'', Heath, Lexington, MA, 1968. MR 39:5777
  • 7. B.Liu, Boundedness for solutions of nonlinear Hill's equations with periodic forcing terms via Moser's twist theorem, J. Differential Equations. 79(1989), 304-315. MR 90k:34050
  • 8. B.Liu and F.Zanolin, Boundedness of solutions for second order quasilinear ODEs, preprint.
  • 9. L.Markus, ``Lectures in Differentiable Dynamics'', A.M.S. Providence, RI, 1971. MR 46:8262
  • 10. J.Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology 21 (1982), 457-467. MR 84g:58084
  • 11. G.Morris, A case of boundedness of Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14(1976), 71-93. MR 53:6019
  • 12. J.Moser, ``Stable and Random Motions in Dynamical Systems'', Princeton Univ. Press, N.J., 1973. MR 56:1355
  • 13. J.Moser, On invariant curves of area-preserving mappings of annulus, Nachr. Akad. Wiss. Gottingen Math.-Phys. kl. II(1962), 1-20. MR 26:5255
  • 14. M.Pei, Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations, J. Differential Equations, 113(1994), 106-127. MR 95h:34060
  • 15. H.Rüssmann, On the existence of invariant curves of twist mapping of an annulus, Lecture Notes in Mathematics, vol. 1007, pp. 677-718. Berlin, Heidelberg, New York: Springer-Verlag, 1981. MR 85f:58048
  • 16. J.You, Boundedness for solutions of superlinear Duffing's equations via twist curves theorems, Scientia Sinica, 35(1992), 399-412. MR 95c:34067

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Additional Information

Bin Liu
Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, P.R.China

Keywords: Boundedness of solutions, quasi-periodic solutions, Moser's small twist theorem
Received by editor(s): January 16, 1997
Published electronically: December 18, 2000
Additional Notes: Supported by NNSF of China
Article copyright: © Copyright 2000 American Mathematical Society

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