Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T04:15:36.914Z Has data issue: false hasContentIssue false
  • English
  • Français

Complete orbits for twist maps on the plane

Published online by Cambridge University Press:  01 August 2008

MARKUS KUNZE  and
RAFAEL ORTEGA
MARKUS KUNZE
Affiliation:
Universität Duisburg-Essen, Fachbereich Mathematik, D-45117 Essen, Germany
RAFAEL ORTEGA
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, E-18071 Granada, Spain (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Twist maps (θ1,r1)=f(θ,r) are considered in this paper, with no assumption on the periodicity of the map in θ. Under appropriate assumptions, the existence of infinitely many bounded (in r) complete orbits is proven. In particular, our results apply to the class of maps where λ>0 and no arithmetic condition has to be imposed on ω1/ω2.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 4 , August 2008 , pp. 1197 - 1213
Copyright
Copyright © 2008 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arnold, V. I.. Mathematical Methods of Classical Mechanics, 2nd edn. Springer, Berlin, 1989.Google Scholar
[2]Golé, Ch.. Symplectic Twist Maps. World Scientific, Singapore, London, 2001.Google Scholar
[3]Hall, G. and Meyer, K.. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Springer, Berlin, 1991.Google Scholar
[4]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[5]Levi, M. and Zehnder, E.. Boundedness of solutions for quasiperiodic potentials. SIAM J. Math. Anal. 26 (1995), 1233–1256.CrossRefGoogle Scholar
[6]Littlewood, J. E.. Unbounded solutions of . J. London Math. Soc. 41 (1966), 491–496.Google Scholar
[7]Liu, B.. Invariant curves of quasi-periodic reversible mappings. Nonlinearity 18 (2005), 685–701.Google Scholar
[8]Morris, G. R.. A case of boundedness in Littlewood’s problem on oscillatory differential equations. Bull. Austral. Math. Soc. 14 (1976), 71–93.Google Scholar
[9]Ortega, R. and Verzini, G.. A variational method for the existence of bounded solutions of a sublinear forced oscillator. Proc. London Math. Soc. 88 (2004), 775–795.CrossRefGoogle Scholar
[10]Verzini, G.. Bounded solutions to superlinear ODEs: a variational approach. Nonlinearity 16 (2003), 2013–2028.CrossRefGoogle Scholar
[11]Zharnitsky, V.. Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi–Ulam problem. Nonlinearity 13 (2000), 1123–1136.Google Scholar