Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-19T01:17:08.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Complete orbits for twist maps on the plane: the case of small twist

Published online by Cambridge University Press:  17 November 2010

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

Abstract

In this article we consider twist maps that are non-periodic (and hence are defined on the plane rather than on the cylinder) and have small twist at infinity. Under natural assumptions the existence of infinitely many bounded orbits is established, and furthermore it is proved that unbounded orbits follow bounded orbits for long times. An application is given to the Fermi–Ulam ping-pong model with a non-periodic moving wall.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 5 , October 2011 , pp. 1471 - 1498
Copyright
Copyright © Cambridge University Press 2010

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]Dolgopyat, D.. Fermi acceleration. Geometric and Probabilistic Structures in Dynamics. Eds. Burns, K., Dolgopyat, D. and Pesin, Y.. American Mathematical Society, Providence, RI, 2008, pp. 149–166.CrossRefGoogle Scholar
[2]Fermi, E.. On the origin of cosmic radiation. Phys. Rev. 75 (1949), 1169–1174.CrossRefGoogle Scholar
[3]Gorelyshev, I. and Neishtadt, A. I.. Jump in adiabatic invariant at a transition between modes of motion for systems with impacts. Nonlinearity 21 (2008), 661–676.CrossRefGoogle Scholar
[4]Kunze, M. and Ortega, R.. Complete orbits for twist maps on the plane. Ergod. Th. & Dynam. Sys. 28 (2008), 1197–1213.CrossRefGoogle Scholar
[5]Kunze, M. and Ortega, R.. Complete orbits for twist maps on the plane: extensions and applications.J. Dynam. Differential Equations to appear.Google Scholar
[6]Laederich, S. and Levi, M.. Invariant curves and time-dependent potentials. Ergod. Th. & Dynam. Sys. 11 (1991), 365–378.CrossRefGoogle Scholar
[7]Moser, J.. Selected Chapters in the Calculus of Variations. Birkhäuser, Basel, 2003.CrossRefGoogle Scholar
[8]Moser, J. and Veselov, A. P.. Two-dimensional ‘discrete hydrodynamics’ and Monge–Ampère equations. Ergod. Th. & Dynam. Sys. 22 (2002), 1575–1583.CrossRefGoogle Scholar
[9]Seba, P.. Quantum chaos in the Fermi-accelerator model. Phys. Rev. A 41 (1990), 2306–2310.CrossRefGoogle ScholarPubMed
[10]Zharnitsky, V.. Instability in Fermi–Ulam ‘ping-pong’ problem. Nonlinearity 11 (1998), 1481–1487.CrossRefGoogle Scholar