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Parabolic fixed points, invariant curves and action-angle variables

Published online by Cambridge University Press:  19 September 2008

Dov Aharonov  and
Uri Elias
Dov Aharonov
Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel
Uri Elias
Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel
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Abstract

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A fixed point of an area-preserving mapping of the plane is called elliptic if the eigenvalues of its linearization are of unit modulus but not ±1; it is parabolic if both eigenvalues are 1 or −1. The elliptic case is well understood by Moser's theory. Here we study when is a parabolic fixed point surrounded by closed invariant curves. We approximate our mapping T by the phase flow of an Hamiltonian system. A pair of variables, closely related to the action-angle variables, is used to reduce T into a twist mapping. The conditions for T to have closed invariant curves are stated in terms of the Hamiltonian.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 2 , June 1990 , pp. 231 - 245
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[1]Arnold, V. I.. Mathematical Methods of Classical Mechanics. Springer-Verlag, Berlin, 1978.CrossRefGoogle Scholar
[2]Moser, J.. On invariant curves of area-preserving mappings of an annulus. Nachr. Math. Wiss., Gottingen II, Math. Phys. K1 (1962), 1–20.Google Scholar
[3]Siegel, C. L. & Moser, J.. Lectures on Celestial Mechanics. Springer Verlag, Berlin, 1971.CrossRefGoogle Scholar
[4]Simo, C.. Stability of degenerate fixed points of analytic area preserving mappings. Astérisque 98–99 (1982), 184–194.Google Scholar
[5]Simo, C.. Invariant curves near parabolic points and regions of stability. Lecture Notes in Mathematics 819 pp. 418–424, Springer Verlag, Berlin, 1980.Google Scholar