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Non-existence of invariant circles

Published online by Cambridge University Press:  19 September 2008

John N. Mather
John N. Mather
Affiliation:
Princeton University, Fine Hall, Princeton, NJ 08544, USA
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Abstract

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The dynamical system associated to the difference equation

has been studied numerically by several authors. On the basis of numerical evidence, they conclude that there exists a number k0 ≈ 0.97 such that there are homotopically non-trivial invariant circles for |k|≤k0 and there are none for |k|>k0. In this note, we give a simple rigorous proof that there are none for |k|>.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 2 , June 1984 , pp. 301 - 309
Copyright
Copyright © Cambridge University Press 1984

References

REFERENCES

[1]Aubry, Le Daeron, & André, . Classical ground states of a one-dimensional model for incommensurate structures. Preprint (1982).Google Scholar
[2]Benettin, G., Galgani, L. & Strelcyn, J.. Kolmogorov entropy and numerical experiments. Phys. Rev. A 14 (1976), p. 2338.CrossRefGoogle Scholar
[3]Birkhoff, G. D.. Surface transformations and their dynamical applications. Acta Math. 43 (1922), 1–119. Reprinted in Collected Mathematical Papers, Vol. II, Amer. Math. Soc.: New York (1950), pp. 111–229.CrossRefGoogle Scholar
[4]Birkhoff, G. D.. Sur quelques courbes fermées remarquables. Bull Soc. Math. de France 60 (1932), 1–26. Reprinted in Collected Mathematical Papers, Vol. II, Amer. Math. Soc.: New York (1950), pp. 418–443.Google Scholar
[5]Chirikov, B. V.. A universal instability of many-dimensional oscillator systems. Physics Reports 52 (1979), 263–379.CrossRefGoogle Scholar
[6]Fathi, A.. Une interprétation plus topologique de la démonstration du théoremè de Birkhoff. Appendix of [9].Google Scholar
[7]Greene, J. M.. A method for determining a stochastic transition. J. Math. Phys. 20 (1979), 1183–1201.CrossRefGoogle Scholar
[8]Hellemann, R. H. G. & Bountis, T.. Periodic solutions of arbitrary period, variational methods. In Stochastic Behaviour in Classical and Quantal Systems, (ed. Casati, G. and Ford, J.). Springer Verlag: New York (1979), p. 353.Google Scholar
[9]Herman, M.. Introduction à l'étude des courbes invariantes par les diffeomorphismes de l'anneau. Asterisque 103–104 (1983).Google Scholar
[10]Mather, J.. Glancing billiards. Ergod. Th. & Dyn. Sys. 2 (1982), 397–403.CrossRefGoogle Scholar
[11]Mather, J.. Non-existence of invariant circles. Preprint (1982).Google Scholar
[12]Powell, G. E. & Percival, J. C.. A spectral entropy method for distinguishing regular and irregular motion of Hamiltonian systems. J. Phys. A 12 (1979), p. 2053–2071.Google Scholar
[13]Titchmarsh, E. C.. The Theory of Functions, (2nd edition). Oxford Univ. Press: Oxford (1939).Google Scholar