Abstract
From the stability condition of the minimum energy states of the Frenkel-Kontorova model and the Mather-Aubry theorem, we get a new computable criterion for estimating the upper bound of kc — the critical value of the perturbation parameter of the 2D standard mapping. By using this criterion, we got the rigorous result \( {k_c} \leqslant \sqrt {{2}} \). Our numerical results confirm the earlier estimation given by Greene, i.e., kc~0.9718....
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References
J.M. Greene, J. Math. Physics, 20, 1183 (1979)
S.J. Shenker and L.P. Kadanoff, J. Stat. Phys., 27, 631 (1982)
S. Aubry and P.Y. Le Daeron, Physica D, 8, 381 (1983)
J.N.Mather, “A Criterion for the Non-existence of Invariant Circles”, preprint
R.B.Griffiths, private communication
J.N.Mather, “Non-existence of inv. circles”, Ergo.Th. and Dynam.Sys., to appear
K. Whiteman, Rep. Prog. Phys., 40, 1033 (1977)
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© 1984 Springer-Verlag Berlin Heidelberg
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Chou, W. (1984). A New Computable Criterion for the Non-Existence of Invariant Circles. In: Horsthemke, W., Kondepudi, D.K. (eds) Fluctuations and Sensitivity in Nonequilibrium Systems. Springer Proceedings in Physics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46508-6_33
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DOI: https://doi.org/10.1007/978-3-642-46508-6_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-46510-9
Online ISBN: 978-3-642-46508-6
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