Abstract
We unify, extend, reinterpret and apply criteria of Birkhoff [1], Herman [9], Mather [2, 3], Aubry et al. [4, 5], and Newman and Percival [6] for the nonexistence of invariant circles for area preserving twist maps. The criteria enable one to establish regions of phase space through which no rotational invariant circles pass. For families of maps the same can be done for regions of the combined space of phase points and parameters. The criteria can be implemented rigorously on a computer, and give a practical method of proving quite strong results. As an example, we present a computer program which proved that the “standard map” has no rotational invariant circles for any parameter value |k|≧63/64.
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Communicated by O.E. Lanford
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MacKay, R.S., Percival, I.C. Converse KAM: Theory and practice. Commun.Math. Phys. 98, 469–512 (1985). https://doi.org/10.1007/BF01209326
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DOI: https://doi.org/10.1007/BF01209326