M. Blank, T. Kruger, L. Pustyl'nikov
A KAM type theorem for systems with round-off errors
(36K, LaTeX)
ABSTRACT. Perturbations due to round-off errors in computer modeling are
discontinuous and therefore one cannot use results like KAM theory
about smooth perturbations of twist maps. We elaborate a special
approximation scheme to construct two smooth periodic on the angle
perturbations of the twist map, bounding the discretized map from
above and from below. Using the well known Moser's theorem we prove
the existence of invariant curves for these smooth approximations. As
a result we are able to prove that any trajectory of the discretized
twist map is eventually periodic. We discuss also some questions,
concerning the application of the intersection property in Moser's
theorem and the generalization of our results for the twist map in
Lobachevski plane.