A KAM type theorem for systems with round-off errors

TL;DR

This paper develops a novel approach to analyze systems with round-off errors by approximating discretized twist maps with smooth perturbations, proving the existence of invariant curves and eventual periodicity of trajectories.

Contribution

Introduces a special approximation scheme for discretized twist maps and extends KAM theory to systems with discontinuous perturbations due to round-off errors.

Findings

Invariant curves exist for smooth approximations of discretized twist maps.

Trajectories of the discretized map are eventually periodic.

Discussion on extending results to Lobachevski plane twist maps.

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.