Rigorous computer assisted application of KAM theory: a modern approach

TL;DR

This paper introduces a rigorous, computer-assisted methodology based on an a posteriori approach to verify the existence of invariant tori in symplectic maps, utilizing Fourier transforms for efficiency.

Contribution

It develops a novel, efficient computer-assisted method to verify KAM theorem hypotheses, enabling rigorous proofs of invariant tori in complex dynamical systems.

Findings

Proved invariant curves for the standard map up to ε=0.9716.

Verified meandering curves for the non-twist standard map.

Established existence of 2D tori for the Froeschlé map.

Abstract

In this paper we present and illustrate a general methodology to apply KAM theory in particular problems, based on an {\em a posteriori} approach. We focus on the existence of real-analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to $\eps=0.9716$), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschl\'e map.