Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach

In this paper, we present and illustrate a general methodology to apply KAM theory in particular problems, based on an 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 [Formula omitted]), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.
Author(s): J.-Ll. Figueras [sup.1] , A. Haro [sup.2] , A. Luque [sup.3] Author Affiliations: (1) Departament of Mathematics, Uppsala University, 0000 0004 1936 9457, grid.8993.b, , Box 480, 751 06, [...]