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Foundations of Computational Mathematics
Erschienen in: Foundations of Computational Mathematics 5/2017

17.11.2016

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

verfasst von: J.-Ll. Figueras, A. Haro, A. Luque

Erschienen in: Foundations of Computational Mathematics | Ausgabe 5/2017

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Abstract

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 \(\varepsilon =0.9716\)), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.
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Metadaten
Titel
Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach
verfasst von
J.-Ll. Figueras
A. Haro
A. Luque
Publikationsdatum
17.11.2016
Verlag
Springer US
Erschienen in
Foundations of Computational Mathematics / Ausgabe 5/2017
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-016-9339-3

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