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Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Univ Barcelona, Dept Matemat Aplicada & Anal, Gran Via 585, E-08007 Barcelona, Spain..
CSIC, Inst Ciencias Matemat, C Nicolas Cabrera 13-15, Madrid 28049, Spain..
2017 (English)In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 17, no 5, 1123-1193 p.Article in journal (Refereed) Published
Abstract [en]

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 ), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschl, map.

Place, publisher, year, edition, pages
SPRINGER , 2017. Vol. 17, no 5, 1123-1193 p.
Keyword [en]
A posteriori KAM theory, Computer-assisted proofs, Russmann estimates, Fast Fourier transform
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-337120DOI: 10.1007/s10208-016-9339-3ISI: 000412483300001OAI: oai:DiVA.org:uu-337120DiVA: diva2:1168556
Available from: 2017-12-21 Created: 2017-12-21 Last updated: 2017-12-21Bibliographically approved

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Figueras, Jordi-Lluis

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