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Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Tillämpad matematik och statistik.
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 (engelsk)Inngår i: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 17, nr 5, s. 1123-1193Artikkel i tidsskrift (Fagfellevurdert) 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.

sted, utgiver, år, opplag, sider
SPRINGER , 2017. Vol. 17, nr 5, s. 1123-1193
Emneord [en]
A posteriori KAM theory, Computer-assisted proofs, Russmann estimates, Fast Fourier transform
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Identifikatorer
URN: urn:nbn:se:uu:diva-337120DOI: 10.1007/s10208-016-9339-3ISI: 000412483300001OAI: oai:DiVA.org:uu-337120DiVA, id: diva2:1168556
Tilgjengelig fra: 2017-12-21 Laget: 2017-12-21 Sist oppdatert: 2017-12-21bibliografisk kontrollert

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