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Foundations of Computational Mathematics

, Volume 17, Issue 5, pp 1123–1193 | Cite as

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

  • J.-Ll. Figueras
  • A. Haro
  • A. Luque
Article

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.

Keywords

A posteriori KAM theory Computer-assisted proofs Rüssmann estimates Fast Fourier transform 

Mathematics Subject Classification

37J40 65G20 65G40 65T50 

Notes

Acknowledgements

We are very grateful to R. Calleja, R. de la Llave and J. Villanueva for useful and fruitful discussions along the last years. We are also grateful to J. B. van den Berg, M. Breden, R. Castelli, A. Celletti, J. Cyranka, J. Gómez-Serrano, J.-P. Lessard, J.D. Mireles-James, K. Mischaikow, C. Simó and P. Zgliczynski for their interest and comments. We would like to acknowledge financial support from the Spanish Grants MTM2012-32541, MTM2015-67724-P (MINECO/FEDER, UE) and the Catalan Grant 2014-SGR-1145. J.-Ll. F. acknowledges the partial support from Essen, L. and C.-G., for mathematical studies. Moreover, A.L. acknowledges support from a postdoctoral position in the ERC Starting Grant 335079. We acknowledge A. Granados and the use of the UPC Dynamical Systems group’s cluster for research computing (see https://dynamicalsystems.upc.edu/en/computing/). Finally, we would like to thank the anonymous referees for helpful comments which led us to write Sect. 7.

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Authors and Affiliations

  1. 1.Departament of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain
  3. 3.Instituto de Ciencias Matemáticas Consejo Superior de Investigaciones CientíficasMadridSpain

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