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The Parameterization Method in KAM Theory

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The Parameterization Method for Invariant Manifolds

Part of the book series: Applied Mathematical Sciences ((AMS,volume 195))

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Abstract

This chapter is devoted to the parameterization method in KAM theory, also referred to as KAM theory without action-angle coordinates. The chapter states and proves a KAM theorem in a posteriori format, with explicit bounds suitable to be applied in an effective and quantitative way. The reader can skip the proof without losing the flavor of the application of the method. We have included full descriptions of the derived algorithms, and applications to the examples that follow, which are: application of the theorem (by hand calculations) to obtain persistence of the golden invariant curve for tiny values of the parameter of the standard map, numerical continuation of this same curve up to values close to the breakdown, and computation of 2D KAM tori in the Froeschlé map.

A.H. acknowledges support from the Spanish grants MTM2009-09723, MTM2012-32541 and MTM2015-67724-P, and by the Catalan grants 2009-SGR-67 and 2014-SGR-1145. A.L. acknowledges support from postdoctoral positions in the Juan de la Cierva Fellowship JCI-2010-06517 (years from 2012 to 2014) and in the ERC Starting grant 335079 (from 2015), the Spanish grant MTM2012-32541, the ICMAT-Severo Ochoa grant SEV-2015-0554 (MINECO), and the Catalan grants 2009-SGR-859 and 2014-SGR-1145.

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Notes

  1. 1.

    In the present chapter, we consider the study of primary or rotational tori (homotopic to the zero section \( \mathbb{T} \times \{ 0\} \)) but all methods and ideas can be directly adapted to deal with secondary tori.

  2. 2.

    A function \( u: \mathbb{R}^{d} \rightarrow \mathbb{R} \) is 1-periodic if \( u(\theta +e) = u(\theta ) \) for all \( \theta \in \mathbb{R}^{d} \) and \( e \in \mathbb{Z}^{d} \). A function \( u: \mathbb{T}^{d} \rightarrow \mathbb{R} \) is viewed as a 1-periodic function \( u: \mathbb{R}^{d} \rightarrow \mathbb{R} \). Similarly, a function \( g: \tilde{\mathcal{A}}\rightarrow \mathbb{R} \) is 1-periodic in x if g(x + e, y) = g(x, y) for all \( x \in \mathbb{R}^{d} \) and \( e \in \mathbb{Z}^{d} \). A function \( g: \mathcal{A}\rightarrow \mathbb{R} \) is viewed as a function \( g: \tilde{\mathcal{A}}\rightarrow \mathbb{R} \) that is 1- periodic in x.

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  1. Barcelona Graduate School of Mathematics Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain

    Àlex Haro

  2. Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain

    Alejandro Luque

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Haro, À., Luque, A. (2016). The Parameterization Method in KAM Theory. In: The Parameterization Method for Invariant Manifolds. Applied Mathematical Sciences, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-29662-3_4

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