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Published in:
An Overview of the Parameterization Method for Invariant Manifolds Read chapter Read first chapter

2016 | OriginalPaper | Chapter

4. The Parameterization Method in KAM Theory

Authors : Àlex Haro, Alejandro Luque

Published in: The Parameterization Method for Invariant Manifolds

Publisher: Springer International Publishing

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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.

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previous chapter The Parameterization Method for Quasi-Periodic Systems: From Rigorous Results to Validated Numerics
next chapter A Newton-like Method for Computing Normally Hyperbolic Invariant Tori
Footnotes
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
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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Metadata
Title
The Parameterization Method in KAM Theory
Authors
Àlex Haro
Alejandro Luque
Copyright Year
2016
Book
The Parameterization Method for Invariant Manifolds

Print ISBN: 978-3-319-29660-9

Electronic ISBN: 978-3-319-29662-3

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-29662-3_4

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