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2016 | OriginalPaper | Buchkapitel

1. An Overview of the Parameterization Method for Invariant Manifolds

verfasst von : Àlex Haro

Erschienen in: The Parameterization Method for Invariant Manifolds

Verlag: Springer International Publishing

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Abstract

This introductory chapter starts by providing an overview of the literature of the parameterization method. After that, it introduces unified formulations of the parameterization method for invariant manifolds of fixed points and for invariant tori in different contexts. These formulations are the basis of the subsequent chapters. This chapter can be considered a reading guide of the rest of the book.

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Nächstes Kapitel Seminumerical Algorithms for Computing Invariant Manifolds of Vector Fields at Fixed Points
Fußnoten
1
In double precision, a \(15000 \times 15000\) matrix uses 1.7 GB of memory and an Intel Core i5 requires more than 4 minutes to solve such a system using the LAPACK library.
 
2
We can also consider the more general case in which \(F: \mathcal{A}_{0} \rightarrow \mathcal{A}_{1}\) is a smooth map between two open sets \(\mathcal{A}_{0},\mathcal{A}_{1} \subset \mathcal{A}\), and \(K:\varTheta \rightarrow \mathcal{A}_{0} \cap \mathcal{A}_{1}\). We do not consider this generality for the sake of notational simplicity.
 
3
In Computer Science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an offline algorithm is given the whole problem data from the beginning and is required to output an answer which solves the problem at hand.
 
4
With a slight abuse of notation, we will not make notational distinctions among coordinates in \(\mathbb{T}^{d}\), \(\mathcal{A}\) and their corresponding covering spaces \(\mathbb{R}^{d}\), \(\tilde{\mathcal{A}}\), and between mappings with those domains and codomains and their corresponding lifts to the appropriate covering spaces.
 
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Metadaten
Titel
An Overview of the Parameterization Method for Invariant Manifolds
verfasst von
Àlex Haro
Copyright-Jahr
2016
Buch
The Parameterization Method for Invariant Manifolds

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

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

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-29662-3_1

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