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An Overview of the Parameterization Method for Invariant Manifolds

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Book cover The Parameterization Method for Invariant Manifolds

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

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.

À. Haro acknowledges support from the Spanish grants MTM2009-09723, MTM2012-32541 and MTM2015-67724-P, and the Catalan grants 2009-SGR-67 and 2014-SGR-1145.

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Notes

  1. 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. 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. 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. 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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    Àlex Haro

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Haro, À. (2016). An Overview of the Parameterization Method for Invariant Manifolds. In: The Parameterization Method for Invariant Manifolds. Applied Mathematical Sciences, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-29662-3_1

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