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Published in: Foundations of Computational Mathematics 6/2017

25-08-2016

Fourier–Taylor Approximation of Unstable Manifolds for Compact Maps: Numerical Implementation and Computer-Assisted Error Bounds

Author: J. D. Mireles James

Published in: Foundations of Computational Mathematics | Issue 6/2017

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Abstract

We develop and implement a semi-numerical method for computing high-order Taylor approximations of unstable manifolds for hyperbolic fixed points of compact infinite-dimensional maps. The method can follow folds in the embedding and describes precisely the dynamics on the manifold. In order to ensure the accuracy of our computations in spite of the many truncation and round-off errors, we develop a posteriori error bounds for the approximations. Deliberate control of round-off errors (using interval arithmetic) in conjunction with explicit analytical estimates leads to mathematically rigorous computer-assisted theorems describing precisely the truncation errors for our approximation of the invariant manifold. The method is applied to the Kot-Schaffer model of population dynamics with spatial dispersion.

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Appendix
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Metadata
Title
Fourier–Taylor Approximation of Unstable Manifolds for Compact Maps: Numerical Implementation and Computer-Assisted Error Bounds
Author
J. D. Mireles James
Publication date
25-08-2016
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 6/2017
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-016-9325-9

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