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Foundations of Computational Mathematics

, Volume 17, Issue 6, pp 1467–1523 | Cite as

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

  • J. D. Mireles JamesEmail author
Article

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.

Keywords

Unstable manifolds Compact maps Infinite dimensional dynamical systems Parameterization Method A posteriori analysis Computer-assisted proof 

Mathematics Subject Classification

37L99 45E10 34C45 37C05 37D10 37M99 65G20 37L65 

Notes

Acknowledgments

The final version of the manuscript was greatly improved thanks to the suggestions of two anonymous referees. Thanks also goes to Mr. Jorge Gonzalez, Mr. Shane Kepley, Mr. Maxime Murray, and Mr. David Blessing for carefully reading the manuscript and for many additional comments and corrections. The author would like to thank Rafael de la Llave, Konstantin Mischaikow, Jan Bouwe van den Berg, Christian Reinhardt, Vincent Naudot, Sarah Day, Bill Kalies, and Jean-Philippe Lessard for helpful discussions, comments, and suggestions during the preparation of this manuscript. The author was partially supported by National Science Foundation Grant DMS 1318172.

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Copyright information

© SFoCM 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesFlorida Atlantic UniversityBoca RatonUSA

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