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Title: | Numerical computation of high-order expansions of invariant manifolds of high-dimensional tori |
Author: | Gimeno, Joan Jorba i Monte, Àngel Nicolás, Begoña Olmedo, Estrella |
Keywords: | Sistemes dinàmics diferenciables Anàlisi numèrica Processament en paral·lel (Ordinadors) Differentiable dynamical systems Numerical analysis Parallel processing (Electronic computers) |
Issue Date: | Sep-2022 |
Publisher: | Society for Industrial and Applied Mathematics. |
Abstract: | In this paper we present a procedure to compute reducible invariant tori and their stable and unstable manifolds in Poincaré maps. The method has two steps. In the first step we compute, by means of a quadratically convergent scheme, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. If the torus has stable and/or unstable directions, in the second step we compute the Taylor--Fourier expansions of the corresponding invariant manifolds up to a given order. The paper also discusses the case in which the torus is highly unstable so that a multiple shooting strategy is needed to compute the torus. If the order of the Taylor expansion of the manifolds is fixed and $N$ is the number of Fourier modes, the whole computational effort (torus and manifolds) increases as $\mathcal{O}(N \log N)$ and the memory required behaves as $\mathcal{O}(N)$. This makes the algorithm very suitable to compute highdimensional tori for which a huge number of Fourier modes are needed. Besides, the algorithm has a very high degree of parallelism. The paper includes examples where we compute invariant tori (of dimensions up to 5) of quasiperiodically forced ODEs. The computations are run in a parallel computer, and the method's efficiency with respect to the number of processors is also discussed. |
Note: | Reproducció del document publicat a: https://doi.org/10.1137/21M1458363 |
It is part of: | SIAM Journal On Applied Dynamical Systems, 2022, vol. 21, num. 3, p. 1832-1861 |
URI: | http://hdl.handle.net/2445/194435 |
Related resource: | https://doi.org/10.1137/21M1458363 |
ISSN: | 1536-0040 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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