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2022-09

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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/194435

Numerical computation of high-order expansions of invariant manifolds of high-dimensional tori

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Recurs relacionat

https://doi.org/10.1137/21M1458363

Resum

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.

Matèries

Sistemes dinàmics diferenciables, Anàlisi numèrica, Processament en paral·lel (Ordinadors)

Matèries (anglès)

Differentiable dynamical systems, Numerical analysis, Parallel processing (Electronic computers)

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Articles publicats en revistes (Matemàtiques i Informàtica)

Citació

GIMENO, Joan, et al. Numerical computation of high-order expansions of invariant manifolds of high-dimensional tori. SIAM Journal On Applied Dynamical Systems. 2022. Vol. 21, num. 3, pags. 1832-1861. ISSN 1536-0040. [consulted: 19 of August of 2026]. Available at: https://hdl.handle.net/2445/194435

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