Numerical computation of high-order expansions of invariant manifolds of high-dimensional tori
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 i...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/194435 |
| Acceso en línea: | https://hdl.handle.net/2445/194435 |
| Access Level: | acceso abierto |
| Palabra clave: | Sistemes dinàmics diferenciables Anàlisi numèrica Processament en paral·lel (Ordinadors) Differentiable dynamical systems Numerical analysis Parallel processing (Electronic computers) |
| Sumario: | 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. |
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