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

Joan Gimeno, Àngel Jorba, Begoña Nicolás, Estrella Olmedo

TL;DR

This procedure to compute reducible invariant tori and their stable and unstable manifolds in Poincar´e maps has a very high degree of parallelism and is suitable for high-dimensional tori for which a huge number of Fourier modes are needed.

Abstract

In this paper we present a procedure to compute reducible invariant tori and their stable and unstable manifolds in stroboscopic 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 O(N log N) q and the memory required behaves as O(N). This makes the algorithm very suitable to compute high-dimensional 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 quasi-periodically forced ODEs. The computations are run in a parallel computer and its efficiency with respect to the number of processors is also discussed.

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

TL;DR

This procedure to compute reducible invariant tori and their stable and unstable manifolds in Poincar´e maps has a very high degree of parallelism and is suitable for high-dimensional tori for which a huge number of Fourier modes are needed.

Abstract

In this paper we present a procedure to compute reducible invariant tori and their stable and unstable manifolds in stroboscopic 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 O(N log N) q and the memory required behaves as O(N). This makes the algorithm very suitable to compute high-dimensional 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 quasi-periodically forced ODEs. The computations are run in a parallel computer and its efficiency with respect to the number of processors is also discussed.
Paper Structure (24 sections, 5 theorems, 73 equations, 3 tables)

This paper contains 24 sections, 5 theorems, 73 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Reducibility of the system
  3. Parametrization of invariant manifolds
  4. Computation of the invariant manifold
  5. Stable invariant manifold
  6. Scaling factor
  7. Multiple shooting
  8. Reducibility and multiple shooting
  9. Multiple shooting applied to invariant manifolds
  10. Computer implementation
  11. Manipulation of Fourier series in several variables
  12. Truncated Fourier series representation
  13. Major orders of Fourier coefficients
  14. Jet transport
  15. Parallelism
  16. ...and 9 more sections

Key Result

Lemma 3.1

Let $B$ be a Floquet matrix associated with the frequency vector $\rho$ on $\mathbb{T} ^d$ and let $|\lambda| \ne 1$ be a real number satisfying that for each eigenvalue $\mu$ of $B$ and a fixed $m \in \mathbb{N}$, $m \geq 2$, Then for all smooth function $g _m$ on $\mathbb{T} ^d$, there exists a unique smooth function $u _m$ such that

Theorems & Definitions (13)

  • Definition 2.1: Reducible skew product
  • Definition 2.2: Discrete reducible torus
  • Remark 2.1: Continuous reducible torus
  • Remark 2.2
  • Lemma 3.1
  • Remark 3.1
  • Lemma 4.1
  • Remark 4.1: Matrix-form
  • Lemma 4.2
  • Lemma 4.3: see §51 in Wilkinson65
  • ...and 3 more