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

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.