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 andtheir stable and unstable manifolds in stroboscopic Poincar\'e maps. The methodhas two steps. In the first step we compute, by means of a quadraticallyconvergent scheme, the Fourier series of the torus, its Floquet transformation,and its Floquet matrix. If the torus has stable and/or unstable directions, inthe second step we compute the Taylor-Fourier expansions of the correspondinginvariant manifolds up to a given order. The paper also discusses the case inwhich the torus is highly unstable so that a multiple shooting strategy isneeded to compute the torus. If the order of the Taylor expansion of themanifolds is fixed and N is the number of Fourier modes, the wholecomputational effort (torus and manifolds) increases as O(N log N) q and thememory required behaves as O(N). This makes the algorithm very suitable tocompute high-dimensional tori for which a huge number of Fourier modes areneeded. Besides, the algorithm has a very high degree of parallelism. The paperincludes examples where we compute invariant tori (of dimensions up to 5) ofquasi-periodically forced ODEs. The computations are run in a parallel computerand its efficiency with respect to the number of processors is also discussed.