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

AUTHORS:
Joan Gimeno^(1), Angel Jorba^(2), Bego\~na Nicolas^(2), Estrella Olmedo^(3)

(1) Department of Mathematics, University of Rome Tor Vergata, Via
    della Ricerca Scientifica 1, 00133 Rome, Italy
(2) Departament de Matematiques i Informatica
    Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
(3) Barcelona Expert Center (BEC), Institute of Marine Sciences (ICM)
    and Consejo Superior de Investigaciones Cient\'ificas (CSIC),
    P. Mar\'itim de la Barceloneta, 37-49, 08003 Barcelona, Spain

E-mails: gimeno@mat.uniroma2.it, angel@maia.ub.es, bego@maia.ub.es,
         olmedo@icm.csic.es

ABSTRACT:
In this paper we present a procedure to compute reducible invariant
tori and their stable and unstable manifolds in Poincar\'e 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 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.