- 09-2 Gemma Huguet, Rafael de la Llave, Yannick Sire
- Fast numerical algorithms for the computation
of invariant tori in Hamiltonian systems
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Jan 1, 09
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Abstract. In this paper, we develop numerical algorithms that use small
requirements of storage and operations for the computation of
invariant tori in Hamiltonian systems (exact symplectic maps and
Hamiltonian vector fields). The algorithms are based on the
parameterization method and follow closely the proof of the KAM
theorem given in [ de la Llave, Gonnzalez, Jorba, Villanueva,
Nonlinearity 2005] and [Fontich, de la Llave, Sire, manuscript 2007] . They
essentially consist in solving a functional equation satisfied by
the invariant tori by using a Newton method.
Using some geometric identities, it is possible to
perform a Newton step using little storage and few operations.
In this paper we focus on the numerical issues of the algorithms
(speed, storage and stability) and we refer to the mentioned papers
for the rigorous results. We show how to compute efficiently both
maximal invariant tori and whiskered tori, together with the
associated invariant stable and unstable manifolds of whiskered
Moreover, we present fast algorithms for the iteration of the
quasi-periodic cocycles and the computation of the invariant bundles, which is
a preliminary step for the computation of invariant
whiskered tori. Since quasi-periodic cocycles appear in other contexts,
this section may be of independent interest.
The numerical methods presented here allow to compute in a unified way
primary and secondary invariant KAM tori. Secondary tori are invariant tori
which can be contracted to a periodic orbit.
We present some preliminary results that ensure that the methods
are indeed implementable and fast. We postpone to
a future paper optimized implementations
and results on the breakdown of invariant tori.