Abstract

This thesis presents an algorithm for the computation of quasi-periodic
invariant tori. The algorithm is based on an invariance equation for
tori which are densely filled by a quasi-periodic orbit. This equation
is derived without introducing (local) torus coordinates, which greatly
simplifies the construction of discretisation methods and distinguishes
the approach discussed here from earlier ones.

Similar to periodic solutions of autonomous systems, a solution of the
invariance equation has a free phase for each unknown basic frequency.
These free phases can be fixed by extending the equation by phase
conditions. The phase conditions given here are generalisations of the
well-known integral condition for periodic orbits. It is shown that an
approximate solution of the extended invariance equation can be computed
using Newton's method for functions.

Concrete algorithms are constructed by discretising the extended
invariance equation using finite-difference and, for comparison,
Fourier-Galerkin methods. These methods are independent of the
stability-type of the torus. Convergence of the finite-difference method
is shown under the restrictions that the system is available in a
partitioned form and that the torus is attractive or attractive after
reversal of time, respectively. The proof of stability is still open for
the extended system.

A pseudo-arc-length continuation based on the methods discussed here (as
correctors) is implemented in the software package {\tt torcont}. It was
successfully tested on numerous examples, some of which are discussed
in this thesis.