  We investigate localization phenomena and stability properties of
quasiperiodic oscillations in $N$ degree of freedom Hamiltonian systems and $N$
coupled symplectic maps. In particular, we study an example of a parametrically
driven Hamiltonian lattice with only quartic coupling terms and a system of $N$
coupled standard maps. We explore their dynamics using the Generalized
Alignment Index (GALI), which constitutes a recently developed numerical method
for detecting chaotic orbits in many dimensions, estimating the dimensionality
of quasiperiodic tori and predicting slow diffusion in a way that is faster and
more reliable than many other approaches known to date.
