  We investigate the behavior of the Generalized Alignment Index of order $k$
(GALI$_k$) for regular orbits of multidimensional Hamiltonian systems. The
GALI$_k$ is an efficient chaos indicator, which asymptotically attains positive
values for regular motion when $2\leq k \leq N$, with $N$ being the dimension
of the torus on which the motion occurs. By considering several regular orbits
in the neighborhood of two typical simple, stable periodic orbits of the
Fermi-Pasta-Ulam-Tsingou (FPUT) $\beta$ model for various values of the
system's degrees of freedom, we show that the asymptotic GALI$_k$ values
decrease when the index's order $k$ increases and when the orbit's energy
approaches the periodic orbit's destabilization energy where the stability
island vanishes, while they increase when the considered regular orbit moves
further away from the periodic one for a fixed energy. In addition, performing
extensive numerical simulations we show that the index's behavior does not
depend on the choice of the initial deviation vectors needed for its
evaluation.
