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 $2leq k leq N$, 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 systems degrees of freedom, we show that the asymptotic GALI$_k$ values decrease when the indexs order $k$ increases and when the orbits energy approaches the periodic orbits 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 indexs behavior does not depend on the choice of the initial deviation vectors needed for its evaluation.
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