On the behavior of the Generalized Alignment Index (GALI) method for regular motion in multidimensional Hamiltonian systems

TL;DR

This paper studies the GALI method's behavior for regular orbits in multidimensional Hamiltonian systems, revealing how its asymptotic values vary with system parameters and confirming its robustness across initial deviations.

Contribution

It provides a detailed analysis of GALI$_k$ behavior near stable periodic orbits in the FPUT $eta$ model, highlighting how it depends on energy and orbit position, and confirms its independence from initial deviation vectors.

Findings

GALI$_k$ decreases with increasing $k$ and energy approaching destabilization.

GALI$_k$ increases as the orbit moves away from the periodic orbit.

GALI$_k$ behavior is independent of initial deviation vectors.

Abstract

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.