Investigating Chaos by the Generalized Alignment Index (GALI) Method

TL;DR

This paper explores the behavior of the Generalized Alignment Index (GALI) method in detecting chaos in high-dimensional dynamical systems, specifically analyzing its dependence on system parameters and initial conditions.

Contribution

It provides a numerical analysis of GALI near stable periodic orbits in the Fermi-Pasta-Ulam-Tsingou lattice, revealing how GALI values vary with stability island width and energy levels.

Findings

GALI values increase near the edge of stability islands at fixed energy.

GALI values decrease as the system's energy increases.

Final GALI values are independent of initial deviation vectors.

Abstract

One of the fundamental tasks in the study of dynamical systems is the discrimination between regular and chaotic behavior. Over the years several methods of chaos detection have been developed. Some of them, such as the construction of the system's Poincar\'e Surface of Section, are appropriate for low-dimensional systems. However, an enormous number of real-world problems are described by high-dimensional systems. Thus, modern numerical methods like the Smaller (SALI) and the Generalized (GALI) Alignment Index, which can also be used for lower-dimensional systems, are appropriate for investigating regular and chaotic motion in high-dimensional systems. In this work, we numerically investigate the behavior of the GALIs in the neighborhood of simple stable periodic orbits of the well-known Fermi-Pasta-Ulam-Tsingou lattice model. In particular, we study how the values of the GALIs depend on the width of the stability island and the system's energy. We find that the asymptotic GALI values increase when the studied regular orbits move closer to the edge of the stability island for fixed energy, while these indices decrease as the system's energy increases. We also investigate the dependence of the GALIs on the initial distribution of the coordinates of the deviation vectors used for their computation and the corresponding angles between these vectors. In this case, we show that the final constant values of the GALIs are independent of the choice of the initial deviation vectors needed for their computation.