The Smaller (SALI) and the Generalized (GALI) Alignment Indices: Efficient Methods of Chaos Detection

TL;DR

The paper introduces and explains the Smaller (SALI) and Generalized (GALI) indices, efficient chaos detection methods based on deviation vectors, applicable to various physical systems for identifying regular and chaotic motion.

Contribution

It presents a concise overview of SALI and GALI methods, highlighting their effectiveness in chaos detection and their applications across multiple scientific disciplines.

Findings

SALI and GALI effectively distinguish between regular and chaotic dynamics.

These indices can identify stable and unstable periodic orbits.

Applications span celestial mechanics, galactic dynamics, and condensed matter physics.

Abstract

We provide a concise presentation of the Smaller (SALI) and the Generalized Alignment Index (GALI) methods of chaos detection. These are efficient chaos indicators based on the evolution of two or more, initially distinct, deviation vectors from the studied orbit. After explaining the motivation behind the introduction of these indices, we sum up the behaviors they exhibit for regular and chaotic motion, as well as for stable and unstable periodic orbits, focusing mainly on finite-dimensional conservative systems: autonomous Hamiltonian models and symplectic maps. We emphasize the advantages of these methods in studying the global dynamics of a system, as well as their ability to identify regular motion on low dimensional tori. Finally we discuss several applications of these indices to problems originating from different scientific fields like celestial mechanics, galactic dynamics, accelerator physics and condensed matter physics.