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

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 finitedimensional 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.

Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method

“…Although the computation of the mLCE is the most widely used technique for characterizing the regular or chaotic nature of orbits, its computational drawbacks, like for example its slow convergence to its limiting value, led to the development of a number of other, efficient chaos detection techniques, which make use of the solutions of the variational equations, like for example the fast Lyapunov indicator (FLI) and its variants [79,80,81,82,83,84], the mean exponential growth of nearby orbits (MEGNO) [85,86,87], the relative Lyapunov indicator (RLI) [88,89,90], the smaller alignment index (SALI) [91,92,93] and its generalization, the GALI [74,75,76,94].…”

Average conservative chaos in quantum dusty plasmas

“…Finally, alignment indexes (SALI and GALI) deserve notification as well, which are mostly used in Hamiltonian systems. These methods are very efficient when studying the global dynamics of the system, and allow to discriminate in a simple way regular motion on low dimensional tori from the unpredictable trajectories of the surrounding chaotic sea [43].…”

Chaoticity in the vicinity of complex unstable periodic orbits in galactic type potentials

“…Thus, in order to evaluate GALI 2 we integrate the equations of motion and the variational equations for two deviation vectors simultaneously. The GALI 2 index behaves as follows (see Skokos and Manos, 2016, and references therein):…”