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We provide a concise presentation of the Smaller (SALI) and the GeneralizedAlignment Index (GALI) methods of chaos detection. These are efficient chaosindicators based on the evolution of two or more, initially distinct, deviationvectors from the studied orbit. After explaining the motivation behind theintroduction of these indices, we sum up the behaviors they exhibit for regularand chaotic motion, as well as for stable and unstable periodic orbits,focusing mainly on finite-dimensional conservative systems: autonomousHamiltonian models and symplectic maps. We emphasize the advantages of thesemethods in studying the global dynamics of a system, as well as their abilityto identify regular motion on low dimensional tori. Finally we discuss severalapplications of these indices to problems originating from different scientificfields like celestial mechanics, galactic dynamics, accelerator physics andcondensed matter physics.

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