  The Smaller Alignment Index (SALI) is a very useful and efficient indicator
that can distinguish rapidly and with certainty between ordered and chaotic
motion in Hamiltonian systems. This is based on the different behavior of the
SALI in the two cases, as it fluctuates around non-zero values for ordered
orbits and converges exponentially to zero for chaotic orbits. In this paper we
present a detailed numerical study, comparing the advantages of the SALI to
those of the Maximal Lyapunov Exponent (MLE), which is the traditional
indicator of chaos detection in dynamical systems. Exploiting these advantages,
we demonstrate how the SALI can be used to identify even tiny regions of order
and chaos in Hamiltonian systems of 2 and 3 degrees of freedom.
