Studying the global dynamics of conservative dynamical systems using the SALI chaos detection method

We use the Smaller ALignment Index (SALI) method of chaos detection, to study the global dynamics of conservative dynamical systems described by differential or difference equations. In particular, we consider the well--known 2 and 4--dimensional symplectic standard map, as well as an autonomous Hamiltonian system of 2 and 3 degrees of freedom describing the motion of a star in models of barred galaxies. The application of SALI helped us to compute rapidly and accurately the percentage of regular and chaotic motion for particular values of the parameters of these systems. We were also able to perform a computationally efficient determination of the dependence of these percentages on the variation of several parameters of the studied models.