Application of the SALI chaos detection method to accelerator mappings

We apply the Smaller ALignment Index (SALI) method to a four-dimensional mapping of accelerator dynamics in order to distinguish rapidly, reliably and accurately between ordered and chaotic motion. The main advantage of this index is that it tends exponentially to zero in the case of chaotic orbits, while it fluctuates around non-zero values in the case of quasiperiodic trajectories. Thus, it avoids the notorious ambiguities concerning the eventual convergence of (maximum) Lyapunov exponents to (positive) non-zero values. Exploiting the different behavior of SALI in these two cases we produce phase space ‘charts’ where regions of chaos and order are clearly identified. Evaluating the percentage of chaotic and escaping orbits as a function of the distance from the origin we are able to estimate rapidly and accurately the boundaries of the dynamical aperture of a proton beam, passing repeatedly through an array of magnetic focusing elements.