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Application of the SALI chaos detection method to accelerator mappings

T. Bountis·Ch. Skokos
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Abstract

We apply the Smaller ALignment Index (SALI) method to a 4--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 {\it 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 {\it dynamical aperture} of a proton beam, passing repeatedly through an array of magnetic focusing elements.

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T. Bountis, Ch. Skokos. 2005-12-13. Application of the SALI chaos detection method to accelerator mappings. https://doi.org/10.1016/j.nima.2006.01.009

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