SALI混沌检测方法在加速器映射中的应用
Application of the SALI chaos detection method to accelerator mappings
摘要：
我们应用小对准指数（SALI）方法的4-维映射的加速器动力学，以区分有序和混沌运动之间的快速，可靠和准确。这个指数的主要优点是，它往往{ \it指数}为零的情况下的混沌轨道，而它的波动在非零值的情况下的准周期轨迹。因此，它避免了关于（最大）李雅普诺夫指数最终收敛到（正）非零值的臭名昭著的模糊性。利用SALI在这两种情况下的不同行为，我们产生相空间“图表”，其中混沌和有序区域被清楚地识别。通过计算混沌轨道和逃逸轨道的百分比随离原点距离的变化，我们能够快速准确地估计质子束在重复通过磁聚焦元件阵列时的{ \it动态孔径}的边界。
英文摘要：
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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