Differential maps, difference maps, interpolated maps, and long term prediction

Mapping techniques may be thought to be attractive for the long term prediction of motion in accelerators, especially because a simple map can approximately represent an arbitrarily complicated lattice. The intention of this paper is to develop prejudices as to the validity of such methods by applying them to a simple, exactly solveable, example. It is shown that a numerical interpolation map, such as can be generated in the accelerator tracking program TEAPOT, predicts the evolution more accurately than an analytically derived differential map of the same order. Even so, in the presence of ''appreciable'' nonlinearity, it is shown to be impractical to achieve ''accurate'' prediction beyond some hundreds of cycles of oscillation. This suggests that the value of nonlinear maps is restricted to the parameterization of only the ''leading'' deviation from linearity. 41 refs., 6 figs