Analytic Estimates for the Dynamic Aperture of Non-Linear Lattices

For an accelerator lattice containing non-linear elements, particle amplitudes outside a "dynamic aperture" are unstable. In this paper an analytic procedure for estimating this aperture not depending on resonance assumptions is described. The amplitude x(N) after N iterations is expanded in powers of the initial amplitude x0. Low order coefficients are obtained using the symbolic algebraic manipulation program REDUCE. For intermediate values of N (e.g., 100) these terms give an asymptotic description which, by particle tracking, is shown to be valid. This leads to an estimate of the dynamic aperture and to its possible enlargement.