Study of the dynamic aperture of the 4D quadratic map using invariant manifolds

A 4D quadratic map can be used to represent the transfer map of a FODO cell with a sextupolar nonlinearity in the kick approximation. This map describes the transverse betatronic motion of a charged p article in a circular accelerator. The dynamic aperture of such a mapping is analysed, i.e., the domain in phase space where stable motion occurs, as a function of the linear tunes. Starting from the study of the stability properties of the fixed points of low period (one or two), it is shown that the dynamic aperture is related to the invariant manifolds emanating from unstalbe points. This repre nsets a generalisation of a similar result obtained for generic two-dimensional sympletic maps.