  The appearance and time variability of accreting millisecond X-ray pulsars
(hereafter AMXPs, e.g. Wijnands & van der Klis 1998) depends strongly on the
accretion rate, the effective viscosity and the effective magnetic diffusivity
of the disk-magnetosphere boundary. The accretion rate is the main parameter
which determines the location of the magnetospheric radius of the star for a
given stellar magnetic field. We introduce a classification of accreting
neutron stars as a function of the accretion rate and show the corresponding
stages obtained from our global 3D magnetohydrodynamic (MHD) simulations and
from our axisymmetric MHD simulations. We discuss the expected variability
features in each stage of accretion, both periodic and quasi-periodic (QPOs).
We conclude that the periodicity may be suppressed at both very high and very
low accretion rates. In addition the periodicity may disappear when ordered
funnel flow accretion is replaced by disordered accretion through the
interchange instability.
