  We perform an extensive and detailed analysis of the generalized diffusion
processes in deterministic area preserving maps with noncompact phase space,
exemplified by the standard map, with the special emphasis on understanding the
anomalous diffusion arising due to the accelerator modes. The accelerator modes
and their immediate neighborhood undergo ballistic transport in phase space,
and also the greater vicinity of them is still much affected ("dragged") by
them, giving rise to the non-Gaussian (accelerated) diffusion. The systematic
approach rests upon the following applications: the GALI method to detect the
regular and chaotic regions and thus to describe in detail the structure of the
phase space, the description of the momentum distribution in terms of the
L\'evy stable distributions, the numerical calculation of the diffusion
exponent and of the corresponding diffusion constant. We use this approach to
analyze in detail and systematically the standard map at all values of the kick
parameter $K$, up to $K=70$. All complex features of the anomalous diffusion
are well understood in terms of the role of the accelerator modes, mainly of
period 1 at large $K\ge 2\pi$, but also of higher periods (2,3,4,...) at
smaller values of $K\le 2\pi$.
