  A famous aspect of discrete dynamical systems defined by area-preserving maps
is the physical interpretation of stochastic transitions occurring locally
which manifest themselves through the destruction of invariant KAM curves and
the local or global onset of chaos. Despite numerous previous investigations
(see in particular Chirikov, Greene, Percival, Escande and Doveil and MacKay)
based on different approaches, several aspects of the phenomenon still escape a
complete understanding and a rigorous description. In particular Greene's
approach is based on several conjectures, one of which is that the stochastic
transition leading to the destruction of the last KAM curve in the standard map
is due the linear destabilization of the elliptic points belonging to a
peculiar family of invariants sets {I(m,n)}
  (rational iterates) having rational winding numbers and associated to the
last KAM curve. Purpose of this work is to analyze the nonlinear phenomena
leading to the stochastic transition in the standard map and their effect on
the destabilization of the invariant sets associated to the KAM curves,
leading, ultimately, to the destruction of the KAM curves themselves.
