  Rotators interacting with a pendulum via small, velocity independent,
potentials are considered. If the interaction potential does not depend on the
pendulum position then the pendulum and the rotators are decoupled and we study
the invariant tori of the rotators system at fixed rotation numbers: we exhibit
cancellations, to all orders of perturbation theory, that allow proving the
stability and analyticity of the dipohantine tori. We find in this way a proof
of the KAM theorem by direct bounds of the $k$--th order coefficient of the
perturbation expansion of the parametric equations of the tori in terms of
their average anomalies: this extends Siegel's approach, from the linearization
of analytic maps to the KAM theory; the convergence radius does not depend, in
this case, on the twist strength, which could even vanish ({\it "twistless KAM
tori"}). The same ideas apply to the case in which the potential couples the
pendulum and the rotators: in this case the invariant tori with diophantine
rotation numbers are unstable and have stable and unstable manifolds ({\it
"whiskers"}): instead of studying the perturbation theory of the invariant tori
we look for the cancellations that must be present because the homoclinic
intersections of the whiskers are {\it "quasi flat"}, if the rotation velocity
of the quasi periodic motion on the tori is large. We rederive in this way the
result that, under suitable conditions, the homoclinic splitting is smaller
than any power in the period of the forcing and find the exact asymptotics in
the two dimensional cases ({\it e.g.} in the case of a periodically forced
pendulum). The technique can be applied to study other quantities: we mention,
as another example, the {\it homoclinic scattering phase shifts}.}
