We introduce an integrable isochronous system and perturb its frequenc
y by an external-deterministic or purely random-noise. Under the pertu
rbation the action variable evolves in time: the corresponding diffusi
on coefficient is exactly computed and its dependence on the magnitude
of the perturbation is carefully investigated. Different behaviors ar
e found and justified: the quasilinear approximation, the superlinear
regime, and the ballistic motion.