  In this paper we study the time evolution of a class of two-level systems
driven by periodic fields in terms of new convergent perturbative expansions
for the associated propagator U(t). The main virtue of these expansions is that
they do not contain secular terms, leading to a very convenient method for
quantitatively studying the long-time behaviour of that systems. We present a
complete description of an algorithm to numerically compute the perturbative
expansions. In particular, we applied the algorithm to study the case of an
ac-dc field (monochromatic interaction), exploring various situations and
showing results on (time-dependent) observable quantities, like transition
probabilities. For a simple ac field, we analised particular situations where
an approximate effect of dynamical localisation is exhibited by the driven
system. The accuracy of our calculations was tested measuring the unitarity of
the propagator U(t), resulting in very small deviations, even for very long
times compared to the cycle of the driving field.
