  In this paper we extend previous results on convergent perturbative solutions
of the Schroedinger equation of a class of periodically time-dependent
two-level systems. The situation treated here is particularly suited for the
investigation of two-level systems exhibiting the phenomenon of (approximate)
dynamical localisation. We also present a convergent perturbative expansion for
the secular frequency and discuss in detail the particular case of
monochromatic interactions (ac-dc fields), providing a complete perturbative
solution for that case. Our method is based on a ``renormalisation'' procedure,
which we develop in a more systematic way here. For being free of secular terms
and uniformly convergent in time, our expansions allow a rigorous study of the
long-time behaviour of such systems and are also well-suited for numerical
computations, as we briefly discuss, leading to very accurate calculations of
quantities like transition probabilities for very long times compared to the
cycles of the external field.
