  We use color-magnitude diagram synthesis together with theoretical relations
from non-linear pulsation models to approach the long-standing problem of the
Oosterhoff dichotomy related to the distribution of the mean periods of
fundamental RR Lyrae variables in globular clusters. By adopting the chemical
composition determined from spectroscopic observations and a criterion to
account for the hysteresis mechanism, we tuned age and mass-loss to
simultaneously reproduce the morphology of both the turn-off and the Horizontal
Branch of a sample of 17 globular clusters of the Milky Way and of nearby dwarf
galaxies in the crucial metallicity range (-1.9<[Fe/H]<-1.4) where the
Oostheroff transition is apparent. We find that the Oosterhoff dichotomy among
Galactic globular clusters is naturally reproduced by models. The analysis of
the relative impact of the various involved parameters indicates that the main
responsibles of the dichotomy are the peculiar distribution of clusters in the
age-metallicity plane and the hysteresis. In particular, there is a clear
connection between the two main branches of the age-metallicity relation for
Galactic globular clusters and the Oosterhoff groups. The properties of
clusters' RR Lyrae belonging to other Oostheroff groups (OoInt and OoIII) are
instead not well reproduced. While for OoIII clusters a larger helium abundance
for a fraction of the cluster's stars can reconcile the model prediction with
observations, some other parameter affecting both the Horizontal Branch
morphology and the RR Lyrae periods is required to reproduce the behavior of
OoInt clusters.
