  This work analyses the dynamical properties of periodically-driven band
models. Focusing on the case of Bose-Einstein condensates, and using a
meanfield approach to treat inter-particle collisions, we identify the origin
of dynamical instabilities arising due to the interplay between the external
drive and interactions. We present a widely-applicable generic numerical method
to extract instability rates, and link parametric instabilities to uncontrolled
energy absorption at short times. Based on the existence of parametric
resonances, we then develop an analytical approach within Bogoliubov theory,
which quantitatively captures the instability rates of the system, and provides
an intuitive picture of the relevant physical processes, including an
understanding of how transverse modes affect the formation of parametric
instabilities. Importantly, our calculations demonstrate an agreement between
the instability rates determined from numerical simulations, and those
predicted by theory. To determine the validity regime of the meanfield
analysis, we compare the latter to the weakly-coupled conserving approximation.
The tools developed and the results obtained in this work are directly relevant
to present-day ultracold-atom experiments based on shaken optical lattices, and
are expected to provide an insightful guidance in the quest for Floquet
engineering.
