  Shaking optical lattices in a resonant manner offers an efficient and
versatile method to devise artificial gauge fields and topological band
structures for ultracold atomic gases. This was recently demonstrated through
the experimental realization of the Harper-Hofstadter model, which combined
optical superlattices and resonant time-modulations. Adding inter-particle
interactions to these engineered band systems is expected to lead to
strongly-correlated states with topological features, such as fractional Chern
insulators. However, the interplay between interactions and external
time-periodic drives typically triggers violent instabilities and
uncontrollable heating, hence potentially ruling out the possibility of
accessing such intriguing states of matter in experiments. In this work, we
study the early-stage parametric instabilities that occur in systems of
resonantly-driven Bose-Einstein condensates in optical lattices. We apply and
extend an approach based on Bogoliubov theory [PRX 7, 021015 (2017)] to a
variety of resonantly-driven band models, from a simple shaken Wannier-Stark
ladder to the more intriguing driven-induced Harper-Hofstadter model. In
particular, we provide ab initio numerical and analytical predictions for the
stability properties of these topical models. This work sheds light on general
features that could guide current experiments to stable regimes of operation.
