Experiments with ultracold atoms in optical lattices usually involve a
weak parabolic trapping potential which merely serves to confine the atoms, but
otherwise remains negligible. In contrast, we suggest a different class of experiments
in which the presence of a stronger trap is an essential part of the set-up. Because
the trap-modified on-site energies exhibit a slowly varying level spacing, similar to
that of an anharmonic oscillator, an additional time-periodic trap modulation with
judiciously chosen parameters creates nonlinear resonances which enable efficient
Floquet engineering. We employ a Mathieu approximation for constructing the nearresonant
Floquet states in an accurate manner and demonstrate the emergence of
effective ground states from the resonant trap eigenstates. Moreover, we show that the
population of the Floquet states is strongly affected by the phase of a sudden turn-on
of the trap modulation, which leads to significantly modified and rich dynamics. As
a guideline for further studies, we argue that the deliberate population of only the
resonance-induced effective ground states will allow one to realize Floquet condensates
which follow classical periodic orbits, thus providing challenging future perspectives
for the investigation of the quantum-classical correspondence.