  We study the quantum dynamics of Bose-Einstein condensates when the
scattering length is modulated periodically or quasi-periodically in time
within the Bogoliubov framework. For the periodically driven case, we consider
two protocols where the modulation is a square-wave or a sine-wave. In both
protocols for each fixed momentum, there are heating and non-heating phases,
and a phase boundary between them. The two phases are distinguished by whether
the number of excited particles grows exponentially or not. For the
quasi-periodically driven case, we again consider two protocols: the
square-wave quasi-periodicity, where the excitations are generated for almost
all parameters as an analog of the Fibonacci-type quasi-crystal; and the
sine-wave quasi-periodicity, where there is a finite measure parameter regime
for the non-heating phase. We also plot the analogs of the Hofstadter butterfly
for both protocols.
