  The bosonic Josephson junction, one of the maximally simple models for
periodic-driven many-body systems, has been intensively studied in the past two
decades. Here, we revisit this problem with five different methods, all of
which have solid theoretical reasoning. We find that to the order of
$\omega^{-2}$ ($\omega$ is the modulating frequency), these approaches will
yield slightly different Floquet Hamiltonians. In particular, the parameters in
the Floquet Hamiltonians may be unchanged, increased, or decreased, depending
on the approximations used. Especially, some of the methods generate new
interactions, which still preserve the total number of particles; and the
others do not. The validity of these five effective models is verified using
dynamics of population imbalance and self-trapping phase transition. In all
results, we find the method by first performing a unitary rotation to the
Hamiltonian will have the highest accuracy. The difference between them will
become significate when the modulating frequency is comparable with the driving
amplitude. The results presented in this work indicate that the analysis of the
Floquet Hamiltonian has some kind of subjectivity, which will become an
important issue in future experiments with the increasing of precision. We
demonstrate this physics using a Bose-Josephson junction, and it is to be hoped
that the validity of these methods and their tiny differences put forward in
this work can be verified in realistic experiments in future using quantum
simulating platforms, including but not limited to ultracold atoms.
