  It is known that there are lattice models in which non-interacting particles
get dynamically localized when periodic $\delta$-function kicks are applied
with a particular strength. We use both numerical and analytical methods to
study the effects of interactions in three different models in one dimension.
The systems we have considered include spinless fermions with interactions
between nearest-neighbor sites, the Hubbard model of spin-1/2 fermions, and the
Bose Hubbard model with on-site interactions. We derive effective Floquet
Hamiltonians up to second order in the time period of kicking. Using these we
show that interactions can give rise to a variety of interesting results such
as two-body bound states in all three models and dispersionless many-body bound
states for spinless fermions and bosons. We substantiate these results by exact
diagonalization and stroboscopic time evolution of systems with a finite number
of particles. We derive a low-energy pseudo-spin-1/2 limit of the Bose Hubbard
system in the thermodynamic limit and show that a special case of this has an
exponentially large number of ground states. Finally we study the effect of
changing the strength of the $\delta$-function kicks slightly away from perfect
dynamical localization; we find that a single particle remains dynamically
localized for a long time after which it moves ballistically.
