  Realizing topological insulators is of great current interest because of
their remarkable properties and possible future applications. There are recent
proposals, based on Floquet analyses, that one can generate topologically
nontrivial insulators by periodically driving topologically trivial ones. Here
we address what happens if one follows the dynamics in such systems.
Specifically, we present an exact study of the time evolution of a
graphene-like system subjected to a circularly polarized electric field. We
prove that, for infinite (translationally invariant) systems, the Chern number
is conserved under unitary evolution. For systems with boundaries, on the other
hand, we show that a properly defined topological invariant, the Bott index,
can change. Hence, it should be possible to experimentally prepare topological
states starting from non-topological ones. We show that the chirality of the
edge current in such systems can be controlled by adjusting the filling.
