  Time-periodic driving provides a promising route to engineer non-trivial
states in quantum many-body systems. However, while it has been shown that the
dynamics of integrable systems can synchronize with the driving into a
non-trivial periodic motion, generic non-integrable systems are expected to
heat up until they display a trivial infinite-temperature behavior. In this
paper we show that a quasi-periodic time evolution over many periods can also
emerge in systems with weak integrability breaking, with a clear separation of
the timescales for synchronization and the eventual approach of the
infinite-temperature state. This behavior is the analogue of prethermalization
in quenched systems. The synchronized state can be described using a
macroscopic number of approximate constants of motion. We corroborate these
findings with numerical simulations for the driven Hubbard model.
