  Driving a quantum system periodically in time can profoundly alter its
long-time dynamics and trigger topological order. Such schemes are particularly
promising for generating non-trivial energy bands and gauge structures in
quantum-matter systems. Here, we develop a general formalism that captures the
essential features ruling the dynamics: the effective Hamiltonian, but also the
effects related to the initial phase of the modulation and the micro-motion.
This framework allows for the identification of driving schemes, based on
general N-step modulations, which lead to configurations relevant for quantum
simulation. In particular, we explore methods to generate synthetic spin-orbit
couplings and magnetic fields in cold-atom setups.
