  We study the breaking of the discrete time-translation symmetry in small
periodically driven quantum systems. Such systems are intermediate between
large closed systems and small dissipative systems, which both display the
symmetry breaking, but have qualitatively different dynamics. As a nontrivial
example we consider period tripling in a quantum nonlinear oscillator. We show
that, for moderately strong driving, the period tripling is robust on an
exponentially long time scale, which is further extended by an even weak
decoherence.
