  Nonlinear classical dissipative systems present a rich phenomenology in their
"route to chaos", including period-doubling, i.e. the system evolves with a
period which is twice that of the driving. However, typically the attractor of
a periodically driven quantum open system evolves with a period which exactly
matches that of the driving. Here we analyze a manybody open quantum system
whose classical correspondent presents period-doubling. We show that by
analysing the spectrum of the periodic propagator and by studying the dynamical
correlations, it is possible to show the occurrence of period-doubling in the
quantum (period-$1$) steady state. We also discuss that such systems are
natural candidates for clean Floquet time crystals.
