Time-periodic  reaction-diffusion equations  can  be discussed  in the
context of  discrete-time  strongly  monotone  dynamical systems.   It
follows from  the general  theory that typical   trajectories approach
stable periodic solutions.   Among these periodic solutions, there are
some that have the same  period as the  equation, but, possibly, there
might   be   others with  larger   minimal  periods  (these are called
subharmonic solutions). The problem of existence of stable subharmonic
solutions is  therefore of fundamental importance  in the study of the
behavior of  solutions.  We  address this problem   for two classes of
reaction  diffusion  equations   under  Neumann  boundary  conditions.
Namely, we consider spatially inhomogeneous  equations, which can have
stable subharmonic solutions  on any domain, and spatially homogeneous
equations,  which  can  have   such solutions  on   some  (necessarily
non-convex) domains.