  For C1-smooth strongly monotone discrete-time dynamical systems, it is shown
that ``convergence to linearly stable cycles" is a prevalent asymptotic
behavior in the measuretheoretic sense. The results are then applied to classes
of time-periodic parabolic equations and give new results on prevalence of
convergence to periodic solutions. In particular, for equations with Neumann
boundary conditions on convex domains, we show the prevalence of the set of
initial conditions corresponding to the solutions that converge to
spatiallyhomogeneous periodic solutions. While, for equations on radially
symmetric domains, we obtain the prevalence of the set of initial values
corresponding to solutions that are asymptotic to radially symmetric periodic
solutions.
