Scalar   one-dimensional  parabolic   equations with      periodically
time-dependent nonlinearities  are considered. For each such equation,
the associated discrete-time dynamical system  is shown to admit Morse
decompositions of the global  attractor whose Morse sets are contained
in a  given,  arbitrarily small   neighborhood of  the  set  of  fixed points.
Existence of such  Morse decompositions implies that the chainrecurrent
set coincides with the  set of fixed points. In  particular, the dynamical
system has  a gradient-like structure. As an application of   these  results,
a  description  of   the  asymptotic behavior  of solutions  of asymptotically
periodic equations  is given: any bounded solution  approaches a set    of
periodic solutions of  the  limiting equation.  Other possible  applications to
nonlocal equations and thin domain problems are discussed.