Discrete, strongly monotone dynamical systems
of class $C^{1,\alpha}$,
with $\alpha>0$, are considered.
Asymptotic
behavior of typical
trajectories, that is, trajectories emanating from a
residual
set of
initial conditions is investigated. The main theorem
asserts
that a
typical trajectory converges to a periodic orbit.
It
is known that,
in general, this result cannot be further
improved
so as to state
typical convergence to fixed points
(unlike
for continuous-time
systems, where typical convergence to equilibria
has been proved).
Applications of the abstract
result
to time-periodic parabolic
equations are given.
As one of the hypotheses, existence of a continuous
separation
for
linearization along compact trajectories is assumed. This
hypothesis
has been later proved to be satisfied in general, see the paper
P. Polacik and I. Terescak,
Exponential separation and invariant bundles for maps in ordered
Banach spaces with applications to parabolic
equations,
J. Dynam. Differential Equations. 5 (1993), 279-303