  We study a discrete non-autonomous system whose autonomous counterpart (with
the frozen bifurcation parameter) admits a saddle-node bifurcation, and in
which the bifurcation parameter slowly changes in time and is characterized by
a sweep rate constant $\epsilon$. The discrete system is more appropriate for
modeling realistic systems since only time series data is available. We show
that in contrast to its autonomous counterpart, when the time mesh size $\Delta
t$ is less than the order $O(\epsilon)$, there is a bifurcation delay as the
bifurcation time-varying parameter is varied through the bifurcation point, and
the delay is proportional to the two-thirds power of the sweep rate constant
$\epsilon$. This bifurcation delay is significant in various realistic systems
since it allows one to take necessary action promptly before a sudden collapse
or shift to different states. On the other hand, when the time mesh size
$\Delta t$ is larger than the order $o(\epsilon)$, the dynamical behavior of
the solution is dramatically changed before the bifurcation point. This
behavior is not observed in the autonomous counterpart. Therefore, the
dynamical behavior of the system strongly depends on the time mesh size.
Finally. due to the very discrete feature of the system, there are no efficient
tools for the analytical study of the system. Our approach is elementary and
analytical.
