  Previous works concerning active galactic nuclei (AGN) variability (e.g.,
Blandford \& McKee 1982) have assumed that the emission characteristics of
illuminated clouds are purely a function of the instant continuum flux to which
they are exposed. This paper shows that this assumption is not necessarily
justified and that the history of exposure accounting for ``local delays" due
to finite cloud equilibrium times can also be relevant. For this reason, a new
formalism is developed in this paper for computing the observational properties
of models which have local delays. The nature of the nonlinear behavior that
results is calculated for some very simple nonlinear cloud line emission
models. It is found that the mean response time is a function of the recent
average value of the continuum. Linear models fit to these nonlinear systems
respond too slowly when there are low-energy (and generally rapid) changes in
the continuum, yet respond too rapidly when there are high-energy (and
generally slow) changes in the continuum. As with systems without local delays,
the expression for the time-dependent line flux contains an integration over
history of the ``spatial" response function, which has structure at lags of the
light travel times of the emission region. However, the kernel of this integral
itself is a function of additional integrations over individual ``cloud"
response functions which have structure at lags of the equilibrium times of the
cloud properties relevant to line emission. If instantaneous or linear response
is incorrectly assumed, local delays and nonlinear response can make a system
appear larger than it actually is. These effects are similar to those that
beaming can cause. Local delays can also be a source of asymmetry about the
peak of the cross-correlation function.
