  Recent work showed that the exact exchange-correlation potential of
time-dependent density functional theory generically displays dynamical step
structures. These have a spatially non-local and time-non-local dependence on
the density in real time dynamics. The steps are missing in the usual
approximations which consequently yield inaccurate dynamics. Yet these same
approximations typically yield good linear response spectra. Here we
investigate whether the steps appear in the linear response regime, when the
response is calculated from a real-time dynamics simulation, by examining the
exact correlation potential of model two-electron systems at various times. We
find there are no step structures in regions where the system response is
linear. Step structures appear in the correlation potential only in regions of
space where the density response is non-linear; these regions, having
exponentially small density, do not contribute to the observables measured in
linear response.
