  We study time-periodic forcing of spatially-extended patterns near a
Turing-Hopf bifurcation point. A symmetry-based normal form analysis yields
several predictions, including that (i) weak forcing near the intrinsic Hopf
frequency enhances or suppresses the Turing amplitude by an amount that scales
quadratically with the forcing strength, and (ii) the strongest effect is seen
for forcing that is detuned from the Hopf frequency. To apply our results to
specific models, we perform a perturbation analysis on general two-component
reaction-diffusion systems, which reveals whether the forcing suppresses or
enhances the spatial pattern. For the suppressing case, our results explain
features of previous experiments on the CDIMA chemical reaction. However, we
also find examples of the enhancing case, which has not yet been observed in
experiment. Numerical simulations verify the predicted dependence on the
forcing parameters.
