  Unlike classical bifurcations, border-collision bifurcations occur when, for
example, a fixed point of a continuous, piecewise $\mathcal{C}^{1}$ map crosses
a boundary in state space. Although classical bifurcations have been much
studied, border-collision bifurcations are not well understood. This paper
considers a particular class of border-collision bifurcations, i.e.,
border-collision period-doubling bifurcations. We apply a subharmonic
perturbation to the bifurcation parameter, which is also known as alternate
pacing, and we investigate the response under such pacing near the original
bifurcation point. The resulting behavior is characterized quantitatively by a
gain, which is the ratio of the response amplitude to the applied perturbation
amplitude. The gain in a border-collision period-doubling bifurcation has a
qualitatively different dependence on parameters from that of a classical
period-doubling bifurcation. Perhaps surprisingly, the differences are more
readily apparent if the gain is plotted vs. the perturbation amplitude (with
the bifurcation parameter fixed) than if plotted vs. the bifurcation parameter
(with the perturbation amplitude fixed). When this observation is exploited,
the gain under alternate pacing provides a useful experimental tool to identify
a border-collision period-doubling bifurcation.
