  Exoplanets sufficiently close to their host star can in principle couple
electrodynamically to the star. This process is known as electrodynamic
star-planet interaction (SPI). The expected emission associated with this
coupling is however difficult to observe due to the bright intrinsic stellar
emission. Identification of time-variability in the stellar lightcurve is one
of the most promising approaches to identify SPI. In this work we therefore
systematically investigate various mechanisms and their associated periods,
which generate time-variability to aid the search for SPI. We find that the
synodic and half the synodic rotation periods of the stars as measured in the
rest frames of the orbiting exoplanets are basic periods occurring in SPI. We
apply our findings to the example of TRAPPIST-1 with its seven close-in planets
for which we investigate the possibility of SPI and the associated
time-variabilities. We show that especially TRAPPIST-1b and c, are very likely
subject to sub-Alfv\'{e}nic interaction, a necessary condition for SPI. Both
planets are therefore expected to generate Alfv\'{e}n wings, which can couple
to the star. The associated Poynting fluxes are on the order of $10^{11}$ to
$10^{15}$ W and thus can hardly be the direct source of currently observable
time-variability from TRAPPIST-1. However these Poynting fluxes might trigger
flares on the star. We find correlations between the observed flares and the
expected planetary induced signals, which could be due to SPI but our findings
are not conclusive and warrant further observations and modelling.
