  This paper considers a game in which a single cop and a single robber take
turns moving along the edges of a given graph $G$. If there exists a strategy
for the cop which enables it to be positioned at the same vertex as the robber
eventually, then $G$ is called cop-win, and robber-win otherwise. We study this
classical combinatorial game in a novel context, broadening the class of
potential game arenas to include the edge-periodic graphs. These are graphs
with an infinite lifetime comprised of discrete time steps such that each edge
$e$ is assigned a bit pattern of length $l_e$, with a 1 in the $i$-th position
of the pattern indicating the presence of edge $e$ in the $i$-th step of each
consecutive block of $l_e$ steps. Utilising the already-developed framework of
reachability games, we extend existing techniques to obtain, amongst other
results, an $O(\textsf{LCM}(L)\cdot n^3)$ upper bound on the time required to
decide if a given $n$-vertex edge-periodic graph $G^\tau$ is cop or robber win
as well as compute a strategy for the winning player (here, $L$ is the set of
all edge pattern lengths $l_e$, and $\textsf{LCM}(L)$ denotes the least common
multiple of the set $L$). Separately, turning our attention to edge-periodic
cycle graphs, we give proof of a $2\cdot l \cdot \textsf{LCM}(L)$ upper bound
on the length required by any edge-periodic cycle to ensure that it is robber
win, where $l = 1$ if $\textsf{LCM}(L) \geq 2\cdot \max L $, and $l=2$
otherwise. Furthermore, we provide lower bound constructions in the form of
cop-win edge-periodic cycles: one with length $1.5 \cdot \textsf{LCM}(L)$ in
the $l=1$ case and one with length $3\cdot \textsf{LCM}(L)$ in the $l=2$ case.
