  We analyze the replicator-mutator equations for the Rock-Paper-Scissors game.
Various graph-theoretic patterns of mutation are considered, ranging from a
single unidirectional mutation pathway between two of the species, to global
bidirectional mutation among all the species. Our main result is that the
coexistence state, in which all three species exist in equilibrium, can be
destabilized by arbitrarily small mutation rates. After it loses stability, the
coexistence state gives birth to a stable limit cycle solution created in a
supercritical Hopf bifurcation. This attracting periodic solution exists for
all the mutation patterns considered, and persists arbitrarily close to the
limit of zero mutation rate and a zero-sum game.
