  We provide a class of examples of interacting particle systems on
$\mathbb{Z}^d$, for $d\in\{1,2\}$, that admit a unique translation-invariant
stationary measure, which is not the long-time limit of all
translation-invariant starting measures, due to the existence of time-periodic
orbits in the associated measure-valued dynamics. This is the first such
example and shows that even in low dimensions, not every limit point of the
measure-valued dynamics needs to be a time-stationary measure.
