  When an amorphous solid is deformed cyclically, it may reach a steady state
in which the paths of constituent particles trace out closed loops that repeat
in each driving cycle. A remarkable variant has been noticed in simulations
where the period of particle motions is a multiple of the period of driving,
but the reasons for this behavior have remained unclear. Motivated by the
mesoscopic features of displacement fields in experiments on jammed solids, we
propose and analyze a simple model of interacting soft spots -- locations where
particles rearrange under stress, and that resemble two-level systems with
hysteresis. We show that multiperiodic behavior can arise among just three or
more soft spots that interact with each other, but in all cases it requires
frustrated interactions, illuminating this otherwise elusive type of
interaction. We suggest directions for seeking this signature of frustration in
experiments, and we describe how to achieve it in designed systems.
