  Athermal disordered systems can exhibit a remarkable response to an applied
oscillatory shear: after a relatively few shearing cycles, the system falls
into a configuration that had already been visited in a previous cycle. After
this point the system repeats its dynamics periodically despite undergoing many
particle rearrangements during each cycle. We study the behavior of orbits as
we approach the jamming point in simulations of jammed particles subject to
oscillatory shear at fixed pressure and zero temperature. As the pressure is
lowered, we find that it becomes more common for the system to find periodic
states where it takes multiple cycles before returning to a previously visited
state. Thus, there is a proliferation of longer periods as the jamming point is
approached.
