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We examine the collective dynamics of disks moving through a square array ofobstacles under cyclic square wave driving. Below a critical density we findthat system organizes into a reversible state in which the disks return to thesame positions at the end of every drive cycle. Above this density, thedynamics are irreversible and the disks do not return to the same positionsafter each cycle. The critical density depends strongly on the angle $\theta$between the driving direction and a symmetry axis of the obstacle array, withthe highest critical densities appearing at commensurate angles such as$\theta=0^\circ$ and $\theta=45^\circ$ and the lowest critical densitiesfalling at $\theta = \arctan(0.618)$, the inverse of the golden ratio, wherethe flow is the most frustrated. As the density increases, the number of cyclesrequired to reach a reversible state grows as a power law with an exponent near$\nu=1.36$, similar to what is found in periodically driven colloidal andsuperconducting vortex systems.

Reversible to Irreversible Transitions for Cyclically Driven Disks on Periodic Obstacle Arrays