  Recent experiments and simulations of amorphous solids plastically deformed
by oscillatory drive have foundsurprising behavior - for small strain
amplitudes the dynamics can be reversible, which is contrary to the usual
notion of plasticity as an irreversible form of deformation. This reversibility
allows the system to reach limit-cycles in which plastic events repeat
indefinitely under the oscillatory drive. Reaching reversible limit-cycles, can
take a large number of driving cycles and it was surmised that the plastic
events encountered during the transient period are not encountered again and
are thus irreversible. Using a graph representation of the stable
configurations of the system and the plastic events connecting them, we show
that the notion of reversibility is more subtle. We find that reversible
plastic events are abundant, and that a large portion of the plastic events
encountered during the transient period are actually reversible, in the sense
that they can be part of a reversible deformation path. We observe that the
transition graph can be decomposed into clusters of configurations that are
connected by reversible transitions. These clusters are the strongly connected
components of the graph and their sizes turn out to be power-law distributed.
The largest of these are grouped in regions of reversibility, which in turn are
confined by regions of irreversibility whose number proliferates at larger
strains. Our results provide an explanation for the irreversibility transition
- the divergence of the transient period at a critical forcing amplitude. Long
transients result from transition between clusters of reversibility in a search
for a cluster large enough to contain a limit-cycle of a specific amplitude.
For large enough amplitudes, the search time becomes very large, since sizes of
the limit cycles become incompatible with the sizes of the regions of
reversibility.
