  We classify periodic Y-systems of rank 2 satisfying the symplectic property.
We find that there are six such Y-systems. In all cases, the periodicity
follows from the existence of two reddening sequences associated with the time
evolution of the Y-systems in positive and negative directions, which gives
rise to quantum dilogarithm identities associated with Donaldson-Thomas
invariants. We also consider $q$-series called the Nahm sums associated with
these Y-systems. We see that they are included in Zagier's list of rank 2 Nahm
sums that are likely to be modular functions. It was recently shown by Wang
that they are indeed modular functions.
