  Observations indicate that the star formation rate (SFR) of nuclear rings
varies considerably with time and is sometimes asymmetric rather than being
uniform across a ring. To understand what controls temporal and spatial
distributions of ring star formation, we run semi-global, hydrodynamic
simulations of nuclear rings subject to time-varying and/or asymmetric mass
inflow rates. These controlled variations in the inflow lead to variations in
the star formation, while the ring orbital period ($18\,{\rm Myr}$) and radius
($600\,{\rm pc}$) remain approximately constant. We find that both the mass
inflow rate and supernova feedback affect the ring SFR. An oscillating inflow
rate with period $\Delta \tau_\text{in}$ and amplitude 20 causes
large-amplitude (a factor of $\gtrsim 5$), quasi-periodic variations of the
SFR, when $\Delta \tau_\text{in} \gtrsim 50\,{\rm Myr}$. We find that the
time-varying ISM weight and midplane pressure track each other closely,
establishing an instantaneous vertical equilibrium. The measured time-varying
depletion time is consistent with the prediction from self-regulation theory
provided the time delay between star formation and supernova feedback is taken
into account. The supernova feedback is responsible only for small-amplitude (a
factor of $\sim 2$) fluctuations of the SFR with a timescale $\lesssim 40\,{\rm
Myr}$. Asymmetry in the inflow rate does not necessarily lead to asymmetric
star formation in nuclear rings. Only when the inflow rate from one dust lane
is suddenly increased by a large factor, the rings undergo a transient period
of lopsided star formation.

Astrophysics of Galaxies