  We present a study of the gas cycle and star formation history in the central
500 pc of the Milky Way, known as Central Molecular Zone (CMZ). Through
hydrodynamical simulations of the inner 4.5 kpc of our Galaxy, we follow the
gas cycle in a completely self-consistent way, starting from gas radial inflow
due to the Galactic bar, the channelling of this gas into a dense, star-forming
ring/stream at ~ 200 - 300 pc from the Galactic centre, and the launching of
galactic outflows powered by stellar feedback. We find that star formation
activity in the CMZ goes through oscillatory burst/quench cycles, with a period
of tens to hundreds of Myr, characterised by roughly constant gas mass but
order-of-magnitude level variations in the star formation rate. Comparison with
the observed present-day star formation rate of the CMZ suggests that we are
currently near a minimum of this cycle. Stellar feedback drives a mainly
two-phase wind off the Galactic disc. The warm phase dominates the mass flux,
and carries 100 - 200 % of the gas mass converted into stars. However, most of
this gas goes into a fountain and falls back onto the disc rather than escaping
the Galaxy. The hot phase carries most of the energy, with a time-averaged
energy outflow rate of 10 - 20 % of the supernova energy budget.
