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.
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