  A unique and often overlooked property of a source loaded with relativistic
protons is that it can become supercritical, i.e. it can undergo an abrupt
transition from a radiatively inefficient to a radiatively efficient state once
its proton energy density exceeds a certain threshold. In this paper, we
investigate the temporal variability of hadronic systems in this hardly
explored regime. We show that there exists a range of proton densities that
prevent the system from reaching a steady state, but drive it instead in a
quasi-periodic mode. The escaping radiation then exhibits limit cycles, even if
all physical parameters are held constant in time. We extend our analysis to
cases where the proton injection rate varies with time and explore the
variability patterns of escaping radiation as the system moves in and out from
the supercritical regime. We examine the relevance of our results to the
variability of the prompt gamma-ray burst emission and show that, at least on a
phenomenological level, some interesting analogies exist.
