  We propose a general mechanism for generating limit cycle (LC) oscillations
by coupling a linear bosonic mode to a dissipative nonlinear bosonic mode. By
analyzing the stability matrix, we show that LCs arise due to a supercritical
Hopf bifurcation. We find that the existence of LCs is independent of the sign
of the effective nonlinear interaction. The LC phase can be classified as a
continuous time crystal (CTC), if it emerges in a many-body system. The bosonic
model can be realised in three-level systems interacting with a quantised light
mode as realised in atom-cavity systems. Using such a platform, we
experimentally observe LCs for the first time in an atom-cavity system with
attractive optical pump lattice, thereby confirming our theoretical
predictions.
