  Self-oscillating systems, described in classical dynamics as limit cycles,
are emerging as canonical models for driven dissipative nonequilibrium open
quantum systems, and as key elements in quantum technology. We consider a
family of models that interpolates between the classical textbook examples of
the Rayleigh and the van der Pol oscillators, and follow their transition from
the classical to the quantum domain, while properly formulating their
corresponding quantum descriptions. We derive an exact analytical solution for
the steady-state quantum dynamics of the simplest of these models, applicable
to any bosonic system---whether mechanical, optical, or otherwise---that is
coupled to its environment via single-boson and double-boson emission and
absorption. Our solution is a generalization to arbitrary temperature of
existing solutions for very-low, or zero, temperature, often misattributed to
the quantum van der Pol oscillator. We closely explore the classical to quantum
transition of the bifurcation to self-oscillations of this oscillator, while
noting changes in the dynamics and identifying features that are uniquely
quantum.
