  We discuss the theory of linear and non-linear spectroscopy of an rf-SQUID
coupled to a Josephson spectrometer. Recent experimental measurements on this
system have shown a strongly non-linear absorption lineshape, whose current
peak maximum undergoes a forward-backward bending transition depending on the
value of the rf-SQUID phase. We show that this transition can be qualitatively
understood by mapping the dynamics of the driven rf-SQUID onto a generalized
Duffing oscillator, with tunable drive and non-linearity, undergoing a
bifurcation. Finally we show that in order to quantitatively reproduce the
experimental data reported in arXiv:2106.02632, it is crucial to include the
feedback from the load-line, leading to an additional source of non-linearity.
