  Nonlinear Josephson circuits play a crucial role in the growing landscape of
quantum information and technologies. The typical circuits studied in this
field consist of qubits, whose anharmonicity is much larger than their
linewidth, and also of parametric amplifiers, which are engineered with
linewidths of tens of MHz or more. The regime of small anharmonicity but also
narrow linewidth, corresponding to the dynamics of a high-$Q$ Duffing
oscillator, has not been extensively explored using Josephson cavities. Here,
we use two-tone spectroscopy to study the susceptibility of a strongly driven
high-$Q$ Josephson microwave cavity. Under blue-detuned driving, we observe a
shift of the cavity susceptibility, analogous to the AC Stark effect in atomic
physics. When applying a strong red-detuned drive, we observe the appearance of
an additional idler mode above the bifurcation threshold with net external
gain. Strong driving of the circuit leads to the appearance of two exceptional
points and a level attraction between the quasi-modes of the driven cavity. Our
results provide insights on the physics of driven nonlinear Josephson
resonators and form a starting point for exploring topological physics in
strongly-driven Kerr oscillators.
