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The addition of nonlinearity to an harmonic resonator provides a route tocomplex dynamical behaviour of resonant modes, including coupling between them.We present a superconducting device that makes use of the nonlinearity ofJosephson junctions to introduce a controlled, tunable, nonlinear inductance toa thin film coplanar waveguide resonator. Considering the device as a potentialquantum optical component in the microwave regime, we create a sensitivebifurcation amplifier and then demonstrate spectroscopy of other resonant modesvia the intermode coupling. We find that the sensitivity of the deviceapproaches within a factor two quantitative agreement with a quantum model byDykman, but is limited by a noise that has its source(s) on-chip.

Bifurcation, mode coupling and noise in a nonlinear multimode superconducting RF resonator