  The Duffing oscillator is a nonlinear extension of the ubiquitous harmonic
oscillator and as such plays an outstanding role in science and technology.
Experimentally, the system parameters are determined by a measurement of its
response to an external excitation. When changing the amplitude or frequency of
the external excitation, a sudden jump in the response function reveals the
nonlinear dynamics prominently. However, this bistability leaves part of the
full response function unobserved, which limits the precise measurement of the
system parameters. Here, we exploit the often unknown fact that the response of
a Duffing oscillator with nonlinear damping is a unique function of its phase.
By actively stabilizing the oscillator's phase we map out the full response
function. This phase control allows us to precisely determine the system
parameters. Our results are particularly important for characterizing nanoscale
resonators, where nonlinear effects are observed readily and which hold great
promise for next generation of ultrasensitive force and mass measurements. We
demonstrate our approach experimentally with an optically levitated particle in
high vacuum.
