  We study the non-linear dynamics of a multimode optomechanical system
constituted of a driven high-finesse Fabry-P\'erot cavity containing two
vibrating dielectric membranes. The analytical study allows to derive a full
and consistent description of the displacement detection by a probe beam in the
non-linear regime, enabling the faithful detection of membrane displacements
well above the usual sensing limit corresponding to the cavity linewidth. In
the weak driving regime where the system is in a pre-synchronized situation,
the unexcited oscillator has a small, synchronized component at the frequency
of the excited one; both large and small amplitude resonator motions are
transduced in a nontrivial way by the non-linear response of the optical probe
beam. We find perfect agreement between the experimental results, the numerical
simulations, and an analytical approach based on slowly-varying amplitude
equations.
