  Optomechanical systems close to their quantum ground state and nonlinear
nanoelectromechanical systems are two hot topics of current physics research.
As high-reflectivity and low mass are crucial features to improve
optomechanical coupling towards the ground state, we have designed, fabricated
and characterized photonic crystal nanomembranes, at the crossroad of both
topics. Here we demonstrate a number of nonlinear effects with these membranes.
We first characterize the nonlinear behavior of a single mechanical mode and we
demonstrate its nonlocal character by monitoring the subsequent
actuation-related frequency shift of a different mode. We then proceed to study
the underlying nonlinear dynamics, both by monitoring the phase-space
trajectory of the free resonator and by characterizing the mechanical response
in presence of a strong pump excitation. We observe in particular the frequency
evolution during a ring-down oscillation decay, and the emergence of a phase
conjugate mechanical response to a weaker probe actuation. Our results are
crucial to understand the full nonlinear features of the PhC membranes, and
possibly to look for nonlinear signatures of the quantum dynamics.
