  We study locking phenomena of two strongly coupled, high-quality factor
nanomechanical resonator modes subject to a common parametric drive at a single
drive frequency. By controlled dielectric gradient forces we tune the resonance
frequencies of the flexural in-plane and out-of-plane oscillation of the high
stress silicon nitride string through their mutual avoided crossing. For the
case of the strong common parametric drive signal-idler generation via
parametric oscillation is observed, analogously to the framework of nonlinear
optical effects in an optical parametric oscillator. Frequency tuning of the
signal and idler resonances is demonstrated. When the resonance frequencies of
signal and idler get closer to each other, partial injection locking, injection
pulling and complete injection locking to half of the drive frequency occurs
depending on the pump strength. Furthermore, satellite resonances,
symmetrically off-set from signal and idler by their beat-note, are observed
which can be attributed to degenerate four-wave-mixing in the highly nonlinear
mechanical oscillations.
