  We present a method to stabilize the frequency of self-sustained vibrations
in micro- and nanomechanical resonators. The method refers to a two-mode system
with the vibrations at significantly different frequencies. The signal from one
mode is used to control the other mode. In the experiment, self-sustained
oscillations of micromechanical modes are excited by pumping at the
blue-detuned sideband of the higher-frequency mode. Phase fluctuations of the
two modes show near perfect anti-correlation. They can be compensated in either
one of the modes by a stepwise change of the pump phase. The phase change of
the controlled mode is proportional to the pump phase change, with the
proportionality constant independent of the pump amplitude and frequency. This
finding allows us to stabilize the phase of one mode against phase diffusion
using the measured phase of the other mode. We demonstrate that phase
fluctuations of either the high or low frequency mode can be significantly
reduced. The results open new opportunities in generating stable vibrations in
a broad frequency range via parametric downconversion in nonlinear resonators.
