  We study the dynamics of a mechanical oscillator with linear and cubic forces
-the Duffing oscillator- subject to a feedback mechanism that allows the system
to sustain autonomous periodic motion with well-defined amplitude and
frequency. First, we characterize the autonomous motion for both hardening and
softening nonlinearities. Then, we analyze the oscillator's synchronizability
by an external periodic force. We find a regime where, unexpectedly, the
frequency range where synchronized motion is possible becomes wider as the
amplitude of oscillations grows. This effect of nonlinearities may find
application in technological uses of mechanical Duffing oscillators -for
instance, in the design of time-keeping devices at the microscale- which we
briefly review.
