We report theory, measurements and numerical simulations on nonlinear piezoelectric ultrasonic devices with stable limit cycles. The devices are shown to exhibit behavior familiar from the theory of coupled auto-oscillators. Frequency of auto-oscillation is affected by the presence of an acoustic cavity as these spontaneously emitting devices adjust their frequency to the spectrum of the acoustic cavity. Also, the auto-oscillation is shown to be entrained by an applied field; the oscillator synchronizes to an incident wave at a frequency close to the natural frequency of the limit cycle. It is further shown that synchronization occurs here with a phase that can, depending on details, correspond to stimulated emission: the power emission from the oscillator is augmented by the incident field. These behaviors are essential to eventual design of an ultrasonic system that would consist of a number of such devices entrained to their mutual field, a system that would be an analog to a laser. A prototype laser is constructed.