We demonstrate a previously unobserved nonlinear phenomenon in an ultra-low loss quartz bulk acoustic wave cavity (Q > 3 > 10 9), which only occurs below 20 mK in temperature and under relatively weak pumping. The phenomenon reveals the emergence of several stable equilibria (at least two foci and two nodes) and jumps between these quasi states at random times. The degree of this randomness as well as separations between levels can be controlled by the frequency of the incident carrier signal. It is demonstrated that the nature of the effect lies beyond the standard Duffing model. © 2014 AIP Publishing LLC.