One interesting class of gravitational radiation sources includes rapidly rotating astrophysical objects that encounter dynamical instabilities .
We have carried out a set of simulations of rotationally induced instabilities in differentially rotating polytropes .
An n =1.5 polytrope with the Maclaurin rotation law will encounter the m =2 bar instability at T / |W| \gtrsim 0.27 .
Our results indicate that the remnant of this instability is a persistent bar-like structure that emits a long-lived gravitational radiation signal .
Furthermore , dynamical instability is shown to occur in n =3.33 polytropes with the j -constant rotation law at T / |W| \gtrsim 0.14 .
In this case , the dominant mode of instability is m =1 .
Such instability may allow a centrifugally-hung core to begin collapsing to neutron star densities on a dynamical timescale .
If it occurs in a supermassive star , it may produce gravitational radiation detectable by LISA .