  Under the right conditions, self sustaining fission waves can form in fertile
nuclear materials. These waves result from the transport and absorption of
neutrons and the resulting production of fissile isotopes. When these fission,
additional neutrons are produced and the chain reaction propagates until it is
poisoned by the buildup of fission products. It is typically assumed that
fission waves are soliton-like and self stabilizing. However, we show that in
uranium, coupling of the neutron field to the 239U->239Np->239Pu decay chain
can lead to a Hopf bifurcation. The fission reaction then ramps up and down,
along with the wave velocity. The critical driver for the instability is a
delay, caused by the half-life of 239U, between the time evolution of the
neutron field and the production of 239Pu. This allows the 239Pu to accumulate
and burn out in a self limiting oscillation that is characteristic of a Hopf
bifurcation. Time dependent results are obtained using a numerical
implementation of a reduced order reaction-diffusion model for a fast neutron
field. Monte Carlo simulations in combination with a linear stability analysis
are used to confirm the results for the full system and to establish the
parameter space where the Hopf occurs.
