  A modification of the quantum kicked rotator is suggested with a
time-dependent delta-kicked interaction parameter which can be realized by a
pulsed turn-on of a Feshbach resonance. The mean kinetic energy increases
exponentially with time in contrast to a merely diffusive or linear growth for
the first few kicks for the quantum kicked rotator with a constant interaction
parameter. A recursive relation is derived in a self-consistent random phase
approximation which describes this superdiffusive growth of the kinetic energy
and is compared with numerical simulations. Unlike in the case of the quantum
rotator with constant interaction, a Lax pair is not found. In general the
delta-kicked interaction is found to lead to strong chaotic behaviour.
