  In aggregation-fragmentation processes, a steady state is usually reached in
the long time limit. This indicates the existence of a fixed point in the
underlying system of ordinary differential equations. The next simplest
possibility is an asymptotically periodic motion. Never-ending oscillations
have not been rigorously established so far, although oscillations have been
recently numerically detected in a few systems. For a class of
addition-shattering processes, we provide convincing numerical evidence for
never-ending oscillations in a certain region $\mathcal{U}$ of the parameter
space. The processes which we investigate admit a fixed point that becomes
unstable when parameters belong to $\mathcal{U}$ and never-ending oscillations
effectively emerge through a Hopf bifurcation.
