Hopf bifurcation in addition-shattering kinetics

In aggregation-fragmentation processes, a steady state is usually reached. This indicates the existence of an attractive fixed point in the underlying infinite system of coupled 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 U of the parameter space. The processes which we investigate admit a fixed point that becomes unstable when parameters belong to U and never-ending oscillations effectively emerge through a Hopf bifurcation.