Steady oscillations in aggregation-fragmentation processes

We report surprising steady oscillations in aggregation-fragmentation processes. Oscillating solutions are observed for the class of aggregation kernels Ki,j = iν jμ + j ν iμ homogeneous in masses i and j of merging clusters and fragmentation kernels, Fij = λKij , with parameter λ quantifying the intensity of the disruptive impacts. We assume a complete decomposition (shattering) of colliding partners into monomers. We show that an assumption of a steady-state distribution of cluster sizes, compatible with governing equations, yields a power law with an exponential cutoff. This prediction agrees with simulation results when θ ≡ ν − μ < 1. For θ = ν − μ > 1, however, the densities exhibit an oscillatory behavior. While these oscillations decay for not very small λ, they become steady if θ is close to 2 and λ is very small. Simulation results lead to a conjecture that for θ < 1 the system has a stable fixed point, corresponding to the steady-state density distribution, while for any θ > 1 there exists a critical value λc, such that for λ<λc, the system has an attracting limit cycle. This is rather striking for a closed system of Smoluchowski-like equations, lacking any sinks and sources of mass.