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 $K_{i,j} = i^{\nu}j^{\mu} + j^{\nu}i^{\mu}$ homogeneous in masses $i$ and $j$ of merging clusters and fragmentation kernels, $F_{ij}=\lambda K_{ij}$, with parameter $\lambda$ 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 simulations results when $\theta \equiv \nu-\mu <1$. For $\theta=\nu-\mu >1$, however, the densities exhibit an oscillatory behavior. While these oscillations decay for not very small $\lambda$, they become steady if $\theta$ is close to two and $\lambda$ is very small. Simulation results lead to a conjecture that for $\theta <1$ the system has a stable fixed point, corresponding to the steady-state density distribution, while for any $\theta >1$ there exists a critical value $\lambda_c(\theta )$, such that for $\lambda < \lambda_c(\theta)$, 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.