  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.

Statistical Mechanics;Chemical Physics