  We consider a system of clusters of various sizes or masses, subject to
aggregation and fragmentation by collision with monomers or by
self-disintegration. The aggregation rate for the cluster of size or mass $k$
is given by a kernel proportional to $k^{a}$, whereas the collision and
disintegration kernels are given by $\lambda k^{b}$ and $\mu k^{a}$,
respectively, with $0\le a,b\le 1$ and positive factors $\lambda$ and $\mu$. We
study the emergence of oscillations in the phase diagram $(\mu,\lambda)$ for
two models: $(a,b)=(1,0)$ and $(1,1)$. It is shown that the monomer population
satisfies a class of integral equations possessing oscillatory solutions in a
finite domain in the plane $(\mu,\lambda)$. We evaluate analytically this
domain and give an estimate of the oscillation frequency. In particular, these
oscillations are found to occur generally for small but nonzero values of the
parameter $\mu$, far smaller than $\lambda$.
