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

Stability condition of the steady oscillations in aggregation models with shattering process and self-fragmentation