Stability condition of the steady oscillations in aggregation models with shattering process and self-fragmentation.
Résumé
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 with a ≥ 0. The collision rate and the disintegration rate are given by λk b and µk b , respectively, with 0 ≤ a, b ≤ 1 and positive factors λ and µ. We study the emergence of oscillations in the phase diagram (λ, µ) 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 (λ, µ). We evaluate analytically this domain in a precise way 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 µ, far smaller than λ.
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