# Hopf bifurcation in addition-shattering kinetics

Source: [Hopf bifurcation in addition-shattering kinetics](<https://www.sota2.com/research/paper/68fa3edb1e209131ce8e8ae0>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-27T04:08:14.972Z

Authors: Stanislav S. Budzinskiy, Sergey A. Matveev, Pavel L. Krapivsky

Published: 2020-12-16T14:56:46.000Z

## Abstract

In aggregation-fragmentation processes, a steady state is usually reached in the long time limit. This indicates the existence of a fixed point in the underlying system of ordinary differential equations. The next simplest possibility is an asymptotically periodic motion. Never-ending oscillations have not been rigorously established so far, although oscillations have been recently numerically detected in a few systems. For a class of addition-shattering processes, we provide convincing numerical evidence for never-ending oscillations in a certain region $\\mathcal{U}$ of the parameter space. The processes which we investigate admit a fixed point that becomes unstable when parameters belong to $\\mathcal{U}$ and never-ending oscillations effectively emerge through a Hopf bifurcation.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2012.09003>)

[PDF](<https://arxiv.org/pdf/2012.09003.pdf>)