Hopf bifurcation in addition-shattering kinetics

TL;DR

This paper provides numerical evidence for the existence of persistent oscillations in addition-shattering aggregation processes, demonstrating a Hopf bifurcation as the mechanism for transition from fixed points to oscillatory behavior.

Contribution

It introduces the first rigorous numerical evidence of Hopf bifurcation leading to sustained oscillations in addition-shattering kinetics.

Findings

Identification of parameter region with oscillations

Demonstration of Hopf bifurcation as the transition mechanism

Evidence of never-ending oscillations in the process

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.