Oscillation and Instability in Chemical Reactions

TL;DR

This paper demonstrates that the diffusive Brusselator model can support complex spatial-temporal wave structures through an equivariant Hopf bifurcation mechanism, extending understanding beyond traditional Turing and temporal oscillation theories.

Contribution

It reveals that diffusion can maintain an equivariant Hopf bifurcation spectral mechanism at any nonzero wave number, unlike previous models limited to specific wavelengths.

Findings

Diffusion acts as a stabilizer and destabilizer depending on context.

Complex oscillations are not solely due to reactant inhomogeneity.

The mechanism occurs around any nonzero wave number.

Abstract

We prove that the famous diffusive Brusselator model can support more complicated spatial-temporal wave structure than the usual temporal-oscillation from a standard Hopf bifurcation. In our investigation, we discover that the diffusion term in the model is neither a usual parabolic stabilizer nor a destabilizer as in the Turing instability of uniform state, but rather plays the role of maintaining an equivariant Hopf bifurcation spectral mechanism. At the same time, we show that such a mechanism can occur around any nonzero wave number and this finding is also different from the former works where oscillations caused by diffusion can cause the growth of wave structure only at a particular wavelength. Our analysis also demonstrates that the complicated spatial-temporal oscillation is not solely driven by the inhomogeneity of the reactants.