Communications in Mathematical Sciences

Volume 17 (2019)

Number 6

Patterns of complex oscillations and instability in chemical reactions

Pages: 1713 – 1736



Jinghua Yao (Department of Mathematics, University of Iowa, Iowa City, Ia., U.S.A.)

Duchao Liu (School of Mathematics and Statistics, Lanzhou University, Lanzhou, China)

Xiaoyan Wang (Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, In., U.S.A.)


We prove that the diffusive Brusselator model can support more complicated spatialtemporal wave structure than the usual temporal-oscillation from a standard Hopf bifurcation. In our current 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.


chemical reaction oscillation, chemical reaction instability, symmetry, spectrum, sectorial operator

2010 Mathematics Subject Classification

34Bxx, 34Gxx, 35Pxx, 37Gxx

Received 23 September 2018

Accepted 23 May 2019

Published 26 December 2019