Hopf Bifurcation in the Brusselator
The Brusselator is a model for chemical oscillation with a limit cycle. The emergence of the limit cycle can be proven by the Andronov–Hopf bifurcation theorem.
The dynamics and chemistry of oscillating reactions has been the subject of study only for the last 50 years, starting with the work of Belousov. The mechanism for the Brusselator is given by . The two species of interest are and , the autocatalytic species. The differential equations given in dimensionless form for these species are and . For this analysis all rate constants except those of the second step are assumed to equal 1 and the reactants and are assumed to be in large enough excess so that their concentrations do not change with time. Both the parameters and are changed in the Demonstration, and as a result concentration-time curves and trajectories (selectivity curves in chemical terms) are shown.
Reference: I. Prigogine and R. Lefever, "Symmetry Breaking Instabilities in Dissipative Systems II," Journal of Chemical Physics, 48, 1968 pp. 1695–1700.