Two-Cell coupled cubic autocatalator: the effect of the uncatalysed reaction

Abstract

A two-cell coupled chemical oscillator based on the cubic autocatalator: P→A A + 2B→ 3B B → C is considered with the inclusion of the extra uncatalysed reaction A→ B. Two cases are considered. In each the cells are assumed to be identical, with the coupling between them taking place by a linear exchange of either reactant A or autocatalyst B. In the former case parameter ranges are found over which it is possible to have one, three or five stationary states; these are discussed in detail, as well as the points of Hopf bifurcation on the various branches of solution. In the latter case, it is shown that the only stationary state possible is the same as for the uncoupled system with Hopf bifurcations also at the same parameter values as for the uncoupled system for all values of the non-dimensional constant describing the coupling between the cells. Asymptotic solutions are derived for large values of the coupling constants in both cases. With coupling through the autocatalyst the system is shown to reduce to the uncoupled system, whilst for coupling through A the former asymmetric solutions are still found, with Hopf bifurcations on these solution branches