The cubic autocatalator: The influence of the quadratic autocatalytic and the uncatalysed reactions

The proptotype chemical reaction scheme, the cubic autocatalator, A + 2B 3B; B C is taken in a closed system, with A formed from the precursor P by the simple step P A. The pooled chemical approximation is invoked whereby the concentration of P can be assumed to remain constant throughout. The effects of allowing the quadratic autocatalytic reaction A + B 2B and the uncatalysed reaction A B in the scheme are considered in detail. The full scheme is described by the non-dimensional parameters µ (measuring the reaction rate of the initiation step) ands andr (measuring the reaction rates of the quadratic autocatalytic and the uncatalysed steps respectively). It is shown, provided only thatr ors (or both) are non-zero, no matter how small, the solution remains bounded for all (positive) values of µ, whereas withr =s = 0 the solution is bounded only for µ > µ₀ (µ₀ = 0.900 32). It is shown that withr = 0 ands 0 the governing equations have a Hopf bifurcation at µ = 1 –s producing a stable limit cycle which exists for all µ in 0 < µ <1 –s. The behaviour of these limit cycles as µ 0 is also discussed.