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) and s and r (measuring the reaction rates of the quadratic autocatalytic and the uncatalysed steps respectively). It is shown, provided only that r or s (or both) are non-zero, no matter how small, the solution remains bounded for all (positive) values of μ, whereas with r =s = 0 the solution is bounded only for μ > μ₀ (μ₀ = 0.900 32). It is shown that with r = 0 and s ≠ 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.