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

✍ Scribed by B. F. Gray; M. J. Roberts; J. H. Merkin

• Book ID: 104633771
• Publisher: Springer
• Year: 1988
• Tongue: English
• Weight: 697 KB
• Volume: 22
• Category: Article
• ISSN: 0022-0833
• DOI: 10.1007/bf00129906

✦ Synopsis

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 p (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 , whereas with r = s = 0 the solution is bounded only for p > ( = 0.900 32). It is shown that with r = 0 and s 0 the governing equations have a Hopf bifurcation at It = 1 -s producing a stable limit cycle which exists for all p in 0 < < 1s. The behaviour of these limit cycles as p 0 is also discussed.