## Abstract

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 > μ_{0} (μ_{0} = 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.

Original language | English |
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Pages (from-to) | 267-284 |

Number of pages | 18 |

Journal | Journal of Engineering Mathematics |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1988 |