## Abstract

The influence of a direct, uncatalysed reaction $A\rightarrow B \text{rate} = k_u a$ on the stationary-state behaviour of the cubic `autocatalator' $A + 2\mathrm{B}\rightarrow 3\mathrm{B} \text{rate} = k_1 ab^2,\\ \mathrm{B}\rightarrow \mathrm{C} \text{rate} = k_2 b$ in an isothermal, well-stirred, open reactor is studied by singularity methods. It is shown that the dependence on the mean residence time of the extent of conversion of A in the stationary state has five different patterns possible. Three of these are similar to those found in the absence of the uncatalysed step: unique, isola and mushroom. The two extra patterns, a breaking wave (a single, S-shaped hysteresis loop) and breaking wave plus isola, require non-zero k$_u$. The uncatalysed reaction also effects the behaviour at long residence time where the system tends to complete conversion. This kinetic scheme represents the simplest, isothermal chemical model with a full unfolding of a winged-cusp singularity and requires only the minimum number of parameters. Sustained oscillatory behaviour, arising from Hopf bifurcations, is also exhibited and is again influenced by the uncatalysed step. The main results can be summarized as follows. $k_u = 0$ multiple stationary states (isola, mushroom) and oscillations; $0 < k_u < \frac{1}{200} k_1 a^2_0$ multiple stationary states (isola, mushroom, breaking wave, breaking wave and isola) and oscillations; $\frac{1}{200} k_1 a^2_0 < k_u<\frac{1}{195} k_1 a^2_0$ multiple stationary states (isola, mushroom, breaking wave, breaking wave and isola), no oscillations; $\frac{1}{195} k_1 a^2_0 < k_u < \frac{1}{27} k_1 a^2_0$ multiple stationary states (breaking wave only), no oscillations; $\frac{1}{27} k_1 a^2_0 < k_u$ no multiple stationary states, no oscillations, (where a$_0$ is the concentration of the reactant A in the feed to the reactor).

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