A simple model for sustained oscillations in isothermal branched-chain or autocatalytic reactions in a well stirred open system I. Stationary states and local stabilities

The behaviour of a simple autocatalytic or branched-chain reaction $A + B \rightarrow 2B; \text{rate} = k_1 ab$, coupled with a surface-controlled decay of the catalyst $B \rightarrow C; \text{rate} = k_2 b/(1 + rb)$, is considered under isothermal, well stirred, open conditions (the c.s.t.r.). This system has a unique physically acceptable stationary state at any given residence time. If the saturation term r is zero, this solution is always stable although it may exhibit damped oscillatory responses characteristic of a stable focus. Small non-zero values of r do not introduce qualitative changes. However for sufficiently large surface effects, such that r > k$_1$/k$_2$, the stable focus may become unstable at a point of Hopf bifurcation. This leads to the appearance of sustained oscillations in the concentrations a and b. For relatively stable catalysts, such that k$_2$ is non-zero but very small compared with k$_1$ a$_0$ (where a$_0$ is the concentration of reactant A in the feed to the reactor) the governing kinetic equations can be reduced to especially simple, asymptotic forms. These in turn allow the analytical determination of the point of Hopf bifurcation in terms of the residence time $t^*_{res}$ at which the change in stability occurs. Numerical solution of the full equations reveals that the asymptotic expressions provide a very good prediction for the dependence of $t^*_{res}$ on k$_1$, a$_0$, k$_2$ and r even when $k_2 \approx k_1 a_0$. Systems which have some autocatalyst B in the feed show an important qualitative change. At long residence times there is a second point of Hopf bifurcation at which the focal stationary state becomes stable again. Oscillations die out at this point and, in the limit of infinitely long residence times, the behaviour approaches that of the corresponding closed system.