Singularity theory for cubic autocatalysis with uncatalysed reaction

Singularity theory for cubic autocatalysis with uncatalysed reaction 

This version of the model, from 6.5, has three unfolding parameters k2, P0, and ku. If x again represents the stationary-state extent of conversion, 1 — ass, then the stationary-state condition (eqn (6.74)) can be written as 

These five equations are all equal to zero simultaneously for one set of parameters only, i.e. for 

The next highest partial derivatives, Fxxx etc., are all non-zero for these values, 

We can find the hysteresis and isola boundaries from the above equations by applying conditions (7.46) and (7.55) respectively. In each case we have three equalities. Two of these can, in principle, be used to determine the values for x and tres. Eliminating these, the third equation then defines a surface in the k2 P0-ku space. Alternatively, we can also specify a particular value for, say, the uncatalysed rate constant ku and then use the third equation to give the boundaries in the k2 P0 plane. This latter procedure has 

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