Singularity theory approach to stationary-state loci

Singularity theory approach to stationary-state loci 

In this chapter, and in the previous one, we have frequently been faced with a stationary-state condition in the form of a polynomial or some other 

Here F represents the functional form of the left-hand sides of the various stationary-state equations, x is the stationary-state solution such as the extent of reaction, the temperature excess, etc., and rres is the parameter we have singled out as the one which can be varied during a given experiment (the distinguished or bifurcation parameter). All the remaining parameters are represented by p, q, r, s,. For example, in eqn (7.21) the role of x could be played by the extent of reaction 1 — ass, with p = 0ad and q = tN for isothermal autocatalysis, x can again be the extent of reaction, with p = P0, q = k2, and r = jcu. 

Our general interest has been to find the conditions, in terms of the extra unfolding parameters p, q, r, etc., at which the qualitative nature of the stationary-state locus changes (e.g. the appearance or disappearance of a hysteresis loop or an isola). In some cases we have been able to make use of special techniques such as factorization or the tangency condition. Now we seek a more widely applicable approach. This will involve the stationary-state condition F = 0 and also a series of equations obtained by differentiation of this expression with respect to the variable x and the parameter tres. 

For instance, if we consider the simple case of the adiabatic non-isothermal CSTR, the stationary-state condition is given by eqn (7.27). Writing x for the extent of reaction, we have 

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