Simple tests for oscillatory instability

Thus the upper end of the oscillatory region lies close to the conditions at which the two stationary-state loci cross, but at slightly lower reactant concentration. The lower end of the unstable region p lies at slightly higher reactant concentrations than that at which the locus ass( p) has its maximum. 

For larger values of the uncatalysed reaction rate constant, i.e. as ku approaches gk2, the two solutions of (2.21) move closer together and so the region of instability and oscillations shrinks. The two points merge when ku = 8 2 when 

The general philosophy of our test for oscillatory instability can be summarized as follows. At the pseudo-steady state, the net rates of change of the intermediate concentrations are zero da/dt = db/dt = 0. If we make a small 

To quantify this idea in mathematical terms, we can recognize that we are really talking about the partial derivative quantities d(da/dt)/da and d(db/dt)/db. Stability has been associated in some sense with these two quantities being negative (i.e. da/dt decreases as a increases, so d(da/dt)/da 0), instability with these being positive. In most normal chemical systems, e.g. those with deceleratory kinetics, the two partial derivatives will be negative. It is a characteristic of autocatalysis, however, that at least one of these may become positive — at least over some ranges of composition and experimental conditions. 

In the present case, the partial derivatives can be obtained from eqns (2.2) and (2.3) as 

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