Thermokinetic oscillator with small parameters

The realm in which relaxation oscillations arise for equations such as the present scheme is that in which the different participants vary on quite different timescales. If we take the rate equations for the concentration of A and the temperature rise in their dimensionless form we have 

The two parameters /t and k appear in the first of these equations, one in each of the two terms on the right-hand side, but not in the second. If we are operating the reaction under conditions such that these are both small quantities, then the two terms in da/dt will both be small. In such a case, the concentration of the intermediate will vary only slowly compared to the rate of change of temperature. 

If we look at the parameter plane, Fig. 5.6, we can see that small values of these parameter s correspond to the region close to the origin, but that this region still includes conditions for which our Hopf analysis has shown that 

If /i and k are particularly small, we can usefully introduce new dimensionless groups M and K defined by 

This new timescale T is a slow time and is the timescale on which the slow variable a is changing. 

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