Hopf bifurcation analysis with Arrhenius model birth and growth of oscillations

Hopf bifurcation analysis with Arrhenius model birth and growth of oscillations 

We have already discussed the expressions resulting from a full Hopf bifurcation analysis of the thermokinetic model with the exponential approximation (y = 0). We may do the same for the exact. Arrhenius temperature dependence (y 0). Although the algebra is somewhat more onerous, we still arrive at analytical, expressions for the stability of the emerging or vanishing limit cycle and the rate of growth of the amplitude and period at 

we recall the conditions for Hopf bifurcation derived in 4.9.2(b). These are 

The evaluation of fi2 follows the same recipe as that given above through the transformation to new variables x and y, the various partial derivatives d2gl/dx2 etc., and then to gl i. This manipulation requires patience or a computer algebra package (preferably both), but really we are only interested in the resulting expression which is 

This rather cluttered equation at least reduces very simply to the correct form for the exponential approximation (eqn (5.43)) in the limit as y- 0. The numerator is a quadratic in 6. We will be interested to see if there are any conditions under which p2 becomes zero. If P2 is negative, the emerging limit cycle is stable as before. If P2 becomes positive, the emerging limit cycle will lose its stability and become unstable. 

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