Behaviour of eigenvalues at Hopf bifurcation

Close to the conditions at which the trace is becoming zero, the eigenvalues A1j2 will have the form of a complex conjugate pair  

We have stressed that both the real and imaginary parts depend on the parameter n because we are imagining experiments where the reactant concentration will be varied whilst k is held constant. If we were doing the experiments another way so that n was held fixed and the dimensionless reaction rate constant varied in the vicinity of the Hopf bifurcation point we would then wish to consider v(/c) and a (/c). 

We must now check that the imaginary part of /,2 is not zero under these conditions. With tr(J) = 0 we have, quite generally, 

The quantity o 0 is of considerable importance. It gives the natural frequency of the oscillations (corresponding to a period of 2n/co0) as they first appear at the point of Hopf bifurcation. 

Next we must check that the real part of the eigenvalues actually passes through zero, as required in 3.4.1. This condition will be satisfied if the derivative of v, or equivalently the trace of the Jacobian, with respect to the parameter being varied (in this case /i) is non-zero when evaluated at fi. For the present model 

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