Well-stirred system has unstable node

If the dimensionless rate constant satisfies inequality (10.78), the well-stirred system again has two Hopf bifurcation points fi and n. However, within the range of reactant concentrations between these, the uniform state also changes character from unstable focus to unstable node at n and n 2, as shown in Fig. 10.10. 

In cases (1) and (5) the eigenvalues to have negative real parts for all ti, so non-uniform perturbations decay. 

For case (3), with /i in the range between ji and fi2, the condition tr(J) = 0, corresponding to eqn (10.80), is of less importance (and hence the locus is shown as a broken curve in Fig. 10.10). As n increases from zero, so the trace does become less positive, but now the determinant can also change sign. The latter occurs first, in fact, so the eigenvalues to change from two, real, and 

Over a range of n between the lower and upper roots of eqn (10.76) the uniform state appears as a saddle point to perturbations with the appropriate wave numbers, so we expect any such spatial non-uniformity to grow. Only for n larger than the upper root of (10.76) do the eigenvalues become both real and negative. 

The neutral stability curve corresponding to the condition det(J) = 0 gives a closed region, within which we expect the appearance of stable (time-independent) spatially non-uniform profiles. 

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