Well-stirred system has unstable focus

When the dimensionless reaction rate constant lies in the range given by eqn (10.77), the well-stirred system has two Hopf bifurcation points /i 2. Over the range of reactant concentration 

There is one solution to eqn (10.75) for each value of /r in the range given by (10.79) the maximum in the locus, and hence the highest mode to which the uniform state is unstable, is given by 

The wave number n can only have integer values, so only discrete values of the ordinate nn/y112 on Fig. 10.9 are allowed. The lowest value for n is unity, 

We can think of the reactant concentration and some initial spatial distribution of the intermediate concentration and temperature profiles specifying a point on Fig. 10.9. If we choose a point above the neutral stability curve, then the first response of the system will be for spatial inhomogeneity to disappear. If the value of /r lies outside the range given by (10.79), then the system adjusts to a stable spatially uniform stationary state. If ji lies between H and n, we may find uniform oscillations. 

however, we start the system with a given non-uniform distribution, corresponding to n = 2 say, and a value for ji such that the initial point lies beneath the neutral stability curve, then the spatial amplitudes will not decay. Rather the positive real parts to the eigenvalues will ensure that the perturbation waveform grows. The system may move to a state which is varying both in time and position—a standing-wave solution. 

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