Oscillatory behaviour Hopf bifurcation analysis

In the previous sections we have implied that the loss of local stability which occurs for a stationary-state solution as the real part of the eigenvalues changes from negative to positive is closely linked to the conditions under which sustained oscillatory responses are born. 

For a two-variable system, the condition for the real part to vanish is simply that the trace of the Jacobian matrix should become zero, with the determinant being positive, as discussed in case (f) of 3.2.1. These requirements are met for the present model at the points /if and /if and we have seen 

Before we can conclude, in general, that a given system will begin to show oscillatory behaviour between two Hopf bifurcation points we must attend to a few additional requirements of the theorem. 

First we must check that the real part of the eigenvalues, Re(A) = tr(J), actually passes through zero and becomes positive for some range. It will not do that if Re(2) has a maximum at zero. To avoid a maximum, we require that at the points n and // the derivative of tr(J) with respect to n should be non-zero, i.e. 

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Hopf bifurcation analysis commonly signals the onset of oscillatory behaviour. This chapter uses a particular two-variable example to illustrate the essential features of the approach and to explore the relationship to relaxation oscillations. After a careful study of this chapter the reader should be able to ... 

In this chapter we give an introduction and recipe for the full Hopf bifurcation analysis for chemical systems. Rather than work in completely general and abstract terms, we will illustrate the various stages by using the thermokinetic model of the previous chapter, with the exponential approximation for simplicity. We can draw many quantitative conclusions about the oscillatory solutions in that model. In particular we will be able to show (i)that the parameter values given by eqns (4.49) and (4.50) for tr(J) = 0 satisfy all the requirements of the. Hopf theorem (ii)that oscillatory behaviour is completely confined to the conditions for which the stationary state is... 

We have now seen how local stability analysis can give us useful information about any given state in terms of the experimental conditions (i.e. in terms of the parameters p and ku for the present isothermal autocatalytic model). The methods are powerful and for low-dimensional systems their application is not difficult. In particular we can recognize the range of conditions over which damped oscillatory behaviour or even sustained oscillations might be observed. The Hopf bifurcation condition, in terms of the eigenvalues k2 and k2, enabled us to locate the onset or death of oscillatory behaviour. Some comments have been made about the stability and growth of the oscillations, but the details of this part of the analysis will have to wait until the next chapter. 

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