Stability of emerging limit cycle

The formulae of 5.2 seem to be less algebraically tractable for the equations appropriate to flow reactors than for those from closed systems. (This is basically because we can eliminate all the terms involving one of the variables by adding the two rate equations together. For flow systems this cancellation does not occur because of the outflow terms.) Nevertheless, the various coefficients can be evaluated implicitly without much problem and the stability parameter p2 calculated. The following situation is found for the particular case of interest here, fi0 = 0. 

If k2 is greater than ys, we know there will be no isola and no Hopf bifurcation point. For k2 /g, but greater than 9/256, P2 is positive. This means that the emerging limit cycle will be unstable. The limit cycle grows as the residence time is reduced below the bifurcation point t s surrounding the upper stationary state which is stable. 

We now know that if a system on the upper branch of the isola, just below the Hopf bifurcation point, is given a small perturbation which remains within the unstable limit cycle, it will decay back to the upper solution. If, however, the perturbation is larger, so we move to a point outside the cycle, we will not be able to get back to the upper solution the system must move to the other stable state, with no reactant consumption. 

The point at which the homoclinic orbit is formed must be calculated numerically, but once it has been located we can show that the limit cycle is still unstable as it approaches the loop formation. We do this by evaluating the trace of the Jacobian matrix for the saddle point solution a2, P2 corresponding to Tres if tr(J) is positive, the limit cycle is unstable (as we always find for this special case of the present model) if tr( J) is negative for the saddle 

We should also consider the behaviour along the top of the isola, on the part of the branch lying at longer residence times than the Hopf point. For Tres t s, and with k2 still in the above range, the uppermost stationary state is unstable and is not surrounded by a stable limit cycle. The system cannot sit on this part of the branch, so it must eventually move to the only stable state, that of no conversion. Thus we fall off the top of the isola not at the long residence time turning point, but earlier as we pass the Hopf bifurcation point. 

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