Growth of stable and unstable limit cycles

The expression n2, which determines the direction in which the limit cycle grows, is 

At the lower Hopf bifurcation, P2 is always negative and n2 is positive. Thus a stable limit cycle emerges from n, growing as the reactant concentration jx is increased. 

Location of degenerate Hopf bifurcation points from eqn (5.55) 

The variation in oscillatory amplitude across the whole range of instability must be completed numerically (a suitable method for this is described in the appendix to this chapter). With the full Arrhenius form there are two possible scenarios, corresponding to the two different types of Hopf bifurcation at p.  

When the Hopf bifurcation at p is supercritical (/ 2 0) the system has just a single stable limit cycle. This emerges at p and exists across the range p p p, within which it surrounds the unstable stationary-state solution. The limit cycle shrinks back to zero amplitude at the lower bifurcation point p%. This behaviour is qualitatively the same as that shown with the simplifying exponential approximation and is illustrated in Fig. 5.4(a). 

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