Hopf bifurcation points transversality

We may use the above description to write down the condition for two Hopf points to merge, by requiring that 

We can now draw the following conclusions. If we choose values of the parameters / 0 and k2 which lie to the right of or above the line H, Hopf bifurcation will not occur as we vary the residence time. Hopf bifurcations are favoured by small values for the decay rate and the autocatalyst inflow 

We have seen that the emerging limit cycle can be stable or unstable, depending on the value of k2, for the case / 0 = 0. The condition for the change in stability is that the exponent / 2 describing the stability of a limit cycle passes through zero at the Hopf point. We can follow this third type of degeneracy as a curve across the parameter plane by specifying that 

These requirements specify two loci one of them, labelled DH l in Fig. 8.12, emanates from the points / = 0, k2 = 9/256, as located in 8.3.6. This curve cuts through the parameter space for isola and mushroom patterns, but always lies below the curve A. (In fact it intersects A at the common point P0 = i(33/2 - 5), k2 = rg(3 - /3)4(1 -, /3)2 where the locus H also crosses.) In the vicinity of DH x, the stationary-state curve has only one Hopf point. This changes from a subcritical bifurcation (unstable limit cycle emerging) for conditions to the right of the curve to supercritical (stable limit cycle emerging) to the left. 

The second locus, DH2, emerges from the point of tangency between the double-zero eigenvalue curve A and the hysteresis line at p0 = k2 = 34/45. 

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Fig. 8.11. The locus H of degenerate Hopf bifurcation points described by the transversality condition (merging of two Hopf points), eqn (8.51). Below this curve, the stationary-state locus exhibits Hopf bifurcation (dynamic instability) at some residence times above it, the system does...

Assume also that x (a) is a critical point of Eq. (29). The eigenvalues Pi( ) Piia),..., pjv(a) will now also depend on the parameter a. If for some values of a, say a < Gq, the critical point is stable, and if in addition a pair of complex conjugate eigenvalues pi(a), p2(o) cross the imaginary axes transversely (d Repi(a)/da a=ao5 0) then we say that a Hopf bifurcation takes place at the value a = Gq. If a Hopf bifurcation occurs in Eq. (29) then there exists a one-parameter family of periodic solutions for a in the neighborhood of ao with a period near 27r/ Im pi(ao)l- K the flow attracts to the critical point x (ao) when a = Gq, then x°(ao) is called a vague attractor. For this case the family of closed orbits is contained m a >Go and the orbits are of attracting type. ... 

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