Degeneracy cycle

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... 

Putting all the stationary-state and Hopf degeneracy loci together on one diagram, Fig. 8.13(a), we find the parameter plane divided into a total of 11 regions. In each of these the pattern drawn out by the stationary-state curve and limit cycles as the residence time varies is qualitatively different. Typical forms for these bifurcation diagrams ((i)-(xi)) are shown in Fig. 8.13(b). 

Fig. 8.13. (a) The division of the fS0 — K1 parameter region into 11 regions by the various loci of stationary-state and Hopf bifurcation degeneracies. The qualitative forms of the bifurcation diagrams for each region are given in fi)—(xi) in (b), where solid lines represent stable stationary states or limit cycles and broken curves correspond to unstable states or limit cycles, (i) unique solution, no Hopf bifurcation (ii) unique solution, two supercritical Hopf bifurcations (iii) unique solution, one supercritical and one subcritical Hopf (iv) isola, no Hopf points (v) isola with one subcritical Hopf (vi) isola with one supercritical Hopf (vii) mushroom with no Hopf points (viii) mushroom with two supercritical Hopf points (ix) mushroom with one supercritical Hopf (x) mushroom with one subcritical Hopf (xi) mushroom with supercritical and subcritical Hopf bifurcations on separate branches. 

Fig 5. Typical pseudo three-level system for an organic molecule. The spin-degeneracy of the triplet state may quite often be ignored, in case only one level in the pumping cycle plays a role. 

The explosion in the amount of vertices for the Simplex methods is an instance of zigzagging, whereas the degeneracy loop is an instance of cycling in a finite set of active constraints (see Section 10.6). 

For straight chains, a modified version of the above method may be used For Ann-carbon chain, inscribe a cycle with 2n- -2 carbons into the circle as before. Projecting out all intersections except the highest and lowest, and ignoring degeneracies gives the proper roots. This is exemplified for the allyl system in Fig. 8-10. 

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