Inclination-flip bifurcation

Fig. 13.6.1. The inclination-flip bifurcation A = 0) is due to a violation of the transversality of the intersection of and at the points of the homoclinic loop F.

Fig. 13.6.4. The bifurcation unfoldings for an orbit- and an inclination-flip bifurcation are identical in the simplest case.

The bifurcation diagram for Case B was proposed, independently, in [126] (see also [127, 129]) and in [77], for Case C — in [119]. Here, we give a unified and self-consistent proof for both cases, including the proof of the completeness of the bifurcational diagram. In the West, the Case B is called the inclination-flip bifurcation and the Case C is called the orbit-flip. 

Another codimension-two homoclinic bifurcation in the Shimizu-Morioka model occurs at (a 0.605,6 0.549) on the curve H2 corresponding to the double homoclinic loops. At this point, the separatrix value A vanishes and the loops become twisted, i.e. we run into inclination-flip bifurcation [see Figs. 13.4.8 and C.7.11]. The geometry of the local two-dimensional Poincare map is shown in Fig. 13.4.5 and 13.4.6. To find out what our case corresponds to in terms of the classification in Sec. 13.6, we need also to determine the saddle index 1/ at this point. Again, as in the case of a homoclinic loop to the saddle-focus, it is very crucial to determine whether 1/ < 1/2 or 1/ > 1/2. Simple calculation shows that z/ > 1/2 for the given parameter values. Therefore,... 

Homburg, A. J., Kokubu, H. and Krupa, M. [1994] The cusp horseshoe and its bifurcations in the unfolding of an inclination flip homoclinic orbit, Ergod, TL Dynam. Syst 14, 667- 93. 

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