Orbit-flip bifurcation

Fig. 13.6.2. The orbit-flip bifurcation — the homoclinic loop F gets closed along the nonleading submanifold at the moment of bifurcation.

We commented at the end of Section 10.2 that a copy of the orbit diagram appears in miniature in the period-3 window. The explanation has to do with hills and valleys again. Just after the stable 3-cycle is created in the tangent bifurcation, the slope at the black dots in Figure 10.4.1 is close to -i-l. As we increase r, the hills rise and the valleys sink. The slope of / (x) at the black dots decreases steadily from -1-1 and eventually reaches -1. When this occurs, a flip bifurcation causes... 

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. 

Fig. 13.6.6. Orbit-flip homoclinic bifurcation the change of the way the separatrix V tends to the saddle results in the change of sign of separatrix A.

Fig. 13.7.22. The bifurcation diagram for the heteroclinic connection in Fig. 13.7.12 when A > 0, A2 < 0, i/i > 1, 1/2 < 1 and 1/11/2 > 1. The system has one simple periodic orbit in regions 1, 2, 3 and 5, two periodic orbits (one simple and one of double period) in region 4, and no periodic orbits elsewhere. The stable periodic orbit loses stability on the curve PD corresponding to a period-doubling (flip) bifurcation. The unstable limit cycle of double period becomes a double-circuit separatrix loop on L. The stable simple limit cycle terminates on Li.

Summary The set of principal stability boundaries of equilibrium states consists of surfaces of three kinds Si, Sr and Ss. Only the Si-like boundaries are safe. As for periodic orbits, there are nine types of principal stability boundaries among them Se, Sg, Sio, Sn are dangerous, while S2, S3, S4 S5 and Si, S2 2ire safe (the latter two correspond to the subcritical Andronov-Hopf and flip bifurcations, respectively). 

Note that the first Lyapunov value is always negative for a flip-bifurcation of any periodic orbit in the logistic map. Indeed, the Schwarzian derivative ... 

Evaluate the values of an that correspond to the flip bifurcations of the orbits of period 16, 32, respectively. Find the corresponding maximal x-coordinates of these cycles and plot them on Fig. C.6.2. ... 

Fig. C.6.6. A part of the bifurcation diagram. AH labels the Andronov-Hopf bifurcation of the non-trivial equilibrium state Qi PD labels a flip-bifurcation of the stable periodic orbits that generates from Oi.

Kokubu, H., Komuro, M. and Oka, H. [1996] Multiple homoclinic bifurcations from orbit flip. I. Successive homoclinic-doublings, Int J. Bif Chaos 6, 833-850. 

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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