Flip bifurcation

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

Figure 10.3.1 also suggests how x itself loses stability. As r increases beyond 1, the slope at x gets increasingly steep. Example 10.3.1 shows that the critical slope f x ) = -1 is attained when r = 3. The resulting bifurcation is called a flip bifurcation. 

Flip bifurcations are often associated with period-doubling. In the logistic map, the flip bifurcation at r = 3 does indeed spawn a 2-cycle, as shown in the next example. 

A cobweb diagram reveals how flip bifurcations can give rise to perioddoubling. Consider any map f, and look at the local picture near a fixed point where / x ) —1 (Figure 10.3.3). 

If the graph of f is concave down near x, the cobweb tends to produce a small, stable 2-cycle close to the fixed point. B ut like pitchfork bifurcations, flip bifurcations can also be subcritical, in which case the 2-cycle exists below the bifurcation and is unstable—see Exercise 10.3.11. 

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... 

When fi=0 the renormalized map (1 undergoes a flip bifurcation. Equivalently, the 2-cycIe for the original map loses stability and creates a 4-cycle. This brings us to the end of the first period-doubling. 

A fixed point of a map is linearly stable if and only if all eigenvalues of the Jacobian satisfy A <1. Determine the stability of the fixed points of the Henon map, as a function of a and b. Show that one fixed point is always unstable, while the other is stable for a slightly larger than Show that this fixed point loses stability in a flip bifurcation (A = -1) at a, = (1 - b. ... 

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

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. 13.7.22. 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.

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. and Krauskopf, B. [1998] Resonant homoclinic flip bifurcations, preprint. Free University-Berlin. 

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