Pitchfork-Hopf bifurcation

8 Pitchfork-Hopf bifurcation (0, i) with symmetry under reflection x3 —x3 

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This route should already be familiar to us from our discussion of the logistic map in chapter 4, Prom that chapter, we recall that the Feigenbaum route calls for a sequence of period-doubling bifurcations pitchfork bifurcations versus the Hopf bifurcations of the Landau-Hopf route) such that if subharmonic bifurcations are observed at Reynolds numbers TZi and 7 2, another can be expected at TZ determined by... 

Like pitchfork bifurcations, Hopf bifurcations come in both super- and subcritical varieties. The subcritical case is always much more dramatic, and potentially dangerous in engineering applications. After the bifurcation, the trajectories must jump to a distant attractor, which may be a fixed point, another limit cycle, infinity, or—in... 

The saddle node catastrophe and the Hopf bifurcation may be shown to be structurally stable. Certain additional conditions (see Sections 5.5.2.2, 5.5.2.3) are imposed on the transcritical bifurcation and the pitchfork bifurcation. The system is structurally stable under perturbations not disturbing these additional conditions on the other hand, when arbitrary... 

Fig. 6.2. Region of transverse stability for synchronous S and antisynchronous A CW solutions, respectively. P denotes the curves of transverse pitchfork bifurcations and H Hopf bifurcations for the parameters a = 2, J = 1, e = 0.03. (b) Zoom of the small part of (a) shows overlapping of the regions.

Fig. 6.3. Two branches of synchronous and antisynchronous CW solutions and the connecting branches of asynchronous solutions. Pa and Ps are pitchfork and Ha, Hg are Hopf bifurcations. Index s stands for the synchronous and a for antisynchronous solutions, respectively, rj = 0.2...

Figure 5. Bifurcation diagram on the plane of the two control parameters p and a. The solid lines 1 and 2 mark the primary instability, where the homogeneous homeotropic orientation becomes unstable. At 1, the bifurcation is a stationary (pitchfork) bifurcation, and a Hopf one at 2. The two lines connect in the Takens-Bogdanov (TB) point. The solid lines 3 and 4 mark the first gluing bifurcation and the second gluing bifurcation respectively. The dashed lines 2b and 3b mark the lines of the primary Hopf bifurcation and the first gluing bifurcation when calculated without the inclusion of flow in the equations.

For 0.33 < X < 0.53, the OFT is a pitchfork bifurcation, and tbe reoriented state is a D state [see the filled circles in Fig. 13(a)], This state loses its stability through a supercritical Hopf bifurcation to an O state [curve 1 in... 

An interesting situation also came to light in the limit of normal incidence. This case was impossible to analyze in the framework of the approximate model, as the modes become large quickly and violate the initial assumptions. It turned out that for a = 0 (which is a peculiar case, since the external symmetry breaking in the x direction vanishes), another stationary instability precedes the secondary Hopf bifurcation that spontaneously breaks the reflection symmetry with respect to x. It is shown by point A in Fig. 18. It is also seen from this figure, that the secondary pitchfork bifurcation is destroyed in tbe case of oblique incidence, which can be interpreted as an imperfect bifurcation with respect to the angle a [43]. 

We would start with the simple examples of (a) tangent bifurcation (b) pitchfork bifurcation and (c) Hopf bifurcation [Figs. 8.3 and 8.4]. 

It is interesting to note that, in this example as well as the two examples that follow, as is gradually increased, the trivial equilibrium point first goes through a supercritical pitchfork bifurcation at Q = cob and then a subcritical pitchfork bifurcation at Q = co where co is the solution of bo = 0. These bifurcations in the amplitude equation correspond to Hopf bifurcations of the original system s trivial equilibrium point. 

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