Supercritical bifurcation

In almost all the measurements the standard reference material 4-methoxybenzyli-dene-4 -n-butyl-aniline (MBBA) or a mixture, Merck Phase V, have been used, sometimes doped with an ionic substance. MBBA is the only room-temperature nematic with dielectric anisotropy < 0 where all the material parameters have been measured. For tabulated values see e.g. Ref. [12]. Unfortunately, it is a Schiffbase and rather imstable when exposed to moisture. Therefore, it is difficult to control the long-time conductivity in situ. Thus the recent successful introduction of the very stable material 4-ethyl-2-fluoro-4 -[2-(rrani-4-n-pentylcyclohexyl)-ethyl]-biphenyl (152) doped with iodine is very promising [50, 51]. This material exhibits at low external frequencies strongly oblique travelling rolls which bifurcate supercritically, leading to a particularly interesting scenario. ... 

The theory can be extended to three-dimensional systems [19, 22, 24]. Although the number of different structures increases, the bifurcation scheme is similar to the 2-D scheme. A 3-D body centered cubic structure involving six pairs of wavevectors and the 2-D hexagonal prismatic columns based on the same set as the 2-D hexagons come successively in a subcritical way (like the 2-D hexagons), losing stability far away from onset. Like stripes in 2-D systems, a 1-D lamellar phase bifurcates supercritically but becomes stable at some distance from onset. 

Since the only equilibrium point E(0 po) in the phase plane becomes unstable for i > ic and the infinity is unstable for any i, we conclude that limit cycles must exist around E(G,p0) for > ic. At the same time, the proven nonexistence of the limit cycles for i < ic implies the supercritical nature of the Hopf bifurcation at = ic in the symmetric case /"(0) = 0. 

Recall that a Hopf bifurcation is termed supercritical if its bifurcation diagram is as shown schematically in Fig. 6.2.2a. Correspondingly, in this case a stable limit cycle is born around the equilibrium, unstable hereon, only at a critical (bifurcation) value of the control parameter A = Ac. In contrast, in the subcritical case (Fig. 6.2.2b), the equilibrium is surrounded by limit cycles already for A < Ac, with an unstable limit cycle separating the stable one from the still stable equilibrium. At the bifurcation A = Ac the unstable limit cycle dies out with the equilibrium, unstable hereon, surrounded by a stable limit cycle. Thus the main feature of the subcritical case (as opposed to the supercritical one) is that a stable equilibrium and a stable limit cycle coexist in a certain parameter range, with a possibility to reach the limit cycle through a sufficiently strong perturbation of the equilibrium. 

We thus observe that M is positive for all w, A > 0 and, therefore, the bifurcation is supercritical. This stands in contrast with the result for the classical Teorell model discussed in the previous section. Recall that there the type of bifurcation depended on the properties of the postulated stationary resistance function /( ), specifically on the sign of / "(0) whenever... 

There are no unstable limit cycles in this model, and the oscillatory solution born at one bifurcation point exists over the whole range of stationary-state instability, disappearing again at the other Hopf bifurcation. Both bifurcations have the same character (stable limit cycle emerging from zero amplitude), although they are mirror images, and are called supercritical Hopf bifurcations. 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

When the Hopf bifurcation at p is supercritical (/ 2 < 0) the system has just a single stable limit cycle. This emerges at p and exists across the range p < p < p, within which it surrounds the unstable stationary-state solution. The limit cycle shrinks back to zero amplitude at the lower bifurcation point p%. This behaviour is qualitatively the same as that shown with the simplifying exponential approximation and is illustrated in Fig. 5.4(a). 

FlO. 5.4. The birth and growth of oscillatory solutions for the thermokinetic model with the full Arrhenius temperature dependence, (a) The Hopf bifurcations /x and ft are both supercritical, with [12 < 0, and the stable limit cycle born at one dies at the other, (b) The upper Hopf bifurcation is subcritical, with fl2 > 0. An unstable limit cycle emerges and grows as the dimensionless reactant concentration ft increases—at /rsu this merges with the stable limit cycle born at the lower supercritical Hopf bifurcation point ft. ... 

Fig. 8.7. Supercritical Hopf bifurcation for cubic autocatalysis with decay and /) = 0, appropriate for small dimensionless decay rate constant k2 < 9/256. A stable limit cycle emerges and grows as the residence time is increased above t s. At higher residence times, this disappears at rj , by merging with an unstable limit cycle born from a homoclinic orbit at t. (With non-zero autocatalyst inflow, (i0 > 0, the stable limit cycle itself may form a homoclinic orbit at long tres.)...

These requirements specify two loci one of them, labelled DH l in Fig. 8.12, emanates from the points / = 0, k2 = 9/256, as located in 8.3.6. This curve cuts through the parameter space for isola and mushroom patterns, but always lies below the curve A. (In fact it intersects A at the common point P0 = i(33/2 - 5), k2 = rg(3 - /3)4(1 -, /3)2 where the locus H also crosses.) In the vicinity of DH x, the stationary-state curve has only one Hopf point. This changes from a subcritical bifurcation (unstable limit cycle emerging) for conditions to the right of the curve to supercritical (stable limit cycle emerging) to the left. 

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. 

Two other points are marked, one along each Hopf curve. These are the degenerate bifurcation points at which the emerging limit cycle changes from stable (supercritical) to unstable (subcritical). These have the locations... 

With p = 0.019, the traverse cuts both Hopf curves, so the stationary-state locus has four Hopf bifurcation points, as shown in Fig. 12.6(c), each one supercritical. There are two separate ranges of the partial pressure of R over which a stable limit cycle and hence sustained oscillations occur. 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

The upper and lower limits in eqn (13.61) correspond to points of supercritical Hopf bifurcation. The stationary state is unstable in the whole of the region, and near to the limits is surrounded by a stable period-1 limit cycle. We now wish to see how the stability of this oscillatory solution, and any higher-order periodicities which might emerge, varies with rN. 

Figure 9. Supercritical antipitchfork bifurcation scenario for symmetric XYX molecules on the left, bifurcation diagram in the plane of energy versus position in the center, typical phase portraits in some Poincare section in the different regimes on the right, the fundamental periodic orbits in position space.

The fact that the antipitchfork bifurcation is supercritical implies that all these new trapped orbits of the invariant set are bom above the bifurcation... 

The classical dynamics follows the first bifurcation scenario we discussed above, i.e., involving a supercritical antipitchfork bifurcation. 

FIGURE 2 The birth and growth of limit cycle oscillations in the I - a, jS, Tr space for a system with non-zero e and k displaying a mushroom stationary-state pattern. Oscillatory behaviour originates from a supercritical Hopf bifurcation along the upper branch and terminates via homoclinic orbit formation. 

Subcritical Hopf transitions are found on the segments HM and GL of the Hopf curve and all other transitions are supercritical. The points H and G in figure 8 are located at (< ] = 0.019308, a2 = 0.030686) and ( i = 0.020668, a2 = 0.018330) respectively, and might be called metacritical. They are bifurcations of codimension two so that we expect only isolated metacritical points on the Hopf curve. These have to satisfy not only the conditions of (42), but also ... 

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