Subcritical bifurcation

When g < 0, stripes and hexagons can bifurcate subcritically and the situation is much more complex than in the previous case. Different types of states can be simultaneously stable with the stationary state [40]. An example of a bifurcation diagram for our basic model at 7 > 7c is given in Figure 5. Moreover, the models which present these subcritical bifurcations also often present multiple homogeneous steady states like our basic model or the Gray-Scott model [31]. Actually, these fully developed structures pertain... 

Fig. 13. Bifurcation diagram for an even wave number and

When <0, the bifurcation diagram is as in Fig. 13. There exists a subcritical region in which three stable steady-state solutions may coexist simultaneously the thermodynamic branch and two inhomogeneous solutions. It must be pointed out that the latter are necessarily located at a finite distance from the thermodynamic branch. As a result, their evaluation cannot be performed by the methods described here. The existence of these solutions is, however, ensured by the fact that in the limit B->0, only the thermodynamic solution exists whereas for B Bc it can be shown that the amplitude of all steady-state solutions remains bounded. 

In Fig. 21 we have drawn the bifurcation diagram of the fundamental steady-state solutions for three values of p [ Kxn is plotted versus UK) as the bifurcation parameter]. There is a subcritical region in the upper or lower branch, depending on the relative height of the peaks in Fig. 20c. The asymptotes K and K" of these branches correspond to half-period solutions of infinite length. When p 2 the asymptote K merges with the w-axis therefore situation 2 above can be viewed as a particular case of situation 3 above, in which the bifurcation point moves to infinity. 

These results are thus in agreement with those of bifurcation theory. In the case of odd wave numbers they demonstrate that in general the bifurcation diagrams have to exhibit a subcritical branch. However, there always exists even for odd wave numbers a value of the parameters such that the bifurcation is soft and this value marks the transition from an upper to a lower subcritical branch (see Fig. 21). This feature was less... 

Fig. 6.2.2. Schematic bifurcation diagrams in a super- (a) and subcritical (b) case.

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. 

Fig. 4.9. The development of oscillatory amplitude Ae and period T across the range of instability, 4.2 x 10 3 = n < n < jx = 0.0195, for the pool chemical model with k = 2x 10-3 and y = 0.21, typical of a system with a subcritical Hopf bifurcation at which an unstable limit cycle emerges at The broken curves give the limiting forms predicted by eqns (4.59)—(4.61).

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...

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. 8.6. Typical arrangement of local stabilities and development of unstable limit cycle, from a subcritical Hopf bifurcation, appropriate to cubic autocatalysis with decay and no autocatalyst inflow and with 9/256 < k2 < 1/16. The unstable limit cycle grows as t, decreases below t m, and terminates by means of the formation of a homoclinic orbit at rf . Stable stationary states, including the zero conversion branch 1 — a, = 0, are indicated by solid curves, unstable states and limit cycles by broken curves.

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... 

Next, consider the case with p = 0.02014. The traverse across Fig. 12.6(a) as r is varied now also cuts the region of multi stability. It passes above the cusp point C (see Fig. 12.5), giving rise to two turning points in the stationary-state locus, but below the double-zero eigenvalue point M. There are still four intersections with the Hopf curve, so there are four points of Hopf bifurcation. The Hopf point at highest r is now a subcritical bifurcation. The dependence of the reaction rate on r for this system is shown in Fig. 12.6(d). 

Figure 10. Same as Fig. 9 for the subcritical antipitchfork bifurcation scenario for symmetric XYX. 

As the energy increases in the interval E < E < Ea, the orbits Y and 2 progressively shift toward the symmetric-stretch orbit 0 and merge at the subcritical antipitchfork bifurcation. Just below this bifurcation, 1 and 2 are elliptic while 0 is still hyperbolic (without reflection). Between and Ea, the periodic orbits 1 and 2 may either remain of elliptic type or become hyperbolic in the energy interval [ , "1 such that E < Edd> < Edd < Ea. 

In the following, we will characterize the different dynamical regimes for CO2 on the LEPS surface. The scenario in a general way resembles the one for Hgl2. However, the initial bifurcations are of the second type described in Section IV.C.2 that is, the symmetric-stretch orbit undergoes a subcritical... 

This dissociative system, which represents the prototype system for chemical reaction dynamics, has been the object of many studies. Child et al. [143] have carried out a detailed analysis of the classical dynamics in a collinear model based on the Karplus-Porter surface. These authors have introduced the concept of PODS and first observed the subcritical antipitchfork bifurcation scenario in this system. 

The bottom of the exit channels is at -3194 cm-1 if the origin corresponds to the saddle of the Karplus-Porter surface. The pair of tangent bifurcations occur at E = 1670 cm 1, which is followed by the subcritical antipitchfork bifurcation at Ea = 2633 cm 1. The bifurcation scenario is thus similar to the CO2 system, and we may expect a three-branch Smale horseshoe in this system as well. 

FIGU RE 10 Illustration of the disappearance of a limit cycle via a turning point on a periodic branch near a subcritical Hopf bifurcation, (a) A stable limit cycle surrounding an unstable focus (b) the unstable focus undergoes a subcritical Hopf bifurcation and leaves an inner unstable limit cycle surrounding a stable focus (c) the two limit cycles combine into a metastable configuration and disappear altogether as the parameter is further increased. 

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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