Bifurcation lines

In the case of strong surfactants the tcp corresponds to the lowest value of Ps for which CI4 vanishes at the bifurcation line for some fluctuation 0(k), with k = k. The value of H4 depends on the form of 0(k). Here we find the approximate position of the tcp, considering a selection of structures... 

C. Stability of the Cubic Phases near the Bifurcation Line... 

The lowest value of Qeff corresponds to different structures for different along the bifurcation line. The sequence of phases is always the same for various strengths of surfactant (with 7 > 27/4) and for increasing p it is L—>G—>D—>P—>C. For 7 = 50 (strong surfactant, like C10E5) the portion of the phase diagram corresponding to the stable cubic phases is shown in Fig. 14(b). For surfactants weaker than in the case shown in Fig. 14 the cubic phases occur for a lower surfactant volume fraction for example, for 7=16 cubic phases appear for p 0.45. 

FIG. 15 The projected surface area per unit volume S, divided by the surfactant volume fraction for different structures along the bifurcation line as a function of surfactant volume fraction Note that due to the geometrical constraints this quantity cannot exceed the length of the surfactant a. Here we set a = 1 for convenience. 

Next we prepare the system in the vicinity of the lower bifurcation line in Fig. 5.9, slightly below a Hopf bifurcation marked by the small rectangle in Fig.5.9 [51]. In the absence of noise the only stationary solution is a sta-... 

Figure 3. Bifurcation lines in the (X2 )plone ofparameter space for a-0.2, P=0.00758, y=0.00213, Y2=0,32, A=2. Plain lines are the locus of...

The ratio E/ps, calculated for different phases below the bifurcation, is shown in Fig. 15. In the special case of the C phase the surface intersects itself therefore, in the computation of S/p we have subtracted the volume occupied along the lines of intersection, since it would be counted twice otherwise. The surface area per volume is an increasing function of the surfactant volume fraction and it determines the sequence of phases. Moreover, we have found that the effect of broadening of the interface on the value S/p in different phases is different, and we have a quantitative... 

Merger of the ALEE and the IRE in 1963 into the Institution of Electrical and Electronics Engineers (IEEE) preseiwed in its name a sense of the earlier bifurcation of the profession, but in fact the lines had become quite blurred and the numerous constituent societies drew and continue to draw from both traditions. At the end of the centuiy, the IEEE was easily the largest professional organization in the world with membership over 350,000. Its thirty-six constituent societies provided an indication of the degree to which electricity has pervaded the modern world, ranging from Communications and Lasers Electron... 

Fig. 20 Reaction pathways in the reduction of methyl (a) and /-butyl chloride (b) by NO". , reactant and products , transition states. In (a) and (b), the full line is the mass-weighted IRC path from the reactant to the product states the dashed line is a ridge separating the Sn2 and ET valleys and the dotted-dashed line is the mass-weighted IRC path from the Sn2 product state to the ET product state (homolytic dissociation). The dotted line in (a) represents the col separating the reactant and the SN2 product valleys. The dotted line in (b) represents the steepest descent path from the bifurcation point, B, to the Sn2 product. In (a), B is the point of the col separating the reactant and the SN2 product valleys where the ridge separating the SN2 and ET valleys starts.

Figure 19. The steady state solutions A0 of the pathway shown in Fig. 18 as a function of the influx vi. For an intermediate influx, two pathways exist in two possible stable steady states (black lines), separated by an unstable state (gray line). The stable and the unstable state annihilate in a saddle node bifurcation. The parameters are k.2 0.2, 3 2.0, K] 1.0, and n 4 (in arbitrary units).

Figure 28. The eigenvalues of the Jacobian of minimal glycolysis as a function of the influence of ATP on the first reaction V (ATP) (feedback strength). Shown is the largest real part of the eigenvalues (solid line), along with the corresponding imaginary part (dashed line). Different dynamic regimes are separated by vertical dashed lines, for > 0 the state is unstable. Transitions occur via a saddle node (SN) and a Hopf (HO) bifurcation. Parameters are v° 1, TP° 1, ATP0 0.5, At 1, and 6 0.8. See color insert.

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

Fig. 16. A cartoon showing the putative channels that provide access to the active site of a large-subunit catalase in A and a small-subunit catalase in B. The main or perpendicular channel is labeled P and the minor or lateral channel, which is bifurcated, is labeled L. A potential channel leading to the proximal side of the heme is shown with a dashed line.

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