Solution lower-branch

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. 

Fig. 23. Bifurcation diagram of KU2 in the case of Fig. 22. The lower branch of solutions corresponds to the thermodynamic branch. It tends to the asymptotic value K 12, which separates it from nonequilibrium types of solutions.

Graessley and co-workers have studied the rheological properties of solutions of branched PVAc in diethyl phthalate (178, 188), using polymer concentrations of 0.17, 0.225, and 0.35 g ml-1. At the lowest concentration, the low shear-rate viscosity was simply related to [17], so that it was lower for branched polymers the equation ... 

Fig. 6.9. Domains of attraction for the two stable branches of stationary-state solution systems which have initial extents of conversion lying within the shaded region evolve to the lower branch those in the unshaded region approach the highest extent of reaction. In the region of multiple stationary states, the middle branch acts as a separatrix.

The requirements for the two changes in the stationary-state locus associated with the appearance of an isola as an isolated point and the birth of a mushroom as an isola grows to touch a lower branch of solutions both have the form... 

If a saturated solution is represented by a point of the lower branch of the solubility curve, this solution becomes non-saturated when the concentration is diminished and supersaturated when the concentration is increased if, on the contrary, a saturated solution is represented by a point on the upper branch C, of the solubility curve, this solution becomes non-saturated when the concentration is increased and supersaturated when the concentration is diminished. 

Otherwise expressed, the unsaturated solutions are represented by the points of the plane TOs (Fig. 56) which are situated bdow the lower branch or above the upper branch Cj of the solubility curve the supersaturated solutions are represented by the points situated between the two branches. 

Eutectic point between ferric chloride hydrates. Investigations of Bakhuis RoozboonL—Of all the solid hydrates which a salt solution may form, ice is always the most hydrated and the anhydrous salt the least hydrated always less rich in water than the first, the solution is always richer in water than the second the solubility curve of the first is reduced to its upper branch, the solubility curve of the other to its lower branch when these two branches meet their point of intersection is necessarily a eutectic point. 

Referring back to Fig. 4-11 and Eq. (4-216), we see that this limiting value for stability is precisely the same as the critical value Rrn,. Thus the lower part of the equilibrium solution curve (the lower branch of solutions) for... 

Consider again the steady-state temperature versus the residence-time curve shown in Figure 6.9 We have labeled seven points on this cun e A, B and C on the lower branch D on the middle branch and E, F and G on the upper branch. The numerical values at these points are listed in Table 6.2. If we first substitute the solution of the mass balance, Equation 6,.36, for CA into the energy balance, Equation 6.37, we obtain... 

The curve yielding Jq as a function of a shows the existence of a phenomenon of bistability in the reduced system when the substrate concentration is held constant. Three distinct values of yo are obtained in fig. 6.10 in the range a linear stability analysis of eqns (6.5) reveals that the steady state on the lower branch of the hysteresis curve is always stable, while it is unstable on the median branch, between points Sj and S2. On the upper branch, the steady state is unstable in the domain ai < a < 2. Two families of periodic solutions, denoted Tj and Fj, appear through a Hopf bifurcation at the points Hj and H2 they disappear at the points H and H 2, of abscissae a l, a 2, when the amplitude of the limit cycle is such that the latter reaches the... 

When the limit point 2 is reached, an increase in a elicits the abrupt transition towards the upper branch of the hysteresis curve, given that the lower branch has now vanished. But the upper branch of the steady-state curve of the (pr, y) system is unstable, and the analysis of the reduced system predicts that oscillations belonging to the branch of periodic solution F2 should occur when a is close to the value a"2. These oscillations correspond to the active phase of bursting represented in fig. 6.9b. 

Figure 1.2 shows the thermodynamic equilibrium for mobile platinum versus potential for an acidic solution in equilibrium with an exposed platinum surface and a lower branch of equilibrium concentration at potentials more positive than approximately 1.1 volts relative to a hydrogen electrode, where the plotted concentration denotes the concentration of mobile species in equilibrium with a platinum oxide layer covering the surface. This shows that excursions to higher potential can rapidly increase the rate of platinum dissolution prior to passivation of the surface. Once the surface is passivated, the dissolution stops and redeposition can occur, albeit incomplete redeposition, as the platinum, once rendered mobile, is free to redeposit on larger particles or diffuse away from the catalyst layer altogether [32]. 

Equation 6 which is the general solution to a variety of exitation schemes, can be solved graphically for ( ij (L). This is shown here for two cases. The solutions for a medium exhibiting Sg — transition (Fig. 2) incorporated in a matched ring resonator. The results for a coherent field pumping are depicted in Fig. The results for are used to calculate the output intensity as a function of lin (Fig. 4). The bistable behavior of this system (lower branch due to almost linear absorption and upper branch due to saturated absorption) can be also demonstrated in many other cases. The solutions for reverse saturable absorber - a medium exhibiting Sg — S. Ay T- —Tjj ( -jn on transitions - in the same optical resonator... 

Equistability of a homogeneous stable stationary state on the upper branch of the hysteresis loop, labelled I in Fig. 11.1, with a homogeneous stable stationary state on the lower branch, labelled II, occurs at one value of the influx coefficient k within the loop. Say that point occms at the location of line A. The predictions of the stationary solution of the master stochastic master equation are (a) the minimmn of the bimodal stationary probability distribution is located on the separatrix, and (b) at equistability the probability of fluctuations P(c) obeys the condition... 

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