A separatrix

Figure Al.2.12. Energy level pattern of a polyad with resonant collective modes. The top and bottom energy levels conespond to overtone motion along the two modes shown in figure Al.2.11. which have a different frequency. The spacing between adjacent levels decreases until it reaches a minimum between the third and fourth levels from the top. This minimum is the hallmark of a separatrix [29, 45] in phase space.

To a first approximation, the composition of the distillate and bottoms of a single-feed continuous distillation column lies on the same residue curve. Therefore, for systems having separatrices and multiple regions, distillation composition profiles are also constrained to lie in specific regions. The precise boundaries of these distillation regions are a function of reflux ratio, but they are closely approximated by the RCM separatrices. If a separatrix exists in a system, a corresponding distillation boundary also exists. Also, mass balance constraints require that the distillate composition, the bottoms composition, and the net feed composition plotted on an RCM for any feasible distillation be collinear and spaced in relation to distillate and bottoms flows according to the well-known lever rule. 

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.

It may also seem sensible, if there are multiple solutions, to ask which of the states is the most stable In fact, however, this is not a valid question, partly because we have only been asking about very small disturbances. Each of the two stable states has a domain of attraction . If we start with a particular initial concentration of A the system will move to one or other. Some initial conditions go to the low extent of reaction state (generally those for which 1 — a is low initially), the remainder go to the upper stationary state. The shading in Fig. 6.9 shows which initial states go to which final stationary state. It is clear from the figure that the middle branch of (unstable) solutions plays the role of a boundary between the two stable states, and so is sometimes known as a separatrix (in one-dimensional systems only, though). 

We have already mentioned the sharp difference in the relaxation times outside the region between the null dines ( 1 s) and inside it (as high as hundreds of seconds). The motion outside this region depends on the "fastest reaction. Inside this region the relaxation rate is dependent on the complicated complex of rate constants, and in the general case we cannot suggest that the reaction rate is limited by some reaction. The common trajectory near which the relaxation is retarded is no more than a specific trajectory that is a separatrix going from the unstable into the stable steady... 

Recently, Maitra and Heller reexamined quantum transport through cantori [93], including the cases in which nh is actually smaller than the classical flux crossing a particular cantorus. In doing so, they used the Whisker map, which can describe the motion within a chaotic layer near a separatrix in a typical nonintegrable system. The Whisker map is given by... 

Figure 13. (The color version is available from the authors.) The previous pair of trajectories as seen in the normal-form coordinates in the hyperbolic direction. The complicated dynamics at the saddle has been smoothed out. The two trajectories approach from the top. Note how the axis acts as a separatrix between them. The reactive one intersects the TS (the diagonal) at the dot. Compare this with Figs. 5, 6, and 7. Primes on 1 3 and p (see the text) have been dropped.

With these properties, if locally separates phase space, as illustrated in the scheme (Figure 1). It is very important to note that even if if has codimension 1 and is locally a separatrix, it does mean in n DOFs that if neither has a simple geometry, because it is subject to stretching and folding because of chaos [24-26], nor separates globally (see Ref. 27). Let us now make a summary of the... 

As for the dynamics of JC under the unperturbed Hamiltonian Hq x,I), we assume that the reaction coordinate jc has a saddle X I) = (Q I),P I)). Its location, in general, depends on the action variables 7. Suppose that the saddle X I) has a separatrix orbit JCo(t,7) connecting it with itself. See Fig. 9 for a schematic picture of the phase space jc = (q,p) under the unperturbed Hamiltonian Hq x,I). Here, we show the saddle X and the separatrix orbit on the two-dimensional phase space jc = (q,p). 

Here, we limit our argument to a system with a homoclinic connection—that is, a separatrix connecting a saddle with itself. The following argument can be straightforwardly extended to a system with a heteroclinic connection— that is, a separatrix connecting different saddles. 

Both methylethylketone (MEK) andmethylisopropylketone (MIPK) form minimum-boihng azeotropes with water (Fig. 13-58b). In this ternary system, a separatrix connects the binary azeotropes and divides the RCM into two regions. The high-boiHng node of Region I is pure water, while the low-boiHng node is the MEK-water azeotrope. 

The same investigation in the case of the Hamiltonian flow (Figure 17) shows exactly the same qualitative situation. Not only we are confident that we are not facing anomalous diffusion but also that the orbits correspond really to chaotic motion and not to regular tori which could have FLI larger than log T because of the proximity to a separatrix. 

The lines dividing the phase plane into regions of a different course of trajectories are called separatrices. For example, the isocline of horizontal tangents, /(x,y) = 0, divides the phase plane into regions in which the solutions y(x) either increase or decrease it thus is a separatrix. We shall... 

Although in classical mechanics E may be varied continuously, the trajectories are displayed only for the values of E that are quantum eigen-energies. The dotted /z(V0 line is a separatrix, which is a dividing surface that no trajectory can cross and which divides the accessible phase space into qualitatively distinct regions (exhibiting normal vs. local mode behavior) filled with qualitatively different trajectories. 

The normal mode representation of the phase space trajectories contains the same information as the local mode representation. However, the resonance region on the normal mode phase space map contains the local mode trajectories (la, lb 2a, 2b) and the stable fixed points Ca and C t,. The trajectories contained within the resonance zone are not free to explore the entire 0 < tp < n range whereas the trajectories outside the resonance zone do explore the 0 < ip < n range and are therefore classified as normal mode trajectories. The fixed point B (Iz = I = +2) is unstable, because it lies on a separatrix, and is located at the north pole of the normal mode polyad phase sphere. The stable fixed point A (7Z = — I = —2) is located at the south pole. 

Hence, the family of all residue curves that originate from one fixed composition point and terminate into another fixed composition point defines a distillation region. Two adjacent regions are separated by a separatrix. In this book we called it simply distillation boundary. 

Briefly, the co-dimension of a submanifold is the dimension of the space in which the submanifold exists, minus the dimension of the submanifold. The significance of a submanifold being "co-dimension one" is that it is one less dimension than the space in which it exists. Therefore it can "divide" the space and act given it is invariant as a separatrix, or barrier, to transport. 

Figure 16 Sketches of numerically generated separatrices on the Poincare map. After extracting the manodromy matrix of a hyperbolic fixed point (p2> the asymptotic eigenvectors W+ and W can be obtained. (A) If no other fix points are nearby in the chaotic sea, the separatrix branch formed by repeated mappings in positive time of points initially on will eventually meet the branch formed by repeated mappings in negative time of points initially on W at a single point hj (called a homoclinic point). The closed curve so generated is the separatrix 5. (B) If a second fixed point (p2> 2)2 associated with a different periodic orbit is nearby, a separatrix 5 may be formed by the intersection of branches arising from the two orbits at two points hi and (called heteroclinic points).

Fig. 5.16 shows residue curve maps for four selected values of the Damkohler number Da at an operating pressure ofp = 0.8 MPa. The residue curve map for distillation without reaction (Da = 0, Fig. 5.16a) shows one saddle point, which is the binary azeotropic point between MeOH and MTBE. The second binary azeotropic mixture of MeOH and IB represents an unstable node. These two points are linked by a separatrix, which acts as a distillation boundary. Consequently, two stable nodes exist in the system one stable node at pure MTBE for initial compositions below the distillation boundary and another stable node at pure MeOH fo mixtures having an initial composition above the distillation boundary. 

The added factor 1/(1 + s/X, s) in Equ. 5.88 represents the toxicity of the substrate at higher concentrations. Let us recall that the condition for calculation of the stationary state with nonvanishing biomass concentration is the relation fx(s) = D. This equation has only one solution if fi(s) is a monotonic function. But with characteristics as in Equ. 5.88, there are two solutions. Together with the washout state ( x, s) we have three stationary states. Two of them are stable ( x, and x, s), one of them is unstable ( x, s). Thus, we have a bistable system. The stationary values of the stable and the unstable stationary state are shown as a function of D in Fig. 6.11. Hysteresis may occur in shift experiments. Figure 6.12 shows how the final biomass concentration depends on the initial concentration. Figure 6.13 demonstrates that the phase plane is divided into two attraction domains. Both domains are touched by a separatrix in which the unstable stationary state lies. Note that, after an external disturbance, the system can cross over the separatrix and shift from one steady state to the other. This bistable behavior is a serious problem in, for example, waste treatment It takes place if substrates such as alcohols, phenols, or hydrocarbons occur in such high concentrations that the utilization of these substrates is inhibited. 

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