Separatrixes

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.

Thus, if both the original and biased systems obey TST so that the above-mentioned derivation holds, hyperdynamics can provide considerable acceleration compared to direct-MD simulations. However, in practice, the applicability of hyperdynamics is limited by the availability of low-overhead bias potentials. Indeed, while some different forms have been proposed in the last few years, often they are computationally expensive, tailored to a limited class of systems or built on sets of restrictive assumptions about the nature of the separatrix. The main challenge, which is the subject of active research in different groups, thus remains the construction of bias potentials that are simple, efficient, generic, and transferable. We present below one recent advance in this area. 

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). 

Separatrix -role in separation process synthesis [SEPARATIONS PROCESS SYNTHESIS] (Vol 21)... 

An even more dramatic effect is observed when comparing the efficiencies of the nonlinear and the linear energy converter. As shown in Fig. 14 the separatrix i nlh = 1, which delimits the regions where the nonlinear and the linear energy converter are more efficient, declines... 

There are also two local bifurcations. The first one takes place for r 13.926..., when a homoclinic tangency of separatrixes of the origin O occurs (it is not shown in Fig. 20) and a hyperbolic set appears, which consists of a infinite number of saddle cycles. Beside the hyperbolic set, there are two saddle cycles, L and L2, around the stable states, Pi and P2. The separatrices of the origin O reach the saddle cycles Li and L2, and the attractors of the system are the states Pi and P2. The second local bifurcation is observed for r 24.06. The separatrices do not any longer reach to the saddle cycles L and L2. As a result, in the phase space of the system a stable quasihyperbolic state appears— the Lorenz attractor. The chaotic Lorenz attractor includes separatrices, the saddle point O and a hyperbolic set, which appears as a result of homoclinic tangency of the separatrices. The presence of the saddle point in the chaotic... 

Figure 21. Structure of the phase space of the Lorenz system. An escape trajectory measured by numerical simulation is indicated by the filled circles. The trajectory of the Lorenz attractor is shown by a thin line the separatrixes T and T2 by dashed lines [182].

Nitromethane shows the simplest residue curve map with one unstable curved separatrix dividing the triangle in two basic distillation regions. Methanol and acetonitrile give rise two binary azeotropic mixtures and three distillation regions that are bounded by two unstable curved separatrices. Water shows the most complicated residue curve maps, due to the presence of a ternary azeotrope and a miscibility gap with both the n-hexane and the ethyl acetate component. In all four cases, the heteroazeotrope (binary or ternary) has the lowest boiling temperature of the system. As it can be seen in Table 3, all entrainers except water provide the n-hexane-rich phase Zw as distillate product with a purity better than 0.91. Water is not a desirable entrainer because of the existence of ternary azeotrope whose n-hexane-rich phase has a water purity much lower (0.70). Considering in Table 3 the split... 

As shown in Fig. 2, the still path obtained experimentally (circles) is in excellent agreement with the still trajectory calculated by simulation. The selected reflux policy permits the still path to cross the separatrix into another distillation region than the initial feed region. Hence, the still path is able to reach the ethyl acetate vertex and this component remains pure into the still at the end of the distillate removal step. Such a behavior is not possible with a homogeneous entrainer that gives rise to a similar residue curve map because the distillation process is restricted to the distillation region where the initial composition of the mixture is located. In this case, ethyl acetate could not be obtained as an isolated product. 

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... 

The phase plane plot of Figure 2 illustrates the behavior of the concentrations of X and Y within this region. Initial concentrations of X and Y corresponding to a point above the broken line will evolve in time to the limit cycle. The broken line represents the separatrix of the middle unstable steady state which has the stability characteristics of a saddle point. Initial values for X and Y corresponding to a point below the separatrix will evolve to the stable state to the right of the diagram. 

Numerical integration of equations (2) and (3) with initial values for X,Y on the limit cycle and with one of the rate constants oscillating as in equation (4) or (5) may result in a transition of the X,Y trajectory across the separatrix towards the stationary state. The occurrence of a transition is dependent on the parameters g, u) and 0. For extremely small amplitude perturbations (g - -0), the trajectory continues to oscillate close to the limit cycle. As g is increased, however, transitions may occur. The time taken for a transition is then primarily a function of the frequency of the perturbation. The time from the onset of the oscillating perturbation to the time at which the trajectory attains the lower steady state (At) is plotted in Figure 3 as a function of with all other parameters held constant. The arrow marks the minimum value for At which occurs when the frequency of the external perturbation exactly equals that of the unperturbed limit cycle itself. The second minimum occurs at the first harmonic of the limit cycle. Qualitatively similar results are obtained when numerical integration is carried out with differing values for g and 0. 

Using the ability to displace the trajectory of a limit cycle across a fixed boundary (the separatrix) as a measure of sensitivity to an external perturbation, it can therefore be seen that nonlinear oscillating reaction systems are able to respond most sensitively to a range of externally applied frequencies close to their endogenous frequency. 

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