Unstable trajectory component

These studies indicate a direct connection between requirements of exponential divergence and adherence to statistical theories. Note, however, that the particular statistical theory obeyed by the dynamics need not be a simple analytic theory such as RRKM or phase space theory. It may appear, therefore, that the an essential simplicity of statistical theory, that is, the ability to bypass long-lived trajectory calculations in favor of an easily computed result, has been lost. This is indeed not the case, that is, it is easy to see that this approach affords a method for obtaining contributions from both the unstable (statistical) trajectory component as well the direct component with a minimum of computation. This technique, the minimally dynamic 33,34 approach, will now be sketched. 

Figure 9.4b displays the RCM for the mixture acetone (562 °C) / chloroform (61.2 °C) / toluene (110.8 °C). The first two components give a maximum-azeotrope (nbp 64.7 °C), but boiling lower than toluene. There are two unstable nodes (acetone and chloroform), one stable node (toluene), and a saddle (azeotrope acetone-chloroform). The residue curves can emanate either from the acetone or from the chloroform vertices, but all terminate in the toluene vertex. The direction of trajectories shows clearly the separation of the... 

At nonsharp separation, the stationary points of section working regions, except the stable node N+, are located outside the concentration simplex (the direction of trajectory from the product is accepted). At sharp separation, other stationary points - trajectory tear-off points x from the boundary elements of concentration simplex - are added to the stable node. These are the saddle points S and, besides that, if the product point coincides with the vertex corresponding to the lightest or to the heaviest component, then this point becomes an unstable node N. ... 

The stationary points of this bundle are located both in the boundary elements of simplex and inside it, at reversible distillation trajectories. The number of such stationary points of the bundle is equal to the difference between the number of the components of the mixture being separated n and the number of the components of section product k plus one. Stationary points of the bundle of top or bottom section are one unstable node A (it exists inside the simplex only in the product point, if product is a pure component or an azeotrope) one stable node A+ (it is located at the boundary element, containing one component more than the product if A < n — 1) the rest of the stationary points of the bundle are saddle points S. The first (in the course of the trajectory) saddle point (5 ) is located at the product boundary element (if product is pure component or azeotrope, then the saddle point coincides with the unstable node N and with product point). The second saddle point (S ) is located at the boundary element, containing product components and one additional component, closest to product... 

Fig. 29 shows that due to the material mismatch, when the distance between the sub-interface crack and the interface 8t approaches zero, the corresponding component at the crack tip is veiy high and is acting in such a direction that the crack tends to deviate away from the interface toward the centerline of the bond. As the distance ht increases, the corresponding K value drops drastically, which suggests that the sub-interface crack will deviate from the interface in a rather gradual fashion. This prediction is consistent with crack trajectory shown in the SEM micrograph of the DCB specimens with directionally unstable cracks in Fig. 10. Since differences in the material mismatch will result in variations in the stress distribution. Fig. 29 also indicates that the crack propagation behavior will also be different for different materials systems.

One of the conclusions from this theorem is that an authentic recurrent trajectory must be unstable. A few exotic examples of dynamical systems on some compact manifolds, called nil-manifolds, are known where all trajectories are recurrent. Moreover, these trajectories are unstable. However, their instability is not exponential but only polynomial. In contrast to an almost-periodic trajectory whose frequency spectrum is discrete, the spectrum of a recurrent trajectory has in addition a continuous component. For further details see [23]. 

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