Rough saddle

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

However, it is possible to have an unstable equilibrium state O such that some trajectories converge to O as t -> -foo. The simplest example is a rough saddle. To prove instability of a saddle in critical cases one can use Chetaev s function where conditions (9.1.6), (9.1.7) and (9.1.10) hold only within some sector adjoining the point O. For details we refer the reader to the book by Khazin and Shnol [75]. 

Fig. 10.2.6. Geometrically, there is no difference between a critical node hp < 0 (a) and a rough stable node. However, a quantitative comparison can be made with respect to the rate of convergence of nearby trajectories to the origin. A similar observation also applies to a rough saddle fixed point and a critical saddle with /2p+i >0 (b).

For bimolecular reactions, reactive species such as radicals may undergo reactions without a barrier—in such cases, no saddle point can be found on the potential energy surface, and more advanced TST methods are needed to compute rate constants. The value shown in the table approaches the diffusion limit indeed, with more accurate rate calculations, barrierless reactions occur even closer to the diffusion limit. Again, heating is needed to accelerate reactions with higher barriers—the case with AE = 20kcal/mol would have a rough Xy2 of 11 h at 150°C. 

Two points should be emphasized First, the previous relationships for pressure drop and holdup are not valid for saddles and Pall rings, but data for these packings in specific situations are found in the literature (R6, D16, W2). Second, as a rough approximation, the equations for downward flow in a packed bubble reactor are still valid for upward flow. [The upflow data for pellets and spheres from Tlirpin and Huntington (T11) and Sato et al. (S8) will be useful for specific cases.]... 

Figure 11. A rough sketch of a conventional transition state (saddle point) between two local minima on the potential energy surface. The transition state is in back of a tall mountain.

Figure 4. Two-parameter skeleton bifurcation diagram of the NDR oscillator model [Eq. (5)] in the t//p parameter plane. Roughly, the locations of saddle-node (sn) and Hopf (h) bifurcations divide the parameter plane into regions with monostable, bistable, and oscdla-tory behavior, respectively.

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