Bifurcating surfaces

S.3 Diels-Alder Reactions Steps toward Predicting Dynamic Effects on Bifurcating Surfaces Reactions 8.10 and 8.11 appear to be ordinary Diels-Alder cycloaddition reactions. One might expect to observe both products and perhaps some selectivity. In fact, the only product observed... 

Another hint toward understanding reaction dynamics on a bifurcating surface was supplied by Singleton in his study of the Diels-Alder cycloaddition of acrolein with methyl vinyl ketone (Reaction 8.12). Recognizing the interconversion of 87 and 88 through a Cope rearrangement along with careful kinetic analysis led to an estimate of the ratio of the rate of formation of 87 88 as 2.5 1. 

A broad array of organic reactions displays bifnrcating PESs. An extensive review of examples was authored by Houk. Tantillo has shown examples of bifurcations in terpene synthesis, and a few examples have been found in substitution chemistry. The importance of dynamics cannot be overstated for reactions on bifurcating surfaces and clearly additional research needs to be done, especially to aid in creating more robust predictive models. 

Rehbein, J., Carpenter, B. K. (2011). Do we fully understand what controls chemical selectivity Physical Chemistry Chemical Physics, 13(41), 20906-20922. Illustrative examples Thomas, J. B., Waas, J. R., Harmata, M., Singleton, D. A. (2008). Control Elements in Dynamically Determined Selectivity on a Bifurcating Surface. Journal of the American Chemical Society, 130(44), 14544—14555. Hong., Y. J., TantiUo, D. J. (2014) Biosynthetic consequences of multiple sequential post-transition-state bifurcations. iVafun Chemistry, 6, 104—111. 

To conclude this section, let us elaborate further on the restrictions (D) and (E). In case (D) the surface corresponding to the double cycle is of codimension-one, and therefore, it divides a neighborhood of the non-rough system Xq into two regions and D. Assume that in the double limit cycle is decomposed into two limit cycles, and that it disappears in D. The situation in -D is simple — all systems there are structurally stable and, moreover, of the same type. As for D the situation is less trivial if (D) is violated, then it is obvious that besides structurally stable systems in there are structurally unstable ones whose non-roughness is due to the existence of a heteroclinic trajectory between two saddles, as shown in Fig. 8.1.6(a). Moreover, this picture takes place in any neighborhood of Xq- In other words, in the region, there exists a countable number of the associated bifurcation surfaces of codimension-one which accumulate to In such cases the surface is said to be unattainable from one side. 

An analogous situation occurs when the system has a separatrix loop to a non-resonant saddle (i.e. its saddle value cr = Ai + A2 0) which is the a -limit of a separatrix of another saddle Oi (see condition (E) and Fig. 8.1.5). In this case, the bifurcation surface is also unattainable from one side, where close nonrough systems may have a heteroclinic connection, as shown in Fig. 8.1.6(b). 

The cases where a bifurcation surface of codimension-one is imattainable from either or both sides are typical for multi-dimensional dynamical systems. 

Fig. 11.1.1. Choice of governing parameter in a family transverse to the bifurcation surface...

Fig. 11.1.2. The bifurcation surface XfV of codimension two is a curve in a three-parameter family.

Such a situation will henceforth be referred to as a a bifurcation of codimension A , and the surface 9Jl is called a bifurcation surface of codimension k (the codimension is equal to the number of the governing parameters). 

The procedure for studying a fc-parameter family is similar to that for the one-parameter case firstly, divide the space of the parameters p into the regions of topologically equivalent behavior of trajectories, and study the system in each of these regions. Secondly, describe the boundaries of these regions (the bifurcation set), and finally study what happens at the bifurcation parameter values. We will see below that in the simplest cases (e.g. an equilibrium state with one zero or a pair of imaginary characteristic exponents, or a periodic orbit with one multiplier equal to 1 or to —1) one can almost always, except for extreme degeneracies, choose a correct bifurcation surface of a suitable codimension and analyze completely the transverse families. Moreover, all of these families turn out to be versal. 

Moreover, in more complex cases the problem of presenting a complete description, or of proving that a family imder consideration is versal, is not even set up. However, the general approach remains the same a bifurcating system is considered as a point on some smooth bifurcation surface of a finite codimension. Then, a transverse family is constructed and the qualitative... 

The bifurcation surface separating these regions is called o swallowtail (see the corresponding picture in Fig. 11.2.14 for I4 > 0). It has a self-intersection... 

Fig. 11.2.14. The bifurcation surface known as a swallowtail. See comments in the text.

In order to understand the structure of the bifurcation surface for period-two orbits we must construct a surface 371 of multiple roots of Eq. (11.4.21), and then select the part that corresponds to positive multiple roots. The surface 371 is defined by the system... 

When the saddle-node is simple, (12.2.22) is just a condition of transversality of the one-parameter family under consideration to the bifurcation surface of systems with the saddle-node, which allows the Poincare map on to be written in the form (12.2.2). 

If the saddle-node L is simple, then all neighboring systems having a saddle-node periodic orbit close to L constitute a codimension-one bifurcational surface. By construction (Sec. 12.2), the function /o depends continuously on the system on this bifurcational surface. Thus, if the conditions of Theorem 12.9 are satisfied by a certain system with a simple saddle-node, they are also satisfied by all nearby systems on the bifurcational surface. This implies that Theorem 12.9 is valid for any one-parameter family which intersects the surface transversely. In other words, our blue sky catastrophe occurs generically... 

Remark 5. The splitting parameter a(/x) may be an arbitrary continuous function of p satisfying the above sign condition. When the system depends on /i smoothly, the family is transverse to the bifurcation surface if a (0) 0. In this case, one can always assume that a(/i) = /i. 

In other words, we assume that the homoclinic loop behaves in one of the two ways described above as it approaches O as t — 00. In principle, this is not a strong restriction since it defines an open and dense subset on the bifurcation surface if F lies in for some system, then by an arbitrarily small smooth perturbation it may be pushed into... 

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