Reaction bifurcation/branching

Quapp, W. How does a reaction path branching take place A classification of bifurcation events. /. Mol. Struct. 2004, 695, 95-101. 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

Figure 5.21 Bifurcation far from equilibrium, (a) Primary bifurcation is the distance from equilibrium, at which the thermodynamic branching of minimal entropy production becomes unstable. The bifurcation point or critical point corresponds to the concentration (b) Complete diagram of bifurcations. As the non-linear reaction moves away from equilibrium, the number of possible states increases enormously. (Adapted, with permission, from Coveney and Highfield, 1990).

Fig. 12.4. Stationary-state solutions and limit cycles for surface reaction model in presence of catalyst poison K3 = 9, k2 = 1, k3 = 0.018. There is a Hopf bifurcation on the lowest branch p = 0.0237. The resulting stable limit cycle grows as the dimensionless partial pressure increases and forms a homoclinic orbit when p = 0.0247 (see inset). The saddle-node bifurcation point is at...

FIGURE 21 Possible bifurcation diagrams surrounding branch set plane of B, the head of reaction, and I + /3, the cooling capacity of feed and coolant. 

Primary bifurcation, the first transition from the reference state on the thermodynamic branch, was defined and discussed in the paper by I. Prigogine. This phenomenon is nowadays well understood. Let us outline briefly its theoretical formulation for the reaction-diffusion equations1-2... 

Dendrimers are hyperbranched polymers that emanate from a single core and ramify outward with each subsequent branching unit [1,2]. In the commonly employed divergent synthesis, dendrimers can be prepared through sequential, alternating reactions of two smaller units, one of which has a point of bifurcation. As is described elsewhere in this volume, several classes of dendrimers are known, including polypropyleneimine (PPI), polyamidoamine (PAMAM), and Frechet-type polyether dendrimers [1,2]. 

Because the boiling temperature of 1,4-BD is much higher than of the two reaction products and the reaction is irreversible, the bifurcation behavior is only affected by the mass transfer coefficient ratio Kwater/KTHF, if kbd is not extremely high or low. There exists a critical value of Kwater/KTHF = 2.1, above which the stable node branch approaches the THF-vertex. 

The pathway bifurcates at chorismate. Let us first follow the prephenate branch (Figure 24,17). A mutase converts chorismate into prephenate, the immediate precursor of the aromatic ring of phenylalanine and tyrosine. This fascinating conversion is a rare example of an electrocyclic reaction in biochemistry, mechanistically similar to the well-known Diels-Alder reaction from organic chemistry. Dehydration and decarboxylation yield phenylpyruvate. Alternatively, prephenate can be oxidatively decarboxylated to p-hydroxyphenylpyruvate. These a-ketoacids are then transaminated to form phenylalanine and tyrosine. 

Fig. 5.43. Schematic bifurcation diagram for CH3CHO + O2 reaction showing separate branches corresponding to dark reaction and steady glow, with upper branch losing stability at a Hopf bifurcation as Ta is reduced to give limit cycle (cool-flame) oscillations. The simple limit cycle also loses stability as is reduced further and a complex oscillation corresponding to the multi-stage ignition will emerge but cannot be adequately represented...

In an experiment, not all of the features might be clearly visible. In particular, the increase in period close to the saddle-loop bifurcation often occurs in such a brief interval that it is easily missed. Furthermore, the stationary state at high overpotentials commonly lies at such positive values that undesired side reactions, such as oxidation of the electrode, take place. Thus, often an experimentalist will avoid crossing the saddle-loop bifurcation and only investigate the first branch of the diagram. 

The main contribution of non-linear chemistry is the pitchfork bifurcation diagram. A stable state becomes uixstable and bifurcates into two new stable branches. We are unable to foresee which one of these states will be chosen by the nature of the physico-chemical reaction. The multiplicity of choices gives its full importance to Ae evolution of the systems. This paper has aimed to show the extent to which the concepts of non-equilibrium and of deterministic chaos sublimate the fundamental physical laws by leading us to the creation of new structures and to auto-organization. Chemistry is no exception to this rule. The final conclusion is given by Jean-Marie Lehn [21], Nobel Prize Winner in... 

As already mentioned, bifurcation of the reaction coordinate takes place in the locally symmetric Jahn-Teller intermediate. Let us assume that one branch of the reaction coordinate leads to the fixation of the 1,4 (or 4,l)-structure in the penultimate chain unit (Scheme 12a) and the other branch causes the formation of the vinyl structure of this unit (Scheme 12b). Then it should be expected that the relative content of these structures in the polymer will not depend on the conditions of the process, in particular on the polymerization temperature (because a plateau exists in Fig. 2). 

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