Catastrophe elementary approach

The way of presentation of the material discussed in Chapter 5 is based on papers of Guckenheimer and on the ideas contained in papers by Stewart. A paper by Nicolis and an article by Othmer in a book published by Field and Burger constitute a very good supplement to these papers. A book by Arnol d (1983), although rather difficult, provides much additional material. The elementary method of analysis of some dynamical catastrophes presented in Section 5.6 is patterned after ArnoPd s approach to the Hopf bifurcation in the van der Pol system described in his book (1975). A book by Gilmore provides basic information on catastrophes in dynamical systems. A paper by Stewart contains another proof (compared to Section 5.5) that Hopf bifurcation is an elementary catastrophy. 

Chemical kinetics equations are commonly nonlinear and may represent diverse phenomena of a catastrophe type. Theoretical studies in this area fall into two groups. Purely model considerations belong to the first group. A certain sequence of elementary reactions — the reaction mechanism, permitted from the chemical standpoint (see the Korzukhin theorem, Chapter 4) is postulated, the corresponding system of kinetic equations is found and its solutions are examined. Such a procedure allows us to predict a possible behaviour of chemical systems. The second approach involves the investigation of a mechanism of a specific chemical reaction, having interesting dynamics. 

In the end, we will describe very briefly the application of catastrophe theory to a description of chemical reactions in terms of individual molecules participating in the reaction. The approach will be based on some methods of quantum chemistry employing solutions to the Schrodinger equation. The description of chemical reactions involving a simultaneous application of the microscopic description, the Schrodinger equation, and of elementary catastrophe theory is notionally similar to the description of diffraction catastrophes for the Schrodinger equation, see Section 3.4.3. 

This quantitative approach is entirely valid for designing the architecture of a system, a level of redundancy and monitoring. Such reasoning is based on the multiplication of elementary probabihties, and thus on an assumption of independence between components. This independence must be verified, mainly by a qualitative judgment. Also, a safeguard is associated with this probability calculation a simple failure (one), regardless of the probabihty of its occurrence, must not lead to catastrophic consequences. 

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