Catastrophe static elementary

In this chapter we shall show how the observed phenomena may be explained by means of elementary catastrophe theory. In principle, the discussion will be confined to examination of non-chemical systems. However, some of the discussed problems, such as a stability of soap films, a phase transition in the liquid-vapour system, diffraction phenomena or even non-linear recurrent equations, are closely related to chemical problems. This topic will be dealt with in some detail in the last section. The discussion of catastrophes (static and dynamic) occurring in chemical systems is postponed to Chapters 5, 6 these will be preceded by Chapter 4, where the elements of chemical kinetics necessary for our purposes will be discussed. 

In Section 1.3 we described the systems in which qualitative and discontinuous changes of state, that is catastrophes, could be observed at a continuous variation in control parameters. The catastrophes occurring in some systems were discussed in terms of elementary catastrophe theory in Sections 3.2-3.6. The discussion was confined to non-chemical systems such a classification (as we shall see later) being rather artificial. Catastrophes (static and dynamic) occurring in chemical systems will be described in Chapters 5, 6. 

The problem of dependence of the type of stationary points and their stability on control parameters c is thus reduced for systems (1.8) to the investigation of a dependence of the type of critical points of a potential function V and their stability on these parameters. The above mentioned problems are, as already mentioned, the subject of elementary catastrophe theory. Owing to the condition (1.9), catastrophes of this type will be referred to as static. A catastrophe will be defined as a change in a set of critical points of a function V occurring on a continuous change of parameters c. As will be shown later, the condition for occurrence of a catastrophe is expressed in terms of second derivatives of a function V, 82V/8il/idil/j. 

When the condition (1.9) is not met in (1.6), we deal with dynamical catastrophes. In some cases, for example for the so-called Hopf bifurcation, dynamical catastrophes may be examined by static methods of elementary catastrophe theory or singularity theory (Chapter 5). General dynamical catastrophes, taking place in autonomous systems, are dealt with by generalized catastrophe theory and bifurcation theory (having numerous common points). Some information on general dynamical catastrophes will be provided in Chapter 5. 

As explained in Section 1.2, the simplest field of applications of catastrophe theory are gradient systems (1.8). In the case of gradient systems, static catastrophes obeying the condition (1.9) can be studied by the methods of elementary catastrophe theory. Let us recall that a fundamental task of elementary catastrophe theory is the determination how properties of a set of critical points of potential function K(x c) depend on control parameters c. In other words, the problem involves an examination in what way properties of a set of critical points (denoted as M and called the... 

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