Elementary catastrophe theory

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 

A special attention in our considerations will be paid to the functions, further called structurally stable whose form, determined by the properties of a set of critical points, does not qualitatively change under not too large a perturbation. As explained in Sections 1.2, 1.3, the significance of 

The fundamental tasks of elementary catastrophe theory may now be formulated  

Equivalent potential functions will be defined as the functions having identical sets of critical points. Such a definition of equivalence of potential functions implies that, as follows from the state equation (1.8) and equations (1.9), (1.10), potential functions of the same form describe physical systems having the same sets of stationary points. 

The problems of structural stability and equivalence are associated with some technical problems which have to be resolved. These are the following problems  

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Numerous rigorous results have been obtained for gradient systems. One natural method of investigation of gradient systems is elementary catastrophe theory the field of catastrophe theory dealing with an examination of gradient systems. In the case of the gradient system of equations (1.8), properties of a stationary state, that is the state invariant with time, may be readily studied... 

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. 

The program of catastrophe theory has been formulated by Thom. The fundamental theorems of elementary catastrophe theory have been proven by Thom, Mather and Arnol d. A large contribution to this theory has been carried in by Zeeman, who also found many practical applications of the theory. Arnol d and Berry have demonstrated the existence of a close relationship between elementary catastrophe theory and optics and found numerous uses for this relationship. 

I. Stewart, Beyond elementary catastrophe theory , Mathematics and Computation, 14, 25 (1984). 

On a continuous change of control parameters a, b evolution of the system on the surface M3 proceeds continuously from point 1 to point 2. However, on a further increase of the parameter b, the trajectory of the system must leave the surface of the potential energy minimum M3. Hence, at point 2 a catastrophe — a qualitative change in the state of the system, takes place. The condition of the potential energy minimum requires the system to be present on the surface M3. The system thus evolves possibly rapidly, from point 2 to point 3 along the path of shortest time. Elementary catastrophe theory does not describe the way of evolution along the path... 

From the standpoint of elementary catastrophe theory, the functions having degenerate critical points are most interesting. As follows from Section 2.2, in gradient systems catastrophes may happen only in a case when the system is described by a potential function having a degenerate critical point. 

As explained earlier, only the term J nm may lead to the occurrence of a catastrophe in elementary catastrophe theory. 

A good introduction to elementary catastrophe theory is a paper by Zeeman in Scientific American. A more systematic lecture on elementary catastrophe theory can be found in Gilmore s or Poston and Stewart s books, as well as in papers by Stewart and Zeeman. A paper by Poston and Woodcock, containing demonstrative graphs, can also be recommended. 

D. R. J. Chillingworth, Elementary catastrophe theory , Bull. Inst. Math. Applic., 11, 155 (1975). 

Applications of Elementary Catastrophe Theory (Non-Chemical Systems)... 

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. 

A classical theory of phase transitions may be formulated by means of elementary catastrophe theory. We shall describe some notions of the theory of phase transitions in terms of elementary catastrophe theory. Next, we shall describe examples of application of catastrophe theory to the description of the liquid-vapour equilibrium. 

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