Thom theorem

The problem of universal unfolding for the functions listed in Table 2.1 may be solved similarly to the problem of finding a form of the function g(x a, b) in (2.8). Universal unfoldings obtained in such a way are compiled in Table 2.2 (this is the first part of the Thom theorem, which pertains to potential functions in one state variable). 

The above considerations for a function of one variable are summarised in the Thom theorem (and more specifically, in its part dealing with functions of one variable) which classifies all structurally stable families of functions. The Thom theorem resolves the problems of determinacy, unfolding and classification. Table 2.2 lists structurally stable families of functions of one variable, containing (for zero values of parameters) the simplest functions with a critical point of a given type (see Table 2.1). Additional information on Thom functions of one variable is provided in Table 2.3. 

As follows from the Thom theorem, each structurally stable function of one state variable, dependent on at most five control parameters, must be equivalent to one of the functions listed in Table 2.2. Recall that functions are considered to be equivalent if they have identical sets of critical points. Another, equivalent definition, which will help to understand better the meaning of local equivalence of a function near to a critical point, is given below. Before that, however, let us examine two examples. 

Subsequently, examples of functions of two variables having degenerate critical points will be examined and the difficulties related to the problems of determinacy and unfolding discussed. We shall give a list of structurally stable families of functions of two variables, having degenerate critical points for some values of parameters on which they depend (this is the second part of the Thom theorem). Finally, we shall examine properties of potential... 

The chapter ends with an Appendix wherein certain very important concepts associated with properties of potential functions describing catastrophes will be given and, to some extent, the Thom theorem substantiated. 

From the Thom theorem (with Arnol d modification) follows the very important conclusion that each function of one or two variables having a critical point, dependent on not more than five parameters and structurally stable must be equivalent, in the sense of the definition given above (see equation (2.12)), to one of the functions given in the list of functions of elementary catastrophes (Tables 2.2, 2.5). 

In the formulation of the Thom theorem occur such important concepts of catastrophe theory as equivalence, determinacy, universal unfolding, codimension. Due to the vital role of these notions in catastrophe theory, we shall try to describe them in more detail. Let us add that the material presented in this Appendix is derived, to a large extent, from the papers of Mather. 

Recall that a function is finitely determined, more specifically k-deter-mined, when it may be locally replaced (that is approximated near to a given point) by the finite, k-term Taylor expansion. All functions of elementary catastrophes given by Thom and Arnol d, see Tables 2.2, 2.5, are k-deter-mined. This is a very important property of a function, allowing us to locally examine its characteristics (and, moreover, this is the necessary condition for a structural stability of the respective family of functions). As demonstrated in Example 2.10, the function x2y is not -determined. Hence, local investigation, in the vicinity of the point (0, 0), of a function whose first non-zero term of the Taylor expansion is x2y without the knowledge of next terms of the Taylor expansion, is impossible. On the other hand, it follows from the Thom theorem that, for example, the function V(x) = x2y + -I- ay4 +. .. is k-determined and subsequent terms of the Taylor series may be neglected. 

The above example shows why a codimension is equal to the number of parameters in a universal unfolding. It follows from the Thom theorem (see Table 2.5) that the universal unfolding of the function F(x, y) = x3 + y3 is, indeed, expressed by the above equation. 

We have demonstrated that Thom s theory of elementary catastrophes finds a direct application in the analysis of structural instabilities which correspond to the making and/or opening of a ring structure. The usefulness of Thom s classification theorem is a consequence of the fact that all the changes in Vp that are involved in such a process occur on a. two-dimensional submanifold of the behaviour space of the electronic coordinates. Clearly, more complex cases of structural changes are to be expected, cases whose complete description will necessitate the use of the full three-dimensional behaviour space. Such a case is illustrated by the formation of a cage structure. 

Difficulties are encountered as more general deformations are considered, deformations of the [l.l.l]propellane molecule which require the use of the full three-dimensional behaviour space. At the present time, Thom s classification theorem does not cover situations which involve more than two... 

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. 

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