Elementary catastrophes

Woodcock, A. E. R. and Poston, T., 1974, A Geometrical Study of the Elementary Catastrophes, Lecture Notes in Mathematics 373. Springer, New York. 

The number of elementary catastrophe types depends upon q. It has been shown that only for values of g < 5 is the number of catastrophe types finite. Thom has classified these types by their co-rank k and co-dimension q for values of g < 4. The concept of an unfolding and the accompanying definitions are illustrated first in terms of the simplest of all catastrophe types, the so-called fold catastrophe for which both the co-rank and co-dimension equal one. 

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. 

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). 

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... 

The fundamental tasks of elementary catastrophe theory may now be formulated ... 

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. 

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