Degenerate critical point

Case I signifies that a function V does not have a critical point at x = 0. Case II corresponds to a so-called nondegenerate critical point, in case III the point x = 0 will be called a degenerate critical point. Further investigations of local properties of a function F(x), aiming at the solution of problems 1-3 (determinacy, unfolding and classification), will account for the usefulness of distinguishing the basic cases I-III. 

It will become evident later that catastrophes are associated with degenerate critical points of functions only in this case may a change of differential type in a function (change in the set of its critical points — a catastrophe) take place on varying control parameters. We shall see that functions having points of type I or II are structurally stable, while... 

It will become evident that a solution of the problem of unfolding leads to a solution of the problem of structural stability it allows, on the basis of knowledge of the effect of any perturbation on a given degenerate critical point IIIA-IIIC (see Table 2.1), to find a family of functions insensitive to... 

The above considerations reveal the significance of the problem of unfolding the knowledge of all essential perturbations of a function having a degenerate critical point of a given type is required. 

It is now evident that the solution to the problem of unfolding (problem 2) would bring us nearer to the solution of the problem of structural stability. For each type of a degenerate critical point (7c) one should find... 

Structural stability of the above families of functions, containing structurally unstable functions with degenerate critical points at zero values of parameters, should now be examined. It will appear that embedding of structurally unstable functions (cases IIIA-IIIC, Table 2.1) in parametrized families of functions increased their structural stability the functions given in Table 2.2 are structurally stable. 

It is useful to distinguish in the catastrophe manifold M a subset I on which the function V has degenerate critical points since, as we shall see later, at these points a catastrophe takes place. The set S is thus given by the following equation ... 

It follows from the examples examined above that in gradient systems catastrophes may occur only in a case when the system is described by a potential function having a degenerate critical point, for in this case the set I delimitating in M the functions of a various differential type is not an empty set. On exceeding by the system the set Z on the catastrophe surface M, the change in a local type of a potential function V(x c), i.e. a catastrophe, takes place at a continuous change of control parameters. 

III) the point x = 0 is a structurally unstable critical point (degenerate critical point) ... 

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. 

Such a function may be included into a structurally stable parameter--dependent family of functions which will be considered to be a potential function. The state of a physical system will be determined from the condition of the minimum of a potential function having a degenerate critical point, defining the catastrophe surface M. 

The critical point x = 0 of a function of many variables V(xu..., x ) may turn out to be a degenerate critical point solely due to dependence on some variables only. In such cases, separation of the function into two terms is possible... 

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... 

On the other hand, the function V may be split into two parts, and NM> see equation (2.26), only the function NM having a degenerate critical point. Transformation (2.35) applied to the function (2.37) reduces it to a simpler form (but not so simple as (2.36))... 

Since det[Ky(0)] = 0, we deal with a degenerate critical point. The eigenvalues of the matrix ViJ are given by the equation... 

If we add terms of higher order to a function of two variables having a degenerate critical point then, in contrast with the case of a function in one variable, the local character of the perturbed function in the neighbourhood of the degenerate critical point may be drastically changed. 

It follows from the above example that in the case of a function having a degenerate critical point, the terms of higher order cannot be arbitrarily... 

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