Functions having degenerate critical points

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. 

The above functions have, at the point (x, y) = (0,0), a degenerate critical point, and for small e the functions V2, V3 may be considered as a perturbation of the function Vv However, the local shape of the functions V3, V2, V3 near to the point (0, 0) is, for an arbitrarily small e, quite different (Fig. 25). 

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 

The above function has a degenerate critical point at (0, 0). Let us consider the following perturbation of this function 

A conclusion may be drawn from the above example that when solving the problem of determinacy and the problem of unfolding for a function of two variables one should not be influenced by the results obtained for a function in one variable. 

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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. 

Let us recall that the Thom functions are structurally stable due to accounting for appropriate lower-order terms (the problem of unfolding) as a result, the insensitivity to perturbations containing terms of low order is achieved, and, due to a proper selection of the form of functions having degenerate critical points (the problem of determinacy, see Tables 2.1, 2.4), the insensitivity to perturbations containing higher-order terms is attained. 

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 simplest functions compiled in Table 2.4, having degenerate critical points of a given type (D4, ..., E6), are structurally unstable. As with to the case of functions of one variable, structurally stable functions having... 

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 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. 

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. 

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))... 

As mentioned above, we shall not describe at this point the method of solving the problem of determinacy for functions of two variables (see Appendix, A2). We shall confine ourselves to providing a list of the simplest potential function, having at x = (0, 0) a regular point, a degenerate critical point, for which the problem of determinacy has been solved. In other words, addition of a perturbation to the functions listed in Table 2.4 must not convert a degenerate critical point into another degenerate critical point. Table 2.4 (functions of two variables) is a counterpart to Table 2.1 (functions of one variable). 

The set Z is such a subset of the set M that the function V has degenerate critical points at points belonging to Z. The sufficient condition for a critical point to be degenerate is vanishing of the Hessian matrix determinant, i.e. vanishing of at least one eigenvalue of the Hessian matrix (the catastrophe functions in two variables have both the eigenvalues equal to zero). 

Similarly, mNk contains such functions that for VemNk the differentials dF(0) = d2F(0) =. .. = d -1 F(0) = 0. Hence, the functions belonging to ntjy3 have at x = 0 a degenerate critical point. 

We shall now proceed to examples of the computation of the codimension of functions of two variables having a degenerate critical point. 

Calculating the Jacobian ideal of the following functions in one and two variables having a degenerate critical point x3, + x4, x5, x6, x7, x2y — y3, x3 + y3, (x2y + y4), x2y + y5, (x3 + y4) according to the rules presented in Section A2.2 we conclude that the above functions are /c-deter-mined, because their codimension is finite. Furthermore, on the basis of the form of Jacobian ideal we establish that the respective universal unfoldings of these functions have a form consistent with the functions compiled in Tables 2.2, 2.5 (in Section A2.2 we computed the Jacobian ideal, among other functions, for the functions x, x2, x3, x4,..., and for x3 + y3, x2y + y4). 

Finally, upon shifting the critical point (pj, xj TJ) = (1,1,1) to the origin (all functions of elementary catastrophes have a degenerate critical point at the origin) ... 

As follows from our previous considerations, the integral (3.67) cannot be computed by the stationary phase method in the vicinity of the caustic due to appearance of the divergence associated with having by the potential function F(x u), i.e. the phase, a degenerate critical point on the caustic (at the return point x0). However, the Airy function can be computed by another method, see Section 3.4.5. The form of the Airy function is shown in Fig. 47. Let us recall that P2( k) is interpreted as the intensity of scattered light or the probability of finding the particle at the point u. 

Let us recall that in elementary catastrophe theory critical points of potential functions are examined. A potential function can have noncritical points, nondegenerate critical points and degenerate critical points. To degenerate critical points correspond sensitive states lying in the state variable and control parameter space in the catastrophe manifold M their... 

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