Structural stability of critical points

The solution of the problems of determinacy and classification does not signify a complete characteristic of critical points. For example, from the intuitive standpoint cases (2.7c), corresponding to inflection points, k = = 3, 5, 7. are not equivalent. The simplest functions representing cases IIIA, IIIB, IIIC (see Table 2.1 above) will also turn out not to be structurally stable. 

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 position of a critical point of the function V1 (x), determined from the equation 

the perturbed function Vv has at x = 0 a nondegenerate critical point — a minimum. Consequently, the function F(x) is not structurally stable, since an addition of a small perturbation ex2 modifies, for an arbitrarily small positive e, its local properties near to the critical point (Fig. 12). 

The above examples throw light on the problem of unfolding (problem 2) and the related problem of structural stability of a function of one variable. Although the function V x) = x3 is structurally unstable, one may hope that its modification having the form 

---