III) the point x = 0 is a structurally unstable critical point (degenerate critical point) [Pg.47]

Since det[Ky(0)] = 0, we deal with a degenerate critical point. The eigenvalues of the matrix ViJ are given by the equation [Pg.56]

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 [Pg.29]

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 [Pg.57]

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 [Pg.40]

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. [Pg.36]

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)) [Pg.55]

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 [Pg.35]

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 [Pg.32]

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. [Pg.47]

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