Catastrophes described by Thom potential functions of one variable

4 Catastrophes described by Thom potential functions of one variable 

To conclude our considerations of functions of one variable, we shall examine the properties of several potential functions given in Table 2.2. In analysing the properties of these functions and the catastrophes described by them, the catastrophe manifold M defined by equation (2.1a) will be helpful. In the case of a function of one state variable it takes the form 

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  

A set containing points in the control parameters space, obtained by projection of points of the set I on the control parameters space, will be referred to as the bifurcation set B. Hence, the set B is given by the following equation  

In addition, the catastrophe mapping /7 — a projection of the point (x c) on the control parameters space 

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