6 - Unfoldings

Summary

‘It sounds high-flown and absurd, but consider the facts!’

(The Naval Treaty)

Let F be a family of functions containing the function f. For example, F might be the family of potential functions of chapter 1 (see 1.1), which depended on two parameters a and b, or it might be the family of distance-squared functions on a curve in ℝⁿ, where there are n parameters given by the coordinates of a point u ∈ ℝⁿ (see 2.13). Changing the parameters a bit will ‘perturb’ f into a nearby function. In fig. 6.1 there are the graphs of some perturbations of f(t) = t⁵ (the axes are not drawn, but the t-axis is in each case horizontal).

The label k at the point of the graph where t = t₀ indicates that the function has a turning point which is an A_k singularity at t = t₀, or equivalently that the tangent line there is horizontal and has (k + 1)-point contact with the curve. Notice that sometimes two turning points can be on the same level, i.e. the function has the same value there, and sometimes, as with two A₂s, this is not possible.

A family of functions containing f is also called an unfolding of f : the family unfolds to reveal all these functions which are f's close relations. Certain unfoldings contain all functions close to f (in a precise sense).