This term I am teaching a course on catastrophe theory for 2nd year students.
When you look up this topic on Wikipedia, you will see words like chaos and singularities mentioned, and you will be shown fancy pictures of the seven elementary catastrophes. But when you try to understand what the actual mathematical content of this theory is, you will be left completely clueless by both the English and the German Wikipedia (at least, I was) – you won’t see any reasonably understandable theorem being written down.
So what is catastrophe theory about? To explain this, consider a smooth function \(f\colon \mathbb{R}^N \to \mathbb{R}\) with a critical point at the origin, i.e., the first partial derivatives vanish at 0. For simplicity assume that f(0)=0, otherwise we would have to introduce a suitable linear translation in all the following considerations.
In Analysis 2 one proves that if the Hessian of f at 0 is positive definite (resp., negative definite), then f has a local minimum (resp., maximum) at 0. This fact can be generalized to the Morse lemma: if the Hessian of f is non-degenerate at 0, then up to a local diffeomorphism around 0 the function f looks like \[\pm x_1^2 \pm \cdots \pm x_N^2\,,\] where the number of plus and minus signs is an invariant of the Hessian matrix (i.e., independent of the local diffeomorphism).
Catastrophe theory now enters the game when the Hessian is not non-degenerate anymore. Is there some sort of standard form for functions with degenerate Hessian? As it turns out, there are now many different possibilities what can happen and the question is how to find suitable invariants that can classify all these possibilities.
An important notion is determinacy: we say that a smooth function f is k-determined if any other smooth function g with the same Taylor polynomial up to degree k as f is, up to a local diffeomorphism, exactly the same function as f (locally around our point 0). Applied to f itself this means that f is, up to a local diffeomorphism, a polynomial function of degree k (which is given by the Taylor expansion of f up to degree k). So k-determined functions are a physicist’s dream: we can just pretend (if we are allowed to do smooth coordinate changes) that they equal their own Taylor polynomials.
The Morse lemma tells us that a function f with a non-degenerate critical point is at that point 2-determined. One can prove that this is actually an if and only if statement, i.e., one can additionally prove that the critical point of a 2-determined function must be non-degenerate. The Morse lemma further gives us the standard forms for 2-determined functions, namely the quadratic forms \(\pm x_1^2 \pm \cdots \pm x_N^2\).
To illustrate now a case with degenerate Hessian, let us consider only functions of two variables, i.e., functions \(f\colon \mathbb{R}^2 \to \mathbb{R}\). As usual, assume that f(0) = 0 and that the first derivatives of f vanish, i.e., \(\partial_x f (0) = \partial_y f (0) = 0\). We will also go full force here by considering only those functions whose Hessian completely vanishes at the origin. Then the functions \[(x,y) \mapsto x^3 – xy^2 \text{ and } (x,y) \mapsto x^3 + y^3\] are 3-determined (and inequivalent to each other, i.e., can not be transformed into each other by a local diffeomorphism). Further, these are the only two possibilities. The concrete statement is the following:
Any smooth function \(f\colon \mathbb{R}^2 \to \mathbb{R}\) vanishing at 0, with vanishing first and second partial derivatives at 0 and which is 3-determined is, up to a local diffeomorphism, exactly one of the two polynomials above.