Catastrophe theory

Remarks

Example: the cusp catastrophe $V(x, a, b) = x^4/4 + a x^2/2 + b x$ with $n = 1$ and $k = 2$. The critical points solve $x^3 + a x + b = 0$, and they are degenerate when also $3x^2 + a = 0$. Eliminating $x$ gives the bifurcation set $4a^3 + 27b^2 = 0$, a curve with a cusp at the origin. Inside the cusp, where $4a^3 + 27b^2 < 0$, the function has two local minima, and outside it has one. Non-example: $V(x, c) = x^2 + c x$ has the single critical point $x = -c/2$, which is nondegenerate for every $c$, so its bifurcation set is empty. Away from the bifurcation set every $V(\cdot, c)$ is a Morse function near its critical points, so the theory is about the degenerate points. Thom's theorem lists the seven generic local forms for $k \le 4$: fold, cusp, swallowtail, butterfly, and the hyperbolic, elliptic and parabolic umbilics.