Umbilic catastrophes

Example

The umbilic catastrophes are the three elementary catastrophes with two state variables $(x, y) \in \R^2$. In their standard forms they are:

1. the hyperbolic umbilic $V = x^3 + y^3 + a xy + b x + c y$, with controls $(a, b, c) \in \R^3$;
2. the elliptic umbilic $V = \tfrac{1}{3}x^3 - x y^2 + a(x^2 + y^2) + b x + c y$, with controls $(a, b, c) \in \R^3$;
3. the parabolic umbilic $V = x^2 y + y^4 + a x^2 + b y^2 + c x + d y$, with controls $(a, b, c, d) \in \R^4$.

When all controls vanish, each $V$ has a critical point at the origin where the whole Hessian matrix is zero. For the hyperbolic umbilic, the Hessian of $x^3 + y^3$ is $\begin{pmatrix} 6x & 0 \\ 0 & 6y \end{pmatrix}$, which is $0$ at the origin. For the elliptic umbilic, the Hessian of $\tfrac13 x^3 - x y^2$ is $\begin{pmatrix} 2x & -2y \\ -2y & -2x \end{pmatrix}$, again $0$ at the origin. For the parabolic umbilic, the Hessian of $x^2 y + y^4$ is $\begin{pmatrix} 2y & 2x \\ 2x & 12y^2 \end{pmatrix}$, again $0$ at the origin. So the degeneracy has corank $2$: no change of coordinates can split off a nondegenerate square in one of the variables. This separates the umbilics from the fold, cusp, swallowtail and butterfly, which have corank $1$ and need only one state variable.

For the elliptic umbilic, the slices $a = \text{const} \ne 0$ of the bifurcation set are curves with three cusps.