Swallowtail catastrophe

Example

The swallowtail catastrophe is the family of functions of one variable $x \in \R$ $$V(x; a, b, c) = x^5 + a x^3 + b x^2 + c x,$$ with three control parameters $(a, b, c) \in \R^3$. For $a = b = c = 0$ the function is $x^5$, whose critical point at $0$ is degenerate: the first four derivatives vanish there.

The critical points solve $V'(x) = 5x^4 + 3a x^2 + 2b x + c = 0$, and such a point is degenerate when also $V''(x) = 20x^3 + 6a x + 2b = 0$. Solving these two equations for $b$ and $c$ in terms of $x$ and $a$ gives $$b = -10x^3 - 3a x, \qquad c = 15x^4 + 3a x^2.$$ So the bifurcation set is the surface in $\R^3$ swept out by these curves as $a$ varies.

Fix $a$ and follow the curve $x \mapsto (b, c)$. Its velocity is $(-30x^2 - 3a, \; 60x^3 + 6a x) = -3(10x^2 + a)\,(1, -2x)$. For $a > 0$ the velocity never vanishes, and the slice is a smooth curve. For $a < 0$ the velocity vanishes at $x = \pm\sqrt{-a/10}$, so the slice has two cusps, and between them the curve crosses itself. This slice, with its two cusps and its crossing point, is the shape that gives the swallowtail its name.