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  "body": "\n## Example\n\nThe *swallowtail catastrophe* is the family of functions of one variable $x \\in \\R$\n$$V(x; a, b, c) = x^5 + a x^3 + b x^2 + c x,$$\nwith three control parameters $(a, b, c) \\in \\R^3$.\nFor $a = b = c = 0$ the function is $x^5$, whose [[critical point|0YTCMFB4HC]] at $0$ is degenerate: the first four derivatives vanish there.\n\nThe critical points solve $V'(x) = 5x^4 + 3a x^2 + 2b x + c = 0$, and such a point is degenerate when also $V''(x) = 20x^3 + 6a x + 2b = 0$.\nSolving these two equations for $b$ and $c$ in terms of $x$ and $a$ gives\n$$b = -10x^3 - 3a x, \\qquad c = 15x^4 + 3a x^2.$$\nSo the [[bifurcation set|JV45S0KQE6]] is the surface in $\\R^3$ swept out by these curves as $a$ varies.\n\nFix $a$ and follow the curve $x \\mapsto (b, c)$.\nIts velocity is $(-30x^2 - 3a, \\; 60x^3 + 6a x) = -3(10x^2 + a)\\,(1, -2x)$.\nFor $a \u003e 0$ the velocity never vanishes, and the slice is a smooth curve.\nFor $a \u003c 0$ the velocity vanishes at $x = \\pm\\sqrt{-a/10}$, so the slice has two cusps, and between them the curve crosses itself.\nThis slice, with its two cusps and its crossing point, is the shape that gives the swallowtail its name.\n\n## Explanations\n\n### L1 human\n\nThe swallowtail is the third of Thom's seven elementary catastrophes, after the fold and the cusp.\nIts potential depends on one state variable and three control knobs.\nThe set of knob settings where the potential has a degenerate critical point is a surface in space.\nCut it by a plane with $a \u003c 0$ and you see a curve with two sharp points and a loop, like the tail of a swallow.\nCrossing this surface changes the number of critical points of the potential by two.\n",
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