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  "body": "\n## Definition\n\nLet $V \\colon \\R^n \\times \\R^k \\to \\R$ be a [[smooth function|0V33BNKZ9S]], written $V(x, c)$, with state variables $x$ and control parameters $c$.\nThe *catastrophe manifold* is the set $M = \\set{(x, c) : \\nabla_x V(x, c) = 0}$ of [[critical points|0YTCMFB4HC]] of the functions $V(\\cdot, c)$.\nThe *catastrophe map* is the projection $\\chi \\colon M \\to \\R^k$, $(x, c) \\mapsto c$.\nThe *bifurcation set* is the set of $c$ for which $V(\\cdot, c)$ has a degenerate critical point, one where the Hessian in $x$ is singular.\n*Catastrophe theory* studies these sets and the local forms of such families up to smooth changes of coordinates.\n\n## Remarks\n\nExample: the cusp catastrophe $V(x, a, b) = x^4/4 + a x^2/2 + b x$ with $n = 1$ and $k = 2$.\nThe critical points solve $x^3 + a x + b = 0$, and they are degenerate when also $3x^2 + a = 0$.\nEliminating $x$ gives the bifurcation set $4a^3 + 27b^2 = 0$, a curve with a cusp at the origin.\nInside the cusp, where $4a^3 + 27b^2 \u003c 0$, the function has two local minima, and outside it has one.\nNon-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.\nAway from the bifurcation set every $V(\\cdot, c)$ is a [[Morse function|KPQ96640ZJ]] near its critical points, so the theory is about the degenerate points.\nThom's theorem lists the seven generic local forms for $k \\le 4$: fold, cusp, swallowtail, butterfly, and the hyperbolic, elliptic and parabolic umbilics.\n",
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