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  "body": "\n## Example\n\nThe *fold catastrophe* is the one-parameter family of functions\n$$V_a(x) = x^3 + a x, \\qquad x, a \\in \\R.$$\nIts [[critical points|0YTCMFB4HC]] solve $V_a'(x) = 3x^2 + a = 0$.\n\n1. For $a \u003c 0$ there are two critical points $x = \\pm \\sqrt{-a/3}$; since $V_a''(x) = 6x$, the point $-\\sqrt{-a/3}$ is a local maximum and $\\sqrt{-a/3}$ is a local minimum.\n2. For $a = 0$ there is one critical point, $x = 0$, and it is degenerate, since $V_0''(0) = 0$.\n3. For $a \u003e 0$ there are no critical points.\n\nSo as $a$ increases through $0$, the minimum and the maximum approach each other, merge at $x = 0$ and disappear.\nThe bifurcation set, the set of parameters where a degenerate critical point occurs, is the single point $a = 0$.\nThe family is the [[universal unfolding|KSAFWZVB8K]] of the germ $x^3$.\nIndeed, the derivative of $V_a$ in $a$ is $x$, and $1$ and $x$ span $\\mathcal{E}_1$ modulo the ideal $(x^2)$.\n\n## Explanations\n\n### L1 human\n\nPicture a ball resting in a dip of a curve that has a dip next to a hump.\nAs the parameter $a$ grows, the dip and the hump become shallower and slide toward each other.\nAt $a = 0$ they merge into a flat inflection, and after that the dip is gone, so the ball suddenly rolls away.\nThis sudden loss of a resting state is the fold catastrophe, and in dynamics it appears as the [[saddle-node bifurcation|MAP5J6HK0T]].\n",
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