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  "body": "\n## Example\n\nThe *cusp catastrophe* is the two-parameter family of functions\n$$V_{a,b}(x) = x^4 + a x^2 + b x, \\qquad x, a, b \\in \\R.$$\nIts [[critical points|0YTCMFB4HC]] solve $4x^3 + 2a x + b = 0$, and such a point is degenerate when also $12 x^2 + 2a = 0$.\n\nBifurcation set: a degenerate critical point $x$ has $a = -6x^2$ and then $b = -4x^3 - 2ax = 8x^3$.\nEliminating $x$ gives the curve\n$$8 a^3 + 27 b^2 = 0,$$\na semicubical parabola with a cusp at the origin, lying in the half-plane $a \\le 0$.\nCheck: with $a = -6x^2$ and $b = 8x^3$, one has $8a^3 = -1728 x^6$ and $27 b^2 = 1728 x^6$.\n\nNumber of critical points: the cubic $4x^3 + 2ax + b$ has discriminant $-16(8a^3 + 27b^2)$.\nSo for $8a^3 + 27b^2 \u003c 0$, inside the cusp, there are three critical points, two local minima and one local maximum between them.\nFor $8a^3 + 27b^2 \u003e 0$, outside the cusp, there is one critical point, a minimum.\nThe family is the [[universal unfolding|KSAFWZVB8K]] of the germ $x^4$, whose codimension is $2$.\n\n## Explanations\n\n### L1 human\n\nNow the curve can have two valleys, and the ball sits in one of them.\nInside the cusp-shaped region of the two knobs $a$ and $b$, both valleys exist; outside it, only one does.\nIf you turn the knob $b$ slowly, the ball stays in its valley until that valley disappears at the edge of the cusp.\nThen it jumps to the other valley.\nTurning $b$ back, the ball does not jump back at the same place but only at the opposite edge, which is called hysteresis.\nThe cusp is the simplest catastrophe with this memory effect, and its [[bifurcation diagram|2XDRZV3GSA]] is the cusp curve.\n",
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