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  "body": "\n## Definition\n\nLet $n, k \\ge 1$ and let $V \\colon \\R^n \\times \\R^k \\to \\R$ be a [[smooth function|0V33BNKZ9S]], written $V(x, c)$ with state $x \\in \\R^n$ and control $c \\in \\R^k$.\nA point $x$ is a *degenerate critical point* of $V(\\cdot, c)$ if:\n\n1. $\\nabla_x V(x, c) = 0$, so $x$ is a [[critical point|0YTCMFB4HC]] of $V(\\cdot, c)$, and\n2. $\\det H_x V(x, c) = 0$, where $H_x V$ is the [[Hessian matrix|0693Y2METT]] of second derivatives in the variables $x$.\n\nThe *bifurcation set* of the family $V$ is the set\n$$B_V = \\set{c \\in \\R^k : V(\\cdot, c) \\text{ has a degenerate critical point}}.$$\n\n## Remarks\n\nExample: the cusp family $V(x, a, b) = x^4/4 + a x^2/2 + b x$ with $n = 1$, $k = 2$.\nClause 1 reads $x^3 + a x + b = 0$ and clause 2 reads $3x^2 + a = 0$.\nThen $a = -3x^2$ and $b = 2x^3$, and eliminating $x$ gives $B_V = \\set{(a, b) : 4a^3 + 27b^2 = 0}$, a curve with a cusp at the origin.\n\nNon-example: for $V(x, c) = x^2 + c x$ the only critical point is $x = -c/2$, and $V'' = 2 \\ne 0$, so clause 2 always fails and $B_V$ is empty.\n\nLet $M = \\set{(x, c) : \\nabla_x V(x, c) = 0}$ and let $\\chi \\colon M \\to \\R^k$ be the projection $(x, c) \\mapsto c$.\nWhen $M$ is a smooth submanifold of dimension $k$, which holds for generic families, $B_V$ is the set of critical values of $\\chi$.\nSome authors also add the Maxwell set, where two critical values of $V(\\cdot, c)$ are equal; this node uses the narrower meaning.\n",
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