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  "body": "\n## Definition\n\nLet $f \\colon \\R \\times \\R \\to \\R$ be [[smooth|0V33BNKZ9S]], and consider the family of scalar equations $x' = f(x, \\mu)$ with parameter $\\mu$.\nAssume $f(0, \\mu) = 0$ for all $\\mu$, so that $0$ is an [[equilibrium point|0G4X36AE0X]] for every $\\mu$.\nThe family has a *transcritical bifurcation* at $(x, \\mu) = (0, 0)$ if:\n\n1. $f_x(0, 0) = 0$;\n2. $f_{x\\mu}(0, 0) \\ne 0$;\n3. $f_{xx}(0, 0) \\ne 0$.\n\n## Remarks\n\nUnder these conditions, besides $x = 0$, the equilibria near $(0, 0)$ form a curve $x = c(\\mu)$ with $c(0) = 0$ and $c'(0) = -2 f_{x\\mu}(0, 0) / f_{xx}(0, 0) \\ne 0$.\nSo this second branch crosses the branch $x = 0$ at $\\mu = 0$.\nSince $f_x(0, \\mu)$ changes sign at $\\mu = 0$, the equilibrium $0$ changes stability there, and the other branch changes stability the opposite way.\n\nExample: $f(x, \\mu) = \\mu x - x^2$.\nThen $f(0, \\mu) = 0$, $f_x(0, 0) = 0$, $f_{x\\mu} = 1$ and $f_{xx} = -2$.\nThe second branch is $x = \\mu$, with $c'(0) = 1 = -2 \\cdot 1 / (-2)$.\n\nNon-example: $f(x, \\mu) = \\mu x - x^3$.\nHere $f_{xx}(0, 0) = 0$, so clause 3 fails; this family has a pitchfork bifurcation instead.\n",
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