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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 $\\mu$.\nWrite $f_x$, $f_\\mu$, $f_{xx}$ for the partial derivatives of $f$.\nThe family has a *saddle-node bifurcation* at $(x, \\mu) = (0, 0)$ if:\n\n1. $f(0, 0) = 0$, so $0$ is an [[equilibrium point|0G4X36AE0X]] for $\\mu = 0$;\n2. $f_x(0, 0) = 0$;\n3. $f_{xx}(0, 0) \\ne 0$;\n4. $f_\\mu(0, 0) \\ne 0$.\n\n## Remarks\n\nUnder these conditions, the equilibria near $(0, 0)$ form a curve $\\mu = m(x)$ with $m(0) = m'(0) = 0$ and $m''(0) = -f_{xx}(0, 0) / f_\\mu(0, 0) \\ne 0$.\nSo for $\\mu$ on one side of $0$ there are two equilibria near $0$, one stable and one unstable, and on the other side there are none.\n\nExample: $f(x, \\mu) = \\mu - x^2$.\nThen $f(0, 0) = 0$, $f_x = -2x$ vanishes at $0$, $f_{xx} = -2 \\ne 0$ and $f_\\mu = 1 \\ne 0$.\nThe equilibria are $\\pm\\sqrt{\\mu}$ for $\\mu \u003e 0$, and there are none for $\\mu \u003c 0$.\n\nNon-example: $f(x, \\mu) = \\mu x - x^2$.\nHere $f_\\mu(0, 0) = 0$, so clause 4 fails; the two equilibria $0$ and $\\mu$ exist for all $\\mu$ and only cross, which is a transcritical bifurcation.\n\nFor a system in $\\R^n$, clause 2 is replaced by a condition on the Jacobian matrix at the equilibrium.\nIt must have a simple eigenvalue $0$ and no other eigenvalue on the imaginary axis.\nThe conditions 3 and 4 are then imposed on the equation reduced to a centre manifold.\n",
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