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  "body": "\n## Definition\n\nLet $X$, $Y$ and $\\Lambda$ be [[Banach spaces|06GZ956657]], let $U \\subseteq X \\times \\Lambda$ be an open neighbourhood of $(0, 0)$, and let $F \\colon U \\to Y$ be [[continuously differentiable|0VW90BBSJ4]] with $F(0, 0) = 0$.\nLet $L = D_x F(0, 0) \\colon X \\to Y$, and assume that $L$ is a Fredholm operator:\n\n1. the [[kernel|0S23E6YE83]] $N = \\ker L$ has finite dimension;\n2. the range $R = L(X)$ is closed and has a complement of finite dimension in $Y$.\n\nChoose closed subspaces $X_1$ and $Y_1$ with $X = N \\oplus X_1$ and $Y = R \\oplus Y_1$, and let $Q \\colon Y \\to R$ be the continuous [[projection|033DA1WE8Y]] onto $R$ along $Y_1$.\nThe restriction of $QL$ to $X_1$ is an isomorphism onto $R$.\nWrite $x = v + w$ with $v \\in N$ and $w \\in X_1$.\nBy the [[implicit function theorem|0BXBADR94C]], the equation $Q F(v + w, \\lambda) = 0$ has a unique solution $w = W(v, \\lambda)$ near $0$.\nHere $(v, \\lambda)$ is near $(0, 0)$, and $W(0, 0) = 0$.\nThe *Lyapunov–Schmidt reduction* of $F = 0$ at $(0, 0)$ is the *bifurcation equation*\n$$(I - Q)\\, F\\big(v + W(v, \\lambda), \\lambda\\big) = 0,$$\nfor $v \\in N$ and $\\lambda$ near $0$.\nIts values lie in $Y_1$.\n\n## Remarks\n\nNear $(0, 0)$, the solutions of $F(x, \\lambda) = 0$ are exactly the points $x = v + W(v, \\lambda)$ with $(v, \\lambda)$ a solution of the bifurcation equation.\nSince $N$ and $Y_1$ are finite-dimensional, this is a system of $\\dim Y_1$ equations in $\\dim N$ unknowns.\n\nExample: $X = Y = \\R^2$, $\\Lambda = \\R$, $F(x, y, \\lambda) = (\\lambda x - x y,\\ y - x^2)$.\nHere $L = \\begin{pmatrix} 0 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$, so $N$ is the $x$-axis and $R$ is the $y$-axis; take $X_1 = R$ and $Y_1 = N$.\nThe equation $Q F = 0$ is $y - x^2 = 0$, so $W(x, \\lambda) = x^2$.\nThe bifurcation equation is $\\lambda x - x^3 = 0$, which has a pitchfork bifurcation at $\\lambda = 0$.\n\nNon-example: on the sequence space $\\ell^2$, the operator $L(x_1, x_2, \\ldots) = (x_1, x_2/2, x_3/3, \\ldots)$ has trivial kernel, but its range is dense and not closed, so clause 2 fails.\n",
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