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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\R^n$ with $n \\ge 2$ and let $f \\colon U \\to \\R^n$ be continuously differentiable.\nLet $\\varphi$ be the [[flow|02903SSA48]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.\nLet $\\Gamma$ be a [[periodic orbit|0AA97XA70K]] of period $T \u003e 0$ and let $x_0 \\in \\Gamma$.\nLet $H$ be a hyperplane through $x_0$ that does not contain the direction $f(x_0)$.\nA transversal section at $x_0$ is the intersection $\\Sigma$ of $H$ with a small open ball around $x_0$.\nThen there are an open neighbourhood $V$ of $x_0$ in $\\Sigma$ and a continuously differentiable function $\\tau \\colon V \\to (0, \\infty)$ with $\\tau(x_0) = T$.\nFor each $x \\in V$, the time $\\tau(x)$ is the first time near $T$ at which the solution through $x$ returns to $\\Sigma$.\nThe Poincaré map of $\\Gamma$ at $\\Sigma$ is\n$$P \\colon V \\to \\Sigma, \\qquad P(x) = \\varphi_{\\tau(x)}(x).$$\n\n## Remarks\n\nThe existence of $\\tau$ follows from the implicit function theorem, because $f(x_0)$ is not parallel to $\\Sigma$.\nThe point $x_0$ is a fixed point of $P$, and fixed points of $P$ correspond to periodic orbits near $\\Gamma$.\nIf the eigenvalues of the derivative $DP(x_0)$ all have absolute value less than $1$, the orbit $\\Gamma$ attracts nearby solutions.\nExample: in polar coordinates let $r' = r(1 - r^2)$ and $\\theta' = 1$, with $\\Gamma$ the unit circle and $\\Sigma$ the positive $x$-axis near $(1, 0)$.\nEvery solution returns to $\\Sigma$ after the time $2\\pi$, and solving the equation for $r$ gives $P(r) = \\left(1 + (r^{-2} - 1) e^{-4\\pi}\\right)^{-1/2}$.\nThen $P(1) = 1$ and $P'(1) = e^{-4\\pi} \u003c 1$, so the circle is attracting.\nNon-example: in the same system, the vertical line $x = 1$ near $(1, 0)$ does not give a transversal section at $(1, 0)$, since $f(1, 0) = (0, 1)$ points along that line.\n\n## Explanations\n\n### L1 human\n\nThe Poincaré map records where a solution crosses a fixed screen after one trip around a closed orbit.\nIt turns the study of the orbit into the study of a fixed point of one map.\n",
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