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  "body": "\n## Statement\n\nLet $f, g \\colon \\R \\to \\R$ be [[continuously differentiable|0VW90BBSJ4]], and let $F(x) = \\int_0^x f(s)\\, ds$.\nAssume:\n\n1. $f$ is an [[even function|0FP31SX5EA]] and $g$ is an odd function;\n2. $g(x) \u003e 0$ for all $x \u003e 0$;\n3. $F$ has exactly one positive zero $a$, with $F(x) \u003c 0$ for $0 \u003c x \u003c a$;\n4. $F$ is increasing on $(a, \\infty)$, and $F(x) \\to \\infty$ as $x \\to \\infty$.\n\nThen the system $x' = y - F(x)$, $y' = -g(x)$, equivalent to $x'' + f(x) x' + g(x) = 0$, has exactly one [[limit cycle|0AAQ73WRFB]].\nThis limit cycle surrounds the origin, and it is stable.\n\n## Remarks\n\nThe proof uses the energy $E(x, y) = \\frac{1}{2} y^2 + G(x)$, where $G(x) = \\int_0^x g(s)\\, ds$.\nAlong solutions, $E' = -g(x) F(x)$.\nBy the symmetry $(x, y) \\mapsto (-x, -y)$, it is enough to follow a solution from a point $(0, y_0)$ with $y_0 \u003e 0$ to its next crossing of the $y$-axis.\nLet $\\Delta(y_0)$ be the change of $E$ on this half turn.\nIt is positive when the half turn stays in $0 \\le x \\le a$.\nFor the half turns that go beyond $x = a$, it decreases strictly in $y_0$ and tends to $-\\infty$.\nSo $\\Delta$ vanishes for exactly one $y_0$, and this gives the unique closed orbit.\n\nFor the van der Pol equation $x'' - \\mu (1 - x^2) x' + x = 0$ with $\\mu \u003e 0$, one has $F(x) = \\mu (x^3/3 - x)$ and $g(x) = x$.\nHere $a = \\sqrt{3}$, $F \u003c 0$ on $(0, \\sqrt{3})$, and $F' = \\mu (x^2 - 1) \u003e 0$ for $x \u003e \\sqrt{3}$.\nSo the van der Pol oscillator has exactly one limit cycle, and it is stable.\n\n## Explanations\n\n### L1 human\n\nLiénard's theorem gives easy conditions under which an oscillator has one and only one self-sustained oscillation.\nThe friction must be negative for small motions and positive for large ones, with a single change of sign.\nThen every motion except the rest state spirals onto the same closed loop.\n\n### L2 human\n\nProving that a limit cycle exists is often done with the [[Poincaré–Bendixson theorem|0EW1X61ZT1]], but that theorem says nothing about how many there are.\nLiénard's theorem gives existence and uniqueness at once for a whole family of equations.\nIts proof is a careful estimate of how the energy changes on a half turn around the origin.\nThe van der Pol oscillator is the main application.\n",
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