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  "body": "\n## Example\n\nLet $\\mu \u003e 0$.\nThe van der Pol oscillator is the equation\n$$x'' - \\mu (1 - x^2)\\, x' + x = 0.$$\nIt is a [[Liénard equation|D2KC8NEAPS]] with $f(x) = -\\mu (1 - x^2)$ and $g(x) = x$, so $F(x) = \\mu (x^3/3 - x)$.\nThe function $F$ has the single positive zero $\\sqrt{3}$, is negative on $(0, \\sqrt{3})$, and is increasing on $(1, \\infty)$.\nBy [[Liénard's theorem|178GG6TXKQ]], the equation has exactly one [[limit cycle|0AAQ73WRFB]], and it is stable.\n\n## Remarks\n\nThe only equilibrium is the origin.\nThe [[linearisation|09ZWYRQ41J]] there is $x'' - \\mu x' + x = 0$, whose characteristic roots $\\frac{1}{2}\\left(\\mu \\pm \\sqrt{\\mu^2 - 4}\\right)$ have positive real part.\nSo the origin is unstable, and every other solution tends to the limit cycle.\nFor $\\mu = 0$ the equation is the [[harmonic oscillator|0YG0ZJPTWG]].\nFor small $\\mu \u003e 0$, the limit cycle is close to the circle of radius $2$ in the $(x, x')$-plane.\nFor large $\\mu$, the oscillation is a [[relaxation oscillation|2NHWCM0DK8]]: slow drifts alternate with fast jumps.\n\n## Explanations\n\n### L1 human\n\nThe van der Pol oscillator is an oscillator with negative friction for small swings and positive friction for large swings.\nSmall motions grow and large motions shrink, so all motions settle on one oscillation of a definite size.\nIt was first studied as a model of electrical circuits with vacuum tubes.\n",
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