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  "body": "\n## Summary\n\nThe Poincaré–Lindstedt method finds approximate [[periodic solutions|FHNDX22WMK]] of weakly nonlinear oscillators.\nIt expands both the solution and its frequency in powers of the small parameter.\nThe frequency corrections are chosen to remove secular terms, which grow without bound in time and spoil a naive expansion.\nThe method shows how nonlinearity shifts the frequency of an oscillation with its amplitude.\n\n## Explanations\n\n### L1 human\n\nA pendulum with a large swing has a slightly different period from one with a tiny swing.\nIf one expands the motion in a small parameter but keeps the old period, the error grows with time.\nThe method lets the period change too, and adjusts it so that the error stays bounded.\nFor $x'' + x + \\varepsilon x^3 = 0$ with amplitude $a$, it gives the frequency $1 + \\frac{3}{8} \\varepsilon a^2$ up to $O(\\varepsilon^2)$.\n",
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