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  "body": "\n## Definition\n\nLet $\\omega_0 \u003e 0$, let $f$ be smooth, and consider the [[ordinary differential equation|0NT64JV96A]]\n$$x'' + \\omega_0^2 x = \\varepsilon f(x, x'),$$\nwith a small parameter $\\varepsilon$, for which one seeks [[periodic solutions|FHNDX22WMK]].\nThe *Poincaré–Lindstedt method* proceeds as follows.\n\n1. Introduce the scaled time $\\tau = \\omega t$, where the unknown frequency is expanded as $\\omega = \\omega_0 + \\varepsilon \\omega_1 + \\varepsilon^2 \\omega_2 + \\cdots$.\n2. Seek $x$ as a $2\\pi$-periodic function of $\\tau$, expanded as $x = x_0(\\tau) + \\varepsilon x_1(\\tau) + \\varepsilon^2 x_2(\\tau) + \\cdots$.\n3. Insert both expansions in the equation, written as $\\omega^2 \\frac{d^2 x}{d\\tau^2} + \\omega_0^2 x = \\varepsilon f\\bigl(x, \\omega \\frac{dx}{d\\tau}\\bigr)$, and collect powers of $\\varepsilon$.\n4. At each order $k \\ge 1$, the equation has the form $\\omega_0^2 (x_k'' + x_k) = g_k(\\tau)$.\n   Choose $\\omega_k$, and the free constants of earlier orders, so that $g_k$ has no Fourier component in $\\cos \\tau$ and $\\sin \\tau$.\n\nStep 4 removes *secular terms* such as $\\tau \\sin \\tau$, so every $x_k$ is periodic.\nThe result is an [[asymptotic expansion|Q26QN6GYZX]] of the periodic solution and its frequency.\n\n## Remarks\n\nExample: the Duffing equation $x'' + x + \\varepsilon x^3 = 0$, with $\\omega_0 = 1$, $x(0) = a$, $x'(0) = 0$.\nOrder $0$ gives $x_0 = a \\cos \\tau$.\nOrder $1$ gives $x_1'' + x_1 = 2 \\omega_1 a \\cos \\tau - a^3 \\cos^3 \\tau = \\bigl(2 \\omega_1 a - \\tfrac{3}{4} a^3\\bigr) \\cos \\tau - \\tfrac{1}{4} a^3 \\cos 3\\tau$, using $\\cos^3 \\tau = \\frac{3}{4} \\cos \\tau + \\frac{1}{4} \\cos 3 \\tau$.\nRemoving the $\\cos \\tau$ term gives $\\omega_1 = \\frac{3}{8} a^2$, so $\\omega = 1 + \\frac{3}{8} \\varepsilon a^2 + O(\\varepsilon^2)$.\n\nNon-example: the naive expansion $x = x_0(t) + \\varepsilon x_1(t)$ in the original time, step 1 omitted, gives $x_1$ containing $-\\frac{3}{8} a^3 t \\sin t$.\nThis term grows without bound, so the expansion is not uniformly valid for $t$ of order $1/\\varepsilon$.\n\n## Explanations\n\n### L1 human\n\nThe method expands both the motion and its frequency, and it fixes the frequency so that no term grows forever.\n",
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