14.2 The Poincaré–Lindstedt method
The cure is to expand the frequency as well. Put \(\tau = \omega t\) with \(\omega = 1 + \varepsilon \omega _1 + \cdots \) and \(x = x_0(\tau ) + \varepsilon x_1(\tau ) + \cdots \), \(2\pi \)-periodic in \(\tau \), and choose each \(\omega _k\) to remove the resonant forcing at the next order.
Theorem 14.2.1. The periodic solution of \(x'' + x + \varepsilon x^3 = 0\) with amplitude \(a\) has frequency \[ \omega = 1 + \frac 38\varepsilon a^2 + O(\varepsilon ^2) . \]
Proof. With \(\tau = \omega t\), \(\omega ^2\ddot x + x + \varepsilon x^3 = 0\), where the dot is \(d/d\tau \). At order \(\varepsilon ^0\): \(\ddot x_0 + x_0 = 0\), \(x_0 = a\cos \tau \). At order \(\varepsilon \): \(\ddot x_1 + x_1 = -2\omega _1\ddot x_0 - x_0^3 = 2\omega _1a\cos \tau - \frac 34a^3\cos \tau - \frac 14a^3\cos 3\tau \). For \(x_1\) to be periodic, the coefficient of the resonant \(\cos \tau \) must vanish: \(2\omega _1a = \frac 34a^3\), so \(\omega _1 = \frac 38a^2\). Then \(x_1 = \frac {a^3}{32}(\cos 3\tau - \cos \tau )\) is periodic, and the expansion is uniformly valid. □
The frequency of a hardening spring (\(\varepsilon > 0\)) increases with amplitude, unlike that of a linear oscillator. Software 14.7.1 checks the formula.
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