Poincaré–Lindstedt method

Definition

Let $\omega_0 > 0$, let $f$ be smooth, and consider the ordinary differential equation $$x'' + \omega_0^2 x = \varepsilon f(x, x'),$$ with a small parameter $\varepsilon$, for which one seeks periodic solutions. The Poincaré–Lindstedt method proceeds as follows.

1. Introduce the scaled time $\tau = \omega t$, where the unknown frequency is expanded as $\omega = \omega_0 + \varepsilon \omega_1 + \varepsilon^2 \omega_2 + \cdots$.
2. Seek $x$ as a $2\pi$-periodic function of $\tau$, expanded as $x = x_0(\tau) + \varepsilon x_1(\tau) + \varepsilon^2 x_2(\tau) + \cdots$.
3. 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 $\varepsilon$.
4. At each order $k \ge 1$, the equation has the form $\omega_0^2 (x_k'' + x_k) = g_k(\tau)$. Choose $\omega_k$, and the free constants of earlier orders, so that $g_k$ has no Fourier component in $\cos \tau$ and $\sin \tau$.

Step 4 removes secular terms such as $\tau \sin \tau$, so every $x_k$ is periodic. The result is an asymptotic expansion of the periodic solution and its frequency.