Poincaré-Lindstedt method

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In perturbation theory, the Poincaré-Lindstedt method is a technique for uniformly approximating periodic solutions to ordinary differential equations when regular perturbation approaches fail.

[edit] Example: the Duffing equation

The undamped, unforced Duffing equation is given by

$\ddot{x} + x + \epsilon x^3 = 0\,$

for $t > 0$ , with $0<\epsilon\ll1$ .^[1] Consider initial conditions

$x(0) = 1, \dot x(0) = 0.\,$

If we try to find an approximate solution of the form $x (t) = x 0 (t) + ε x 1 (t) + ...$ , we obtain

$x(t) = \cos t + \epsilon\left( \frac{1}{32}\left( \cos 3t - \cos t \right) - \frac{3}{8}t \sin t \right).\,$

This approximation grows without bound in time, which is inconsistent with the physical system that the equation models. The term responsible for this unbounded growth, called the secular term, is $t sin t$ . The Poincaré-Lindstedt method allows us to create an approximation that is accurate for all time, as follows.

In addition to expressing the solution itself as an asymptotic series, form another series with which to scale time $t$ :

$\tau = \omega t,\,$ where

$\omega = 1 + \epsilon \omega_1 + ...\,$ .

(Here we take $ω 0 = 1$ because the leading order of the solution's frequency is $1 / 2π$ .) Then the original problem becomes

$\omega^2 x''(\tau) + x(\tau) + \epsilon x^3(\tau) = 0\,$

with the same initial conditions. If we search for a solution of the form $x (τ) = x 0 (τ) + ε x 1 (τ) + ...$ , we obtain $x 0 = cosτ$ and

$x_1 = \frac{1}{32}\cos 3\tau + \left( \omega_1 - \frac{3}{8} \right)\tau\sin\tau.\,$

So a secular term can be removed if we choose $ω 1 = 3 / 8$ . We can continure in this way to higher orders of accuracy; as of now, we have the approximation

$x(t)=\cos\left(1 + \frac{3}{8}\epsilon \right) t + \frac{1}{32}\epsilon\cos 3\left(1 + \frac{3}{8}\epsilon \right)t. \,$