Poincaré-Lindstedt method

The Poincare-Lindstedt method fixes a specific embarrassment of naive perturbation theory for periodic motion: a small nonlinearity slightly changes an oscillator's frequency, but if you expand at the original frequency the correction terms grow with time and eventually wreck the approximation, even though the true motion is a perfectly bounded oscillation. The cure is to expand the frequency itself along with the solution, so the periodic motion is described in its own correctly tuned time.

The mechanism: for a weakly nonlinear oscillator like the Duffing equation y'' + y + epsilon y^3 = 0, introduce a stretched time tau = omega t with an unknown frequency omega = 1 + epsilon omega_1 + epsilon^2 omega_2 + ..., and expand the solution y = y_0 + epsilon y_1 + ... in this strained time. At each order, resonant forcing terms (those proportional to cos tau or sin tau, matching the natural frequency) would generate secular terms that grow like tau times a sinusoid. You then choose the frequency corrections omega_1, omega_2, ... precisely so that these resonant terms cancel. That requirement both removes the unbounded growth and delivers the genuine amplitude-dependent frequency of the nonlinear oscillation.

Historically this is how Lindstedt and Poincare salvaged the perturbation series of celestial mechanics, where naive expansions of planetary orbits contained time-growing terms that falsely predicted instability. It captures the hallmark of nonlinear oscillation: the period depends on the amplitude. The honest caveat: the basic method assumes the motion is strictly periodic with a single frequency, so it does not handle damping or amplitude decay (the orbit must close on itself). For damped or slowly modulated systems you need the more general multiple-scale analysis, of which Poincare-Lindstedt is essentially the periodic, single-frequency special case.