= Poincaré-Lindstedt method
{c}
For a weakly perturbed <harmonic oscillator>, expand both its periodic solution and its <angular frequency>, using the stretched time $\tau=\omega t$. At each order, choose the frequency correction to remove fundamental-harmonic forcing that would produce a <secular term>. For a self-excited <periodic orbit>, cancellation of the independent sine and cosine terms can determine both the leading <amplitude> and the frequency correction. Actual existence follows when the associated <Poincaré return map> has a simple displacement zero, by the <implicit function theorem>.
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