= Resonant oscillator return translation
{title2=$P(x,y)=(x,y+\gamma T/2)$}
For $x''+\omega^2x=\gamma\cos\theta$, $\dot\theta=\omega\ne0$, use the section $\theta=0$ modulo $2\pi$ and $T=2\pi/|\omega|$. Exact <resonance> produces the secular term $\gamma t\sin(\omega t)/(2\omega)$, giving the displayed return translation. Nonzero forcing makes every section orbit unbounded. At zero forcing the map is the identity. The assumption $\omega\ne0$ is essential: at zero frequency no transverse phase section exists and the motion is constant-acceleration or free-particle motion.
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