Resonant oscillator return translation ($P(x,y)=(x,y+\gamma T/2)$)

For $x^{\prime \prime }+{\omega }^{2}x=\gamma \cos \theta$, $\dot{\theta }=\omega \ne 0$, 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 $\omega \ne 0$ is essential: at zero frequency no transverse phase section exists and the motion is constant-acceleration or free-particle motion.