Subharmonic solutions of the forced pendulum equation: a symplectic approach

Using the Poincaré-Birkhoff fixed point theorem, we prove that, for every $\beta > 0$ and for a large (both in the sense of prevalence and of category) set of continuous and $T$-periodic functions $f: \mathbb{R} \to \mathbb{R}$ with $\int_0^T f(t)\,dt = 0$, the forced pendulum equation $x'' + \beta \sin x = f(t)$ has a subharmonic solution of order $k$ for every large integer number $k$. This improves the well known result obtained with variational methods, where the existence when $k$ is a (large) prime number is ensured.