Subharmonic solutions of the forced pendulum equation: a symplectic approach

Using the Poincaré–Birkhoff fixed point theorem, we prove that for every β > 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.