We give a new (and unified) proof of a local result of existence and multiplicity for periodic solutions of the pendulum equation. We also show that, under a suitable assumption, there can be generically a finite number of geometrically distinct solutions.
Flavio Donati. "Some remarks about periodic solutions to the forced pendulum equation." Differential Integral Equations 8 (1) 141 - 149, 1995. https://doi.org/10.57262/die/1369143788