Periodic Motions of Linear Impact Oscillators via the Successor Map

We investigate the existence and multiplicity of nontrivial periodic bouncing solutions for linear and asymptotically linear impact oscillators by applying a generalized version of the Poincaré–Birkhoff theorem to an adequate Poincaré section called the successor map. The main theorem includes a generalization of a related result by Bonheure and Fabry and provides a sufficient condition for the existence of periodic bouncing solutions for Hill’s equation with obstacle at x = 0.