Periodic Motions of Linear Impact Oscillators via the Successor Map

Abstract

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 $x\not =0$.