A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam

Abstract

This paper delineates a class of time-periodically perturved evolution equations in a Banach space whose associated Poincar´e map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form x˙ = f0(x) + "f1(x, t), where x˙ = f0(x) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.