Archive for Rational Mechanics and Analysis

, Volume 76, Issue 2, pp 135-165

First online:

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

  • Philip HolmesAffiliated withTheoretical and Applied Mechanics, Cornell University
  • , Jerrold MarsdenAffiliated withDepartment of Mathematics, University of California

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This paper delineates a class of time-periodically perturbed evolution equations in a Banach space whose associated Poincaré 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=f o(X)+εf 1(X,t), where f o(X) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.