Archive for Rational Mechanics and Analysis

, Volume 76, Issue 2, pp 135–165

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


  • Philip Holmes
    • Theoretical and Applied MechanicsCornell University
  • Jerrold Marsden
    • Department of MathematicsUniversity of California

DOI: 10.1007/BF00251249

Cite this article as:
Holmes, P. & Marsden, J. Arch. Rational Mech. Anal. (1981) 76: 135. doi:10.1007/BF00251249


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=fo(X)+εf1(X,t), where fo(X) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.

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© Springer-Verlag GmbH & Co 1981