Archive for Rational Mechanics and Analysis
, Volume 76, Issue 2, pp 135165
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 timeperiodically 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.
 Title
 A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam
 Journal

Archive for Rational Mechanics and Analysis
Volume 76, Issue 2 , pp 135165
 Cover Date
 198106
 DOI
 10.1007/BF00251249
 Print ISSN
 00039527
 Online ISSN
 14320673
 Publisher
 SpringerVerlag
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 Authors

 Philip Holmes ^{(1)}
 Jerrold Marsden ^{(2)}
 Author Affiliations

 1. Theoretical and Applied Mechanics, Cornell University, Ithaca, New York
 2. Department of Mathematics, University of California, Berkeley