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Published June 1981 | Updated
Journal Article Open

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.

Additional Information

© 1981 Springer. 1981. This version: July 20, 1994. Communicated by D. D. Joseph. Research partially supported by NSF Contract DMS 89-19074 and CTS 89-06343. We thank Mary Silber and Vivien Kirk for helpful discussions on the Hamiltonian structure of normal forms. Research partially supported by DOE Contract DE-FGO3-92ER25129.

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August 19, 2023
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