A Partial Differential Equation with Infinitely Many Periodic Orbits:
Chaotic Oscillations of a Forced Beam

Holmes, P. J. and J. E. Marsden

Arch. Rational Mech. Anal., 76, 135-166

Abstract:

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 $\dot{x}$ = f₀(x) + $\varepsilon$f₁(x, t), where $\dot{x}$ = f₀(x) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.