Andrew Belmonte^1, Michael J. Shelley^2, Shaden T. Eldakar^1, and Chris
H. Wiggins^2

When shaken periodically at one end, a hanging chain or string displays
a startling variety of distinct dynamic

behaviors, depending on its length and the amplitude and frequency
of the shaking. We find experimentally that

instabilities occur in tongue-like regions of parameter space. The
unstable states observed include swinging and rotating

pendular motion, and also more complex, chaotic states. Mathematically,
the dynamics are described by a nonlinear wave

equation. Linear stability analysis predicts instabilities within the
well-known resonance tongues; their boundaries

agree well with our experiment. Full numerical simulations of the 3D
dynamics reproduce and elucidate many aspects of

the experiment, indicating for instance that the kinetic energy of
the entire chain is periodically zero in the pendular

state, and that sharp gradients in tension occur in the complex states.
Experimentally the chain is also observed to tie

knots in itself, some quite complex, which modify its subsequent dynamics;
however the trefoil knot does not remain tied,

but slips off the free end. These occurrences are beyond the reach
of the current analysis and simulations.

^1 W. G. Pritchard Laboratories, Department of Mathematics, Pennsylvania State University, University Park, PA 16802

^2 The Courant Institute of Mathematical Sciences, New York University,
New York, NY 10012