Dynamic patterns and self-knotting of a driven hanging chain, Phys

Abstract: When shaken vertically, a hanging chain displays a startling variety of distinct behaviors. We nd experimentally that instabilities occur in tongue-like bands of parameter space, to swinging or rotating pendular states, or to chaotic states. Mathematically, the dynamics are described by a nonlinear wave equation. A linear stability analysis predicts parametric instabilities within the well-known resonance tongues; their boundaries agree very well with experiment. Fully nonlin-ear simulations of the 3D dynamics reproduce and elucidate many aspects of the experiment. However, the chain also ties knots in itself, which is beyond the current reach of either the analysis or the simulations. Spatially-extended dissipative systems driven out of equilibrium give rise to fas-cinatingly complex dynamics in which the interplay of multiple spatial and temporal scales is a central feature; examples range from chemical pattern formation to uid turbulence [1]. Mathematical descriptions of these systems are usually posed in terms of partial dierential equations (PDEs), with one spatial dimension (1D) of-ten being sucient to capture some of the complexity characteristic of these system