The Pendulum Weaves All Knots and Links

The behavior of a dissipative chaotic dynamical system is determined by the skeleton of its strange attractor, which consists of an uncountably infinite set of unstable periodic orbits. Each orbit is a topological knot, and the set is an infinite link. The types of knots and links supported by the system may be determined by collapsing the attractor along its local stable manifolds to form a template, a branched two manifold with boundary that supports the same set of knots and links as the original attractor. We show that the strange attractor of a chaotic, vertically forced physical pendulum can be collapsed to a template that supports all knots and links. Thus, one of the simplest and most well known dynamical systems is capable of the most complex behavior possible. 1 The vertically forced pendulum The physical vertically forced pendulum with velocity dependent damping is well modeled by the non-dimensional equation of motion θ̈ = −ρθ̇ − sin θ(1− α cos wt) where θ is the angular position measured from the straight down position, ρ is the damping parameter, α is the forcing amplitude, ω the forcing frequency, and overdots denote derivatives with respect to time. This system is capable of chaotic motion in two distinct modes: one where the pendulum bob does not go over the top, and one where it does. The non-over-the-top mode has an attractor that lives in the standard phase plane (θ, θ̇), but the attractor of the over-thetop mode crosses the periodic boundary θ(−π) = θ(π). Therefore, its surface of section (Poincaré plane) in standard phase space coordinates θ(t) = (θ(t), θ̇(t)) lives in an annulus. When we must build a phase space using experimental data, we frequently use time delay coordinates θ(t) = (θ(t), θ(t − τ)) [5] 1 , so the surface of section lives on a torus because both boundaries are periodic. 1Time delay coordinates are used to reconstruct an attractor in phase space when only one dynamical variable is available, e.g., from an experimental time series. A time delay