Chaotic Elastica

This demo due to Francis Moon consists of a flexible beam in the presence of two attracting magnets. The magnets are very strong and are each pulling the beam in the opposite direction. Thus, if the beam is exactly in the middle it will be at rest, but the slightest perturbation near one of the magnets will cause the beam to transition to a rest state with the beam bending towards one of the magnets (as indicated in the first frame of the movie below). These two equilibrium states are stable to local perturbations. Next we add a periodic forcing to the beam (by attaching a motor at the top with an offset flywheel) which perturbs the beam. For low frequencies the beam stays at the local stable equilibrium, but as the forcing frequency is increased a bifurcation occurs and the beam oscillates chaotically between the two equilibrium states. This is an example of how a well-behaved 2-dimensional deterministic system can become chaotic when it is forced (mathematically, the forcing increases the dimension of the state space from 2 to 3). This is a mechanical anaolgue for the forced Duffing equations.

Here is an interesting graphic related to this model. The picture below is an animation of the Poincare Sections for the forced Duffing model. They are generated by writing the 2D driven model as a 3D autonomous model (where z = t) and then considering slices in z. Each image represents a slice between 0 and 2 Pi. This shows some interesting structure within the chaotic dynamics.

If we mod out the z variable by 2 Pi we create a fractal torus defined by the Poincare sections (i.e., the frames above represent what you would see upon taking vertical slices of the image below). We used z-hue shading to reveal some of the structure.