This demo is a mechanical analogue for the forced Duffing equations. It consists of a flexible beam in the presence of two attracting magnets. The magnets are pulling the beam in opposite directions. 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 an equilibrium state bending towards one of the magnets. These two equilibrium states are stable to small perturbations. Next we add a periodic forcing to the beam (by attaching a motor with an offset flywheel) which perturbs the beam. For low frequencies the beam remains at the local stable equilibrium, but as the forcing frequency increases a bifurcation occurs and the beam begins to oscillate chaotically between the two equilibrium states. This model illustrates 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).
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 horizontal 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 a z-hue shading to reveal some of the structure.
