To understand this one, you’re going to need an antigravity chamber. Pin one end of a metal bar to a wall (with a frictionless connection), and periodically give the bar a short pulse of gravity with strength given by a constant \(K\). This bar will have a very strange path over time, but rather than drawing that path, we’re going to plot something at a higher level: a phase plot of the bar's angle and momemtum. If the bar has initial angle \(\theta\) and angular momentum \(p\), then we plot the point \((\theta, p)\). We then briefly switch on the gravity, find the new values of \(\theta\) and \(p\), and plot that point too. Repeating this for dozens or hundreds of iterations, we begin to get a sense of an orbit of the system. For example, if \(K = 0\), then the gravity does nothing, so the initial value of \(p\) will be the only thing modifying the angle — therefore, the orbits will be horizontal lines in phase space, since \(p\) is never changed. Since a full rotation of the bar (\(\theta = 2 \pi\)) is the same as no rotation at all, we treat the phase plot as being on a torus, which just means both \(p\) and \(\theta\) loop from \(2 \pi\) back to \(0\) as if they were on a circle. Because of this, those horizontal lines are really loops.
When \(K > 0\), though, things are more complicated. The loops are now much stranger, and eventually, as \(K\) approaches a particular number whose value is around \(.971635\), the system descends into chaos, evidenced by orbits looking more like static than lines. As \(K\) continues to increase, this butterfly effect compounds, and eventually nearly every inital condition leads to chaos. This system — called the Kicked Rotator — may be a hard thing to visualize, but it’s undeniable that the picture is a beautiful one.