# The Kicked Rotator

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.