Double pendulum fractal

The double pendulum fractal is a map of starts. Each point is one double pendulum, the two angles its upper arm $\theta_1$ and lower arm $\theta_2$ make with straight down, let go from rest; its colour is the time at which either arm first swings over the top. Bright points flip early, dark ones late or not by $t = 60$. It is a first-passage picture, the same kind as the escape-time colouring of the Mandelbrot set. Here you can watch the map develop, set two neighbouring starts side by side, and magnify it.

The model: two equal point masses on two equal massless rods, no friction, with $m = l = g = 1$ so that time is in units of $\sqrt{l/g}$.

The lens in the middle never lights up, and it is not a matter of waiting longer. From rest the energy is all height, measured from hanging still,

and it stays the same as the pendulum moves. With the lower arm pointing straight up the height is at least 2, with the upper arm up at least 4. So wherever $E \le 2$, that is $2\cos\theta_1 + \cos\theta_2 \ge 1$, neither arm can ever pass over the top. The faint curves inside the lens join starts of equal energy, from $E = 0.25$ near the middle out to its edge at $E = 2$: how far short of the top those pendulums fall. Enough energy is not a promise, though: many starts just outside the lens have not flipped by $t = 60$.

Wide smooth bands lie next to fine filaments because of what happens before the flip. In a band the lower arm goes over on its first or second big swing; the motion before that is short and regular, and starts close together stay close until they flip together. The edge between two such bands is made of starts that only just reach the top: on one side the arm gets over, on the other it falls back and goes over a swing later. In a filament the pendulum has already tumbled for a long time before it flips, small differences grow quickly, and a hair’s difference in the start decides which swing finally carries an arm over.

Magnify a filament and it is not the same pattern again: it splits into more, thinner, nearly parallel bands, layer after layer. The edge between two early bands, magnified, stays a clean line. The page stops at ten thousand times. Beyond that, a growing share of the cells that flip late change with the method of computation, and the picture would no longer show only the pendulum.