a very good and valuable work. maybe we can imagine the "Wirklichkeit" behind the reality. a sea of possibilities, they can all become real. Every point is an "Wirks" (according to Heisenberg and Dürr), a particle of effect, which is displayed to us like waves. Everything is vibration, everything is potentially possible. The laws of physics coagulate from this.
Some ideas: (A) 0:53 shows the state space is connected like a torus. You could put 9 copies of your 1:45 time evolution animation into a grid so you can see the connectedness at what are currently the "edges". (B) For any given state or subspace of interest, such as the "mostly stable" region, you could highlight when the system gets near that state or subspace via a different color mapping - instead of having a continuous mapping, make just two colors with a very brief gradient, and assign one color to all the angle states that are inside / close to the state of interest, and the other color to all the states that are outside. Then on the time evolution picture you will more sharply see when a subset of all the initial pendulums reaches that state of interest.
1:58 this gave me an idea. Because of how absolutely chaotic the fractal is, it will shuffle through every possible combination it can, however it I see limited to a few colours and we only have a few pixels. There is a chance (thanks to it’s limited pixels) that they will all fall into the same pixel colour with every position to be the same. The chances are near 0 but not impossible.once it happens everything will be one colour, still unpredictable but much easier to calculate.
Since the chaos of two revealed a commonality represented by a vesica piscis, would a triple pendulum produce commonalities representable by something resembling a triquetra? There's something harmonic appearing here despite the chaos. I suspect a vesica piscis would appear in many similar scenarios involving two component chaos. I wonder if the border of the central shape changes to match the equation. Very fun stuff, worth further investigation. Thank you.
An interesting variation would be to visualise how sensitive a certain initial condition is to perturbation, or more loosely spoken, how quickly it succumbs to chaos. This can use the same data, no recomputation needed. (it's a static image however)
I wonder what kind of fractals we could come up with if a damping term was added, so that the pendulum eventually comes to rest. Perhaps color the pixels depending on how many rotations one of the angles goes through, or how long it takes to settle within a certain range close to the resting position.
