Exploring the Fractal Boundary Between Order and Chaos in the Logistic Map 🌌

I made this video to visually demonstrate how the boundary between order and chaos is fractal. This is not just true for the logistic map, but for many other complex systems that may also have sensitive dependence on initial conditions (SDIC). A profound result, that between order and chaos there is infinite repetition and emergence of new detail. This video only zooms in enough to make the point, but you could keep going forever and the small parts would keep resembling the whole. The details that make up the entire structure are just scaled down versions of the whole - a fractal.

It is worth mentioning that this is deterministic chaos, which means there is no randomness, and it is exactly repeatable / computable given the rules of the system and the initial condition(s). In the chaotic regions, even the slightest change of the initial value X0 will lead to a very different value after the same number of iterations. This rate of divergence of trajectories can be characterised for different systems by a measure known as the Lyapunov exponent. The logistic map with alpha = 4 was used as one of the earliest (pseudo) random number generators on computers, though it is not perfectly uniform (white noise) as there is bias near the extreme values (0 and 1).

If you wish to reproduce this video, here are some details:
X(i + i) = a * X(i) * (1 - X(i))
X(0) = 0.5

_0 denotes first frame
_1 denotes last frame

a_start_0 = 2.98
a_end_0 = 4.00
x_start_0 = 0.00
x_end_0 = 1.00

a_start_1 = 3.5699455
a_end_1 = 3.5699458
x_start_1 = 0.49998
x_end_1 = 0.500047

N_pre_iters_0= 2048
N_pre_iters_1= 1024000
Npts_0 = 512
Npts_1 = 768000

Draw area size
1920 x 1080

Note that the pre-iterations are required to get past the initial transients. The map shows the converged values. Not having enough pre-iterations leads to oscillations around the bifurcation points. More iterations are needed the further you zoom. The last frames took a while to render! Note that zooming is an exponential process, so the width and height of the zoom window is scaled by a power law, i.e. W(i) = W(0) * zoomfactor ^ i .

Coded in LabVIEW, background music (more like a "soundscape") made using Reason. Video put together in Hitfilm Express.