Bifurcation Diagram

2017-03-25, post № 164

mathematics, PIL, programming, Python, #alpha, #chaos, #chaos theory, #delta, #Feigenbaum, #fractal, #iterations, #Mandelbrot set, #modelling, #population

Generating the famous fractal, which can be used to model populations with various cycles, generate pseudo-random numbers and determine one of nature’s fundamental constants, the Feigenbaum constant 𝛿.
The fractal nature comes from iteratively applying a simple function, $x\mapsto\lambda\cdot x\cdot (1-x)$ with $0\leq\lambda\leq 4$ , and looking at its poles.
The resulting image looks mundane at first, when looking at $0\leq\lambda\leq 3$ , though the last quarter section is where the interesting things are happening (hence the image below only shows the diagram for $2\leq\lambda\leq 4$ ).
From 𝜆 = 𝟥 on, the diagram bifurcates, always doubling its number of poles, until it enters the beautiful realm of chaos and fractals.

For more on bifurcation, fractals and 𝛿, I refer to this Wikipedia entry and WolframMathworld.

Source code: bifurcation-diagram.py

Extra assets: bifurcation-diagram-0.png, bifurcation-diagram-2.png, bifurcation-diagram-3.png