Jonathan. Frech’s WebBlog

Bifurcation Diagram (#164)

Jonathan Frech

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)$$x\mapsto\lambda\cdot x\cdot (1-x)$ with $0\leq\lambda\leq 4$$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$$0\leq\lambda\leq 3$, though the last quarter section is where the interesting things are happening (hence the image below on­ly shows the diagram for $2\leq\lambda\leq 4$$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