# 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, with , and looking at its poles.

The resulting image looks mundane at first, when looking at , though the last quarter section is where the interesting things are happening (hence the image below only shows the diagram for ).

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.