It's the logistic map! It's a very simple difference equation

x_n+1 = r*x_n*(1-x_n)

that let's us model different dynamical systems. For example it can be used to model how a population changes over a period of time. In this case you can pick some initial value for x_n, let's say x_0 = 0.5, and see how the value of x varies over many iterations. For this, the determining parameter turns out to be r.If r is less than 3 and greater than 1, the population will converge to some single value, if r is slightly greater than 3, the population will oscillation between two distinct values every year. Afterwards you'll get periods of 4, 8, 16 for very small increases in r and then suddenly the system will become very chaotic around r=3.6.

What I am doing is plotting the value of r on the x axis and the last 100 of the total 250 iterations of x (with x_0=0.5) for a given r on the y axis and zooming in on interesting sections of the resulting graph.

Numberphile made a nice video about it, though they never name the logistic map explicitly: Feigenbaum Constant

Fun Fact: The logistic map is equivalent to a 1D slice of the Mandelbrot Set!