Robert May's bifurcation graph

In a 1976 paper titled "Simple mathematical models with very complicated dynamics", Lord Robert May demonstrated in 1976 how an apparently simple equation, like a highly simplified population growth model, could exhibit very complex behaviour.

The model is a logistic map, and the picture on this page demonstrates the behaviour of the model applied to a selected set of initial conditions. Here's how to do it:

For each x, set y to 0.5 and iterate

yn+1 = xyn(1-yn)


a few thousand times, skipping the first few hundred values. Plot the rest of the values along the y axis. The result, for the most interesting interval of x=[2.5..4], looks like this:

 

Robert May's bifurcation graphRobert May's bifurcation graph

 

Interestingly, when you zoom in on some non-trivial part of the image, you will find the same kind of complexity, as well as distorted copies of the same shapes at every level. The bifurcation graph is an excellent, very simple illustration of fractality and of chaos theory. Check out the applet, and read more at Wikipedia!