Bifurcation Demo

Bifurcation diagrams (aka Feigenbaum plots) are one way of demonstrating the nature of chaos. They show how an iterative function may, for certain values of a constant, converge to a steady state over time, while for other values it may oscillate between a number of different states, and for yet others, vary chaotically without ever settling into a pattern. This applies to certain population models, such as the logistic equation, ax(1-x), which is included in the demo below. By just changing the constant a, the long-term behavior of the system can be drastically altered.

In the graphs below (generated when you click the "start" button), the x-axis represents the value of the constant a. In the upper graph, the y values represent the steady state or oscillatory states to which the system tends for those values of a. Where the system is chaotic, hitting an infinite number of points, you'll see a mishmash of points on the graph.

The lower graph depicts the curves formed by taking a critical point of f_a(x) (a point where the derivative is 0) and iterating it a few times. One curve is the first iteration for various values of a, one is the second, etc. These are displayed in different colors so you can tell which is which. These curves tend to reflect the general shape/behavior of the bifurcation diagram, and if you look closely, you can see some of them showing up in the upper graph, faint lines discernible within the chaotic mishmash. This is because the values that the equation hits tend to be more concentrated around those curves.

You can customize the graphs by changing values for hspan (the span of a values shown), vspan, number of iterations (for the lower graph), and by choosing a different function from the pull-down list.

Here's the source code.

You need a Java-enabled browser to run this applet.