Iterating a formula, it is possible that the results walk to a fixed value. Raising a parameter the behavior changes to two periodically repeating values. With a more raised parameter you can see another bifurcation into four different values, that change to the next at every new iteration. A little bit later, there is one more bifurcation, etc. Beginning with a critical parameter-value no more order can be seen; the output values are chaotical and the next iteration result depends on minimal variations behind the decimal point in the input.
My program Feigenbaum explores formulas in the described way and also allows time-diagrams, graphical iteration, phase diagrams in two or three dimensions and even Liapunov-Pictures. You can order the program for personal use only by email (please send me an aquivalent: ideas, pictures, programs, education-concepts, etc. with your mail).
More information: Watch the essays (written in German)
The physicist Mitchell Feigenbaum (b. 1944) studied turbulences in liquids and developed the Logistic map (Feigenbaum Diagram), which shows a quantitative universality for a class of nonlinear transformations.