Population growth kinetics x = rx(1-x)
The famous P.F.Verhulst dynamics for the growth of a population of simple organisms ('logistic map', above is a 'Feigenbaum' plot, stem and roots to the left are cut off). For every value r from 0 to 4 the recursive function x[i+1] = r*x[i]*(1 - x[i]) ( i= 1(1)...) is run to "end" (called 'orbit'), meaning to a stationary point at i Þ ¥, if it exists. This leads to a single point below r = 3.0 (only a small red segment visible in the picture left of r1, left ordinate), then to bifurcations (more than one stationary points in pairs) and, finally, chaotic behaviour (many 'stationary' points, ~ randomly distributed) when the growth parameter r reaches r¥ ~3.56 and beyond. The Dynamics package, with tutorial programs, allows an exhaustive interactive investigation of this case (and several others) with the most essential tools of modern chaos research. A description of these is to be found in the Kinetics Manual, p. 55-71.
One of them is Ljapunow instability theory which is demonstrated here by computing and superposing the Ljapunow exponent (white "curve", ordinate at right) during the development of the dynamics. It marks the regular bifurcations r1, r2, r3, ..., r¥ (exponent=0) and their transition to chaotic orbits (exponent > 0). Stable orbits have an exponent < 0 and occur at seemingly random r-locations also in the chaotic region!). Below r=3.0 no bifurcations occur. Since Verhulst (in 1845) did not go beyond about r=2.5 he failed to discover bifurcations or chaotic behaviour (understood after 1976 by Grossmann & Thomae, Feigenbaum and others)!
There is a wealth of information on the Web, including many excellent animations of the Feigenbaum(-plot). Take Google for a ride on 'Verhulst', 'Feigenbaum', or 'logistic map'.
Here's my simple program to produce the Verhulst dynamics on a Windows box (on Linux: wine prep0.exe. You also need mfc42.dll from Windows).