The Lyapunov Exponent: Executable or Text Version
Certain mathematical equations vary in their reaction to errors in initial conditions. The Lyapunov Exponent provides a way to map this susceptibility graphically. Alexander Lyapunov was a Russian mathematician who died in 1918. Sensitivity to errors is one of the hallmarks of mathematical chaos, along with repeated cycles and "mixing" behaviour.
The simplest situation showing a very complex pattern of chaotic / non-chaotic behaviour is recursive application of the Logistic Equation:
P(X) = p.X.(a-X)
Those interested in the detailed mathematics explained accessibly should read the article Extracting Beauty from Chaos by Cambridge University researcher Andy Burbanks. Just as Andy Burbanks does, my program uses TWO versions of the above Logistic Equation, P(X) and Q(X), with a 2-D picture plotted of p versus q.