A little bit of chaos
Supreme court justice Potter Stewart once said, "... [chaos] is impossible to define, but I know it when I see it," or at least it went something like that. So here it is: chaos in pictures. Everything below was generated from java programs that were written without any external libraries or code.
These two chaotic functions are identical except for an input value that differs by .01(Pickover attractor (-1.2,2,2.32,.5), Pickover attractor (-1.2,2,2.31,.5)). Chaotic functions are extremely sensitive to their initial conditions.
As a function transitions into chaos, the initial conditions provide less information about which region they are going to end up in. The colors of the points represent their initial locations, and the more chaotic the function, the less points of the same color are grouped together (Standard map:k=.05, Standard map: k=.5, Standard map: k=1, Standard map: k=2).
Some functions can be represented by coloring the input values that make them chaotic. Blue represents chaos, and yellow represents stability. (Lyapunov fractal BBAABBA, Lyapunov fractal BBAABBA)