Algodoo

This scene contains both the Logistic Equation Xn+1=Xn*r*(1-Xn) and the with it connected Feigenbaum diagram, drawing the state of the 50 points for all values of 0<r<4. If you like to find out more about these Functions, simply google in the internet, but here is a "rough" description:

The Logistic Equation is a recursive equation usually used to simulate the growth of an object in a limited enviroment. The parameter R is a unification of both growth and deathrate, and the 1 in this particular equation symbolices a maximum capacity of 1. This form of logistic equation however shows special behavior once reaching r>2.5. If one watches the curve, they will see how the line suddently zips up into 2 seperate stable lines, periodically switching between each other. After r>3.4, this process repeats, resulting in 4, then 8, and then the other powers of 2 in such an extremely short timespan that one barely notices. After reaching about 3.45, the whole equation switches into a special state, which(guess what) is called chaos, an elemental part of Maths, or to be exact, the Chaos Research, which is a part of Maths that works with so called determined chaos, which is an algorithm that turns chaotic, but remains calculatable.
If one would take all the points of the graph(usually the areas where most points clutter up), and then mark these points in a R-Y diagram(X axis shows value of R, y axis shows the "stable areas"), they will end up with the so called Feigenbaum-diagram, which is an essential object in the chaos research. The Feigenbaum diagram starts with a line, that starts unfocused(no real point of concentration, goes towards 0), and then quickly turns into a straight line. Then this line suddently splits in 2 parts, and in more shorter time steps, keeps duplicating(periodic duplication), until it reaches the point of chaos, where basically no more stable points seem to exist. In this area of chaos however, sometimes appear so called "periodic windows", areas where the chaos quits, and catches itself in a set amount of stable points(largest is the period-3-window). These, like the whole diagram itself, duplicate, and then turn into new chaotic areas. After reaching R>=4, the whole diagram dissolves into infinity, resulting in a sudden stop of the diagram. If one zooms close to the diagram, they will see straight lines running trough the chaotic scribble. These are called "supertracks", areas where chaotic numbers are a bit more dense, and basically overlap from seperate parts. If these tracks collide with a periodic window, they mark the area where a new period will start.
The whole process that can be witnessed in the diagram is called "Bifurcation" and is visible in many totally different chaotic algorithms, and even in real life(Benárd-cells). That's why this diagram is an importand part of chaos research.