Lorenz Attractor: Dependence on Initial Conditions

Fractals have a heavy dependence on initial conditions. That means that if you change anything about the initial conditions, even something so small as changing the millionth decimal place, the resulting motion will take a completely different path than the original.

Here is a video of the Lorenz attractor with three particles (red, white, and blue). They all appear to start at the same point and take the same path. However, it does not take long before the particles diverge and take completely different paths.

Here's the most surprising bit: The initial positions of the particle differ by 0.0000006 units! that's 6 ten-millionths of a unit! This was the smallest change in position I could use with my program without suffering the effects of rounding error.

To make it more clear that the particles are moving in different paths, here is another video with the lines turned off: