Lorenz atractor
the butterfly effect in a strange atractor
an applet made by António Miguel de Campos (Portugal)

Lorenz atractor is generated by the following noninear diferential equations:

dx/dt = s ( y - x )
dy/dt =
r x - y - xz
dz/dt = xy -
b z

In the screen we see the representation of the dynamic behavior of the system in 3D phase space, and, initially, for r=28, s = 10, b = 8/3. The starting point is indicated at bottom left (x=0.0, y=1.0, z=0.0). If you click at the bottom right (New Start Point) you see the evolution starting from a different starting point.

If you click in the middle, where it is written Rho=28, you can change the value of r (the applet chooses a random value between 9 and 29).
If you click in the screen where it says CHAOS, you will simultaneously see two trajectories of evolution (one blue and the other yellow) starting from 2 different very close starting points (they differ only by 0.00001 in the x-coordinate. Notice the butterfly effect. (Click again in CHAOS to come back to the normal functioning).
If you click in the screen where it says ROSS, will see the atractor of Rossler that is generated by following the 3 connected nonlinear diferential equations:

dx/dt = - y - z
dy/dt = x + 0,2 y
dz/dt = 0,2 + z (x - c)

To come back to the atractor of Lorenz, click again in ROSS.
 

Mandelbrot set and Julia sets  

Fractal figures generated using L-systems 

Wall paper-errors due to finite precision

A fractal leaf and the Sierpinski triangle