The Lorenz attractor
The Lorenz system is a simplified model of atmospheric air flow. It is defined by the following three differential equations:
dx/dt = -sigma*(x-y)
dy/dt = r*(x-z) - y
dz/dt = x*y - b*z
The applet below simulates the Lorenz equations for simga=10, b=8/3 and for 0 <= r <=28; by varying r (using the scrollbar) you can see how the system changes and bifurcations occur.
The top two graphs show part of the phase trajectory; the top left graph is an xy projection of the trajectory and the top right an xz projection. The bottom graph shows a timeseries plot of x. The graphs should be synchronized.
r should start at 28; for this value the system behaves chaotically, as you can see from the waveform. If you drag r down to zero, all three states quickly settle to zero.
At r=1 a pitchfork bifurcation occurs; the attracting fixed point at x=y=z=0 becomes unstable and two new, stable fixed points are 'born'; you can see this by increasing r to about 10 (press bump if nothing happens).
At about r=24, the two fixed points which are created at r=1 become unstable and the only attractor is the chaotic attractor; this attractor is partially shown by the three graphs above; a more complete part of the attractor is shown in the static pictures below:
The Lorenz attractor
The Lorenz attractor - xy projection
The Lorenz attractor - xz projection