The Lorenz Model - Demo 6

Poincare Section

Chaos is not just randomness - there is a lot of structure to the dynamics as well. The full structure in the three dimensional phase space is hard to visualize, and the projections we have looked at in demonstrations 1 and 2 are deceptive in apparently showing crossings of the phase space trajectories. Instead, looking at the intersections of the orbit with a particular plane - the Poincare section - is useful. Here we look at the intersection with the plane Z=31:

The intersection has the appearance of lines, which would correspond to a planar structure in the 3-d phase space. This cannot be quite right, since again that would give crossing trajectories. But clearly the orbit is not space filling, i.e. three dimensional. Actually the structure is a "fractal" with nonintegral dimension, that is actually very close to two, and we can think of this as the definition of a strange attractor:

The orbit at long times lies in a subset of the phase space with lower, non-integral dimension (a fractal) known as a strange attractor.

If you study the appearance of the dots on the Poincare section, from the location of one dot you can begin to predict roughly where the next dot will appear. There is a great deal of predictability in the chaotic dynamics! One way of showing this is to plot a "one dimensional return map" of successive values of one of the coordinates. You can do this by choosing the "Return Map" diagnostic choice. Lorenz used a slightly different version, which turns out to be preferable for this case. This is shown in the next demonstration.

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Last modified Thursday, November 18, 1999
Michael Cross