# 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