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 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 because of the strong contraction of volumes for the parameters used.

From the location of one dot on the Poincare section you can 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. Lorenz used a slightly different version, which turns out to be preferable for this case.

Return Map

Lorenz noted that the maximum values of the Z variable obtained on successive orbits around one or other fixed point seemed to have some predictability. He therefore plotted a return map of successive maximum values of this variable i.e. plot the (n+1)th value of Zmax against the nth value of Zmax

The return map (filled in after some long iteration time) has the appearance of a function, and leads to the study of "iterated one dimensional maps". (Again, looking on a very fine scale we would find that the points do not strictly define a function - there is some "fuzziness" to the line.) A great deal of information about the Lorenz dynamics can be intuitively understood from the return map. For example the intersection of the "function" with the diagonal line corresponds to a fixed point; the fact that the magnitude of the slope of the function is greater than unity shows that the fixed point is unstable. Indeed we can associate the chaotic motion with the fact that the magnitude of the slope is greater than unity everywhere leading to the sensitive dependence on initial conditions.

One Dimensional Maps