Page:Lawhead columbia 0054D 12326.pdf/211

 can only move in areas of each plane that have some special property—suppose that we cut different shapes into each of the sheets of paper, and mandate that the pencil isn’t allowed to tear any of the sheets. The presence of the cut-out sections on each sheet represents the constraints based on the patterns present on the system’s time-evolution in each state-space: the pencil is only allowed in areas where the cut-outs in all three sheets overlap.

Suppose the cut-outs look like this. On the top sheet, almost all of the area is cut away, except for a very small circle near the bottom of the plane. On the middle sheet, the paper is cut away in a shape that looks vaguely like a narrow sine-wave graph extending from one end to another. On the bottom sheet, a large star-shape has been cut out from the middle of the sheet. Which of these is the most restrictive? For most cases, it’s clear that the sine-wave shape is: if the pencil has to move in such a way that it follows the shape of the sine-wave on the middle sheet, there are vast swaths of area in the other two sheets that it just can’t access, no matter whether there’s a cut-out there or not. In fact, just specifying the shape of the cut-outs on two of the three sheets (say, the top and the middle) is sometimes enough to tell us that the restrictions placed on the motion of the pencil by the third sheet will likely be relatively unimportant—the constraints placed on the motion of the pencil by the sine-wave sheet are quite stringent, and those placed on the pencil by the star-shape sheet are (by comparison) quite lax. There are comparatively few ways to craft constraints on the bottom sheet, then, which would result in the middle sheet’s constraints dominating here: most cutouts will be more restrictive than the top sheet and less restrictive than the middle sheet

The lesson here is that while the state of any given system at a particular time has to be

201