PCT in Phase space

From Tom Bourbon [932812.1545]

In my post (Tom Bourbon [931227.1213]) addressed to [Martin Taylor
931221.1400], I proposed to translate Martin's ball-in-a-bowl example, in
his post on "information theory, dynamical analysis, and PCT," into a
target-cursor-tracking example. Privately, Martin confirmed that my
proposed method looked satisfactory, so I will proceed. I am undertaking
this project to assure myself that I understand the points Martin made in
his post on the unity of the three theories. If anyone else wants to see
the results, I will post them on the net, otherwise I will post directly
to Martin.

There is one change to the example in my original post. I composed the
figure rotated 90 degrees counterclockwise from the way I posted it. When I
rotated the figure prior to posting, I forgot to alter the labels on the
axes. The corrected figure is included below.

Here is the revised version of my project description. It assumes a
stationary target, a cursor that moves vertically relative to the target and
jumps to a position either above or below the target at the beginning of the
time shown in the example, and a person who moves the cursor back to the
position of the target. (The target is at the intersection of the two axes:
0 velocity, and 0 displacement from itself.) In the translation (which I
will complete later) I will use this example, and variations on it, while I
go point by point through Martin's integrative post.

Martin, in your post on "IT, dynamics, PCT," you used the example of a
phase-space analysis of the momenta and locations of a ball in a bowl to
illustrate basic concepts in dynamical analysis and information theory. To
test my understanding of your presentation, I am trying to convert your
discussion to one of (what else?) the position of a cursor relative to that
of a target. Before I proceed, would you consider it appropriate for me to
think of the phase-space in terms of (d) displacement of the cursor from the
target and (v) velocity of change in displacement? For example, if the
cursor is suddenly displaced positively on the axis d to point A relative to
the stationary target, the cursor accelerates in v, then decelerates, while
moving back to the position of the target in d; similarly for a sudden
negative displacement of the cursor to point B on the axis d. In either
case, the cursor "ends up" at the position (in d) and velocity (in v) of
the stationary target. (In the plot below, the locations in phase-space of
the cursor in example A are shown as periods; those for example B, as zeros;
and the point of 0-displacement and 0-velocity [when the cursor matches the
target] as @, to represent both a period and a zero. Would you consider the
point @ to represent a "point attractor?")

                         +d (displacement of cursor from target)
                         . A
                       . |
                      . |
                       . |
            -v ----------@---------- +v( velocity of change in displacement)
                         > @
                         > @
                         > @
                       B @

(Time runs from A down to the @; for B, time runs up to the @. From A, the
person accelerates the cursor downward [-] to the target; from B, upward
[+] to the target.)

If anyone wants to see the next stages in this translation, let me know.

Until later,