| 1. For the logistic map select s = 3.5. Interpret the eventual behavior
of the iterates using the histogram, time series, and Kelly plot. |
| Here are the pictures. |
| Kelly plot After some initial blue squares, the picture settles into a
repeating pattern of length four: purple, white,
pink, and purple. Note that
even though only three colors are repeated, the pattern has length four. |
| Time series After some initial variation, the time series appears to settle into a
pattern of four repeated values: highest, lowest, second highest, second lowest. Can you approximate
these values? |
| Histogram Ignoring a few scattered dots, the bulk of the histogram
settles down into four bars. |
| All three assays suggest a 4-cycle. |
| 2. For the logistic map select s = 3.7381. Interpret the eventual behavior
of the iterates using the histogram, time series, and Kelly plot. |
| Here are the pictures. |
| Kelly plot This is much less clear than problem 1. At the start and occasionally
later there appears to be a repeating pattern of five colors: grey,
purple, dark blue,
pink, and purple. Despite
this pattern, certainly a 5-cycle is not present. |
| Time series If the Kelly plot appears to give a 5-cycle at the beginning,
why do we see what appears to be four repeated values in the time series? Try magnifying horizontally.
Are all the lines segments straight? Here is a
magnification. |
| Histogram The points seem to cluster about five bars, but with many more points
scattered between. Rerun the program taking the initial point as close as possible to the
value of one the histogram bars. |
| 3. For the logistic map select s = 3.7425. Interpret the eventual behavior
of the iterates using the histogram, time series, and Kelly plot. |
| Here are the pictures. |
| Kelly plot The Kelly plot shows a repeating pattern of five colors: grey,
purple, dark blue,
pink, and purple. This could indicate a 5-cycle, but the width of the
bins can hide more complicated behaviors. |
| Time series The time series looks like a 5-cycle, as does a magnification at first glance. However, a
careful comparison
of the heights reveals at least 9 values, and maybe 10. This is clearest in the next-to-the-lowest line. |
| Histogram The histogram reveals the clearest interpretation. The three middle values clearly separate into pairs, as does
the lowest value, although there the histogram bars are adjacent. The top appears to be a single bar, but having length twice that of the
other bars. Recalling that the histogram bars correspond to bins of nonzero width, we see the top bin must contain two cycle points. |
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