# Exercises for Chaos Under Control

## Chapter 6: Close Pairs

14. Suppose we have a data set x1, ..., x20, made from a 5-cycle x1, x2, x3, x4, and x5. (So x6 = x1, x7 = x2, x8 = x3, and so on.) Suppose we have taken a filter smaller than |x1 - x2|, |x1 - x3|, |x1 - x4|, |x1 - x5|, |x2 - x3|, |x2 - x4|, |x2 - x5|, |x3 - x4|, |x3 - x5|, and |x4 - x5|. Sketch the close-pairs plot for this data set. Answer

15. How would the close-pairs plot of Exercise 14 change if |x1 - x2| is less than the filter value? Answer

16. Make a close pairs plot for the time series given in Exercise 8. Use a filter value of 0.5. How would the plot change if you used a filter value of 0.1?

17.[C] Use ClosePairs to analyze 100 points of the tent map in the following parameter ranges: 0.8, 1.1, 1.3, 1.5, 1.7, 1.9. In each case, take the initial point to be 0.5 and drop the first 100 points. Experiment with different filter values: 0.1, 0.05, and 0.01. Answer

18.[C] Use ClosePairs to analyze 100 points of the logistic map in the following parameter ranges: 0.5, 1.5, 2.5, 3.1, 3.45, 3.55, 3.65, 3.75, 3.85, 3.95. In each case, take the initial point to be 0.5 and drop the first 100 points. Experiment with different filter values: 0.1, 0.05, and 0.01. Answer

19.[C] Investigate the effect on Close-Pairs plots of sensitive dependence on initial conditions. Use ClosePairs to analyze 100 points of the logistic map for the parameter 3.95 and with initial point 0.5, and dropping the first 100 points. Repeat with initial point 0.4. Describe the similarities and differences of the two plots. Try different filter values -- 0.1, 0.05, 0.01, and 0.005. Answer

20.[C] What would happen if you had performed the experiment of exercise 19 for the parameter value 3.2? Do not try the experiment first, but think and describe what you expect to see. Then use ClosePairs to check your answer. Answer

21.[C] Use ClosePairs to analyze 100 points from a uniform random sequence, and 100 points from a Gaussian random sequence. Experiment with different filter values: 0.1, 0.05, and 0.01. Describe the differences between the plots for the Gaussian random distribution and the uniform random distribution. Try several runs for a given filter, noting the effect of different seed values.

22.[C] Design your own time series: construct a time series so the close-pairs plot will draw a particular picture. Use ClosePairs to read your series and draw your picture. If you didn't get what you expected, comment on why you didn't.

23.[C] For the logistic map, find s values (VERY) near, but less than, the opening of the 3-cycle window. Do close-pairs plots for these s-values, and for s = 3.8284 (just inside the opening of the 3-cycle window). Does the close-pairs plot reveal any hint of the 3-cycle window before it opens up? Use ClosePairs.

24.(a) What, if anything, does a vertical line of dots mean in a close-pairs plot? Does it mean a near-fixed point?

(b) Suppose a close-pairs plot has a vertical line of dots: dots at entry number 1 and delays 1, ..., 10. Must there be dots at entry number 2 and delays 1, ..., 9? Explain your answer. Answer

25.[C] Try a close-pairs plot for "pseudo-1/f" with different filter values. Try different runs to see what features reflect 1/f-ness, what features reflect the particular run. Use ClosePairs.