Exercises for Chaos Under Control

Chapter 5: Universality

25.[A] What is the general expression for the maximum value of x that can be attained via iteration in the sine map? For what value of s does the sine map "pop" through the roof of the unit square? (That is, for what value of s does the maximum value of x become greater than 1?) How did you find this? Answer

26.[C] Using SINE/BIFURCATION, magnify the sine map in the range 2.9464 ² s ² 2.9784, 0.1522 ² x ² 0.2013 and compare this with the logistic map in the range 3.8284 ² s ² 3.8570, 0.1310 ² x ² 0.1755 (using LOGISTIC/BIFURCATION). Find several analogous locations in the bifurcation diagrams of the logistic and sine maps and compare the magnifications. Record your observations, including, of course, the ranges of x and s values in the regions you magnified.

27.[C] (a) Using LOGISTIC/BIFURCATION find the height (range of x-values) and width (range of s-values) for the three mini-bifurcation diagrams in the 3-cycle window of the logistic map. For each, compute the quotient

(range of s-values)/(range of x-values) Answer

(b) Repeat this for sine map using SINE/BIFURCATION. Answer

(c) Comparing the results of (a) and (b), can you conclude anything?

28.[C] Compute the Feigenbaum quotients for the logistic map in this way. Denote by s1 the s-value where the stable fixed point curve in the bifurcation diagram intersects the line x = 1/2, by s2 the s-value where a branch of the stable 2-cycle curves intersects the line x = 1/2, and so on for s3 (4-cycle), s4 (8-cycle), and s5 (16-cycle). (Do this by using LOGISTIC/BIFURCATION, and Magnify.) Now compute the quotients

(s2 - s1)/(s3 - s2),

(s3 - s2)/(s4 - s3), and

(s4 - s3)/(s5 - s4).

What pattern do you see? Answer

29.[C] Repeat Exercise 25 for the sine map using SINE/BIFURCATION. Answer

30.[C] Compare the quotients of exercise 28 with those of exercise 29. What do you conclude?

31. (a) What is a "period-doubling cascade" and why is such a phenomenon important or interesting?

(b) What is Feigenbaum's number and why is it important or interesting?

(c) From the plot find an approximate value of Feigenbaum's number. Answer

32.[E] A dripping faucet can be made to undergo a period doubling sequence by adjusting the flow rate. See if you can observe at least the first period doubling. At the very lowest drip rate successive drops should have a fixed interval. You can hear the periodicity by placing a pie tin under the faucet. Very gradually increase the flow rate. Listen for a change in the rhythm of the drops. Perhaps you can make the observation quantitative by timing the drop intervals with a digital stopwatch.

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