Exercises for Chaos Under Control

Chapter 7: Recoding the Logistic Map

23. Show where some of the "good" (those that don't wind up at -infinity) values of x are in Figure 7.16. Convince yourself that these might form a Cantor set of some sort. Do not try to compute the dimension: this is not a standard Cantor set.

24.[A] Show equation (6) follows from (5). (Hint: use the quadratic formula.)

25.[A] Show that the fixed points of the Mandelbrot map (7) occur at

zf = (1 ± sqrt(1 - 4*c))/2 Answer

26.[A] For what value of c is there a single fixed point? What is zf there? Answer

27.[N] Let w = 2 - 3*i and z = 4 + i. Compute

(a) w + z Answer

(b) w - z Answer

(c) w*z Answer

(d) w/z, being sure to put the answer into standard form. Answer

28.[N] Compute the magnitude of the answers to each part of exercise 27. Answer

29.[A] Show that the relation c2 = (a2 - b2) + (2*a*b)*i, where c = a + b*i, follows from the definition for complex multiplication.

30.[A] Show, for any complex numbers w and z, the magnitudes satisfy |w*z| = |w|*|z|.

31. Show that the complex numbers z lying on the unit circle satisfy |z| = 1. Show those lying inside the unit circle satisfy |z| < 1, and those outside satisfy |z| > 1.

32.[N] Show that the (filled-in) Julia set for c = 0 is the unit disc by completing the following parts. (Observe, for c = 0, the Mandelbrot map (8) reduces to zn+1 = zn2.)

(a) First, note that iteration of the Mandelbrot map produces the sequence z2, z4, z8, z16, ... .

(b) Using a calculator with an xy key, compute 1.1n for n = 2, 4, 8, 16, 32, 64, 128. Deduce if r > 1, then r2k runs away to infinity as k increases.

(c) Now compute 0.9n for n = 2, 4, 8, 16, 32, 64, 128. Deduce in 0 < r < 1, then r2k goes to 0 an k increases.

(d) Using the answer to exercise 30, deduce that if the complex number z satisfies |z| > 1, then |z2k| runs away to infinity as k increases; and if |z| < 1, then |z2k| goes to 0 as k increases.

(e) If |z| = 1, deduce |z2k| = 1 for all k.

(f) Deduce the Julia set for c = 0 consists of all z with |z| ² 1.

(g) Using exercise 31, identify the Julia set.

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