Exercises for Chaos Under Control

Chapter 7: Julia Sets

As mentioned in this section, most Julia sets are difficult to determine. Nevertheless, for some values of the control parameters, we can find parts of the Julia sets. Exercises 14 - 19 explore this for control settings with b = 0.

14.[A] (a) Verify that if b = 0 and y0 = 0, then the Mandelbrot map gives y1 = 0, y2 = 0, ... . Answer

(b) Under these assumptions, b = 0 and y0 = 0, show the Mandelbrot map reduces to the simpler relation

(S) xn+1 = xn2 + a, yn+1 = 0. Answer

15.[A] (a) Show that the simplified Mandelbrot map (8) has fixed points x+ and x- given by

x+ = (1 + sqrt(1 - 4*a))/2 and

x- = (1 - sqrt(1 - 4*a))/2.

(Hint: recall the quadratic formula.) Answer

(b) Locate the fixed points for a = 1/4, 0, -3/4, and -2. Answer

16. Use graphical iteration to show that for a > 1/4, the iteration scheme (8) takes every x0 to infinity.

17. (a) For a = 1/4, use graphical iteration to show that any x0 between -x+ and x+ remains bounded under iteration of (8). From this, show the Julia set for a = 1/4 (and b = 0, remember) contains the interval between -x+ and x+ along the x-axis.

(b) What happens to x0 outside this interval?

(c) Conclude that this Julia set intersects the x-axis only in this interval.

18. Repeat exercise 17 for

(a) a = 0

(b) a = -3/4

(c) a = -2.

What can you conclude in general, for -2 <= a <= 1/4?

19. For a < -2, use graphical iteration to show the Julia set intersects the x-axis in only a Cantor set. Answer

20. Pictured here are two Julia sets, (a) and (b). For both these Julia sets, state whether the associated control parameters are taken from inside or from outside the Mandelbrot Set. Answer

21. Suppose (a, b) is preperiodic. Show that the launch point (x=a, y=b) belongs to the Julia set for the control parameters (a, b). (Hint: to show (a, b) belongs to the Julia set, show the orbit of (a, b) remains bounded.) Answer

22.[C] (a) Select JULIA. Look at Julia sets for points inside the buds around the body of the M Set.

(b) For each of these points, look at the Julia set for a point nearby, but outside of the M Set.

(c) Do you see any relations between these pictures?

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