# Exercises for Chaos Under Control

## Chapter 8: Basins of Attraction

1.[E] Build a replica of the magnetic pendulum toy described in the text. Tape the lying magnets to a sheet of paper and attempt to map out the various basins of attraction as shown in Figure 8.4. Is the intermingling of the basins more or less complicated than the roots of unity basins shown in Figure 8.5?

2. What are the basins of attraction of 0 and of -infinity in the tent map when s < 1? Answer

3. What is the basin of attraction of the strange attractor in the tent map when s=1.9? (Hint: how many fixed points are there?) Are there other attractors for that value of s? What are their basins (if any) and what makes up the boundary? Answer

4. (a) What is the basin of attraction of -infinity for the tent map when s > 2? Answer

[A] (b) What is the similarity dimension of all points not in the basin of attraction of -infinity for the tent map when s>2? (Your answer should contain s.) Answer

5. (a) Sketch the graph of the map xn+1 = xn3. (Hint: compute xn+1 for xn = 0, ±1/2, ±1, ±3/2, ±2.)

(b) Find the fixed points of the map. (There are three.) Answer

(c) Which of these fixed points are stable; which are unstable? Answer

(d) Use graphical iteration to find the basins of attraction of the stable fixed point(s). Answer

(e) Use graphical iteration to determine the fate of the points not in the basins of attraction of the stable fixed point(s). Answer