# Exercises for Chaos Under Control

## Chapter 4: The Tent Map

22.[A] (a) Show the maximum height of the tent map graph is s/2. Answer

(b) Show the nonzero fixed point occurs at xf = s/(1+s). Answer

23.[A] For s ³ 1, show the first iterate of s/2 has the value s*(1 - (s/2)). Answer

24.[A] (a) Using the results of 22 and 23, above, show that the condition on s for band merging is equivalent to setting s3 - s2 - 2s + 2 = 0. Answer

(b) Show that the numerical value of s where the chaotic bands merge in the bifurcation diagram of the tent map is sqrt(2). Answer

25. (a) Use graphical iteration to locate four points which the tent map takes to the fixed point w of the Figure.

(b) Numerically calculate the four values of x you found in (a) when s = 2.

(c) Show how to locate infinitely many points which the tent map takes to w. Answer

26. (a) Draw a graph of the second return map for s = 1, s = 3/2, and s = 2. (You can check your graphs using the IterateAgainSam program. Do you see how?) Answer

(b) Locate the 2-cycle for each of these values of s. Answer

(c) Show that the x-values of tent map 2-cycle are s/(1+s2) and s2/(1 + s2). Answer

27. Consider the Tent Map for 0 ² s ² 2. For which values of s does the tent map exhibit sensitive dependence on initial conditions? Answer