# Exercises for Chaos Under Control

## Chapter 5: Surprises in the Dynamics of the Logistic Map

14.[C] Select LOGISTIC/BIFURCATION. Using Magnify, determine the approximate s-value at which the 3-cycle window opens (the first appearance of the 3-cycle) and the value at which the 3-cycle window closes (where full-blown chaos returns). Find these values to three digits to the right of the decimal point. Answer

15.[C] (a) Select LOGISTIC/BIFURCATION. Using Magnify, locate a 4-cycle window.

(b) Find the size of this window (the s-range between where the window opens and where full-blown chaos returns) by repeating the instructions for exercise 14. Answer

16.[C] Select LOGISTIC/BIFURCATION. Locate three different 5-cycle windows in the Bifurcation Diagram of the logistic map. Answer

17.[C] Select LOGISTIC/BIFURCATION. Locate the 3-cycle window and the three small copies of the bifurcation diagram in this window.

(a) Using Magnify, find the x-range of the top copy at the right edge of the window. Answer

(b) Repeat (a) for the middle copy. Answer

(c) Repeat (a) for the bottom copy. Answer

18.[C] Select LOGISTIC/BIFURCATION. Locate the 4-cycle window and the four small copies of the bifurcation diagram in this window.

(a) Using Magnify, find the x-range of the top copy at the right edge of the window. Answer

(b) Repeat (a) for the second copy. Answer

(c) Repeat (a) for the third copy. Answer

(d) Repeat (a) for the fourth copy. Answer

19.[C] Comparing exercises 17 and 18, what deduction can you make?

20.[A] Find the equation for the nonzero fixed point curve of the bifurcation diagram of the logistic map. Answer

21.[C] Select LOGISTIC/BIFURCATION.

(a) With the Seed value at 0.5 (the default setting), plot the bifurcation diagram dropping 0 points and iterating 20.

(b) Repeat (a) with the Seed value set to 0.01. Describe the differences you see.

(b) Repeat (a) with the Seed value set to 0.01. Describe the differences you see.

(d) Repeat (c) with the Seed value set to 0.01. Describe the differences you see.

22.[N] Consider the logistic map with s = 4. Taking two points near 1/2, say x0 = 0.490 and x'0 = 0.491, what happens to the distance between these points under one iteration of this logistic map. How is this result consistent with sensitive dependence on initial conditions?

23. Here is a picture of part of the bifurcation diagram of the logistic map. Match each of the labelled s values A, B, C, D, and E with the caption most closely describing the dynamics there. Answer

____ Here is a 3-cycle window

____ The nonzero fixed point is attracting.

____ Here is a period-doubling bifurcation.

____ There is an attracting 2-cycle.

____ Here is chaos.

24. For each of the following dynamical systems give a brief description of the associated attractor:

(a) tent map with s=0.5; Answer

(b) tent map with s=1.9999; Answer

(c) logistic map with s=0.5; Answer

(d) logistic map with s=2.0; Answer and

(e) logistic map with s= 3.2. Answer