Exercises for Chaos Under Control

Chapter 5: Fixed Points of the Logistic Map

4. The picture here is the second return map graph for the logistic map at an s-value where the logistic map has a stable 2-cycle. Use graphical iteration to determine where the points a and b go under iteration of the second return map. Answer

5. Label all the stable fixed points in the diagram above with S and all the unstable fixed points with U. Answer

6. What cycle is depicted by the histogram in the Figure? Answer

7.[C] Select LOGISTIC/HISTOGRAM. Determine which n-cycle is produced by the logistic map for these parameter values: 2, 3.1, 3.52, 3.55, 3.567, 3.74, 3.83, and 3.845. Check using LOGISTIC/TIMESERIES. Answer

8. (a) For 1 < s < 3, for almost all x in the interval [0,1] the sequence generated by iterating the logistic map converges to the fixed point 1 - 1/s. Find those x for which the sequence does not converge to 1 - 1/s. Answer

(b) For 3 < s < 3.449 the logistic map has an attracting 2-cycle. Show there are infinitely many points not converging to this 2-cycle. Answer

9.[C] We know for 1 < s < 3 the logistic map has an attracting non-zero fixed point, and from almost any starting point the sequence produced by iterating the logistic map converges to that fixed point. Select LOGISTIC/TIMESERIES.

(a) For s = 1.5 determine which starting values give a sequence always increasing to the fixed point.

(b) Repeat (a) for s = 2.

(c) Repeat (a) for s = 2.5. (Be careful with this one.)

(c) Repeat (a) for s = 2.5. (Be careful with this one.)

10.[N] A slight variation on the logistic map is given by

xn+1 = s*xn*(1 - xn2).

(a) Is the graph of xn+1 as a function of xn symmetric about xn = 1/2? (If it is then the values of xn+1 for xn = 1/3 and 2/3 are the same, for example. Are they?)

(b) Make plots of xn+1 as a function of xn for s = 1/2, 1, and 2, for xn between 0 and 1. Answer

(c) From your plots estimate the value of xn for which xn+1 is maximum. Answer

(d) Using the value determined in (c), estimate the maximum value s can be and still have the dynamics bounded between 0 and 1. Answer

(e) For what values of s does this map have a single fixed point? Is it stable or unstable? For what values of s does this map have more than one fixed point? How many does it have? What can you say about the stability of these fixed points? (Make an estimate of the ranges of s for which the fixed points are stable and unstable.) Answer

11.[N] Another slight variation on the logistic map is given by

xn+1 = s*xn2*(1 - xn).

(a) Is the graph of xn+1 as a function of xn symmetric about xn = 1/2? (If it is then the values of xn+1 for xn = 1/3 and 2/3 are the same, for example. Are they?)

(b) Make plots of xn+1 as a function of xn for s = 2, 4, and 6, for xn between 0 and 1. Answer

(c) From your plots estimate the value of xn for which xn+1 is maximum. Answer

(d) Using the value determined in (c), estimate the maximum value s can be and still have the dynamics bounded between 0 and 1. Answer

(e) For what values of s does this map have a single fixed point? Is it stable or unstable? For what values of s does this map have more than one fixed point? How many does it have? What can you say about the stability of these fixed points? (Make an estimate of the ranges of s for which the fixed points are stable and unstable.) Answer

12.[A] (a) Solve, analytically, for the fixed points of the map in exercise 10. (That is, find values of xf in terms of s.) Answer

(b) Determine an expression for the slope of the tangent line to the graph of xn+1, for any value of xn. Answer

(c) From your result in (b), what is the slope of the tangent line to the graph of xn+1 at xn = 0 and 1? Answer

(d) From your result in (b), determine the value of xn for which xn+1 is a maximum. Answer

(e) What is the maximum value of s for which iterations of this map remain bounded between 0 and 1? Answer

(f) From your result in (b), comment on the the stability of the fixed points you found in (a). Answer

13.[A] (a) Solve, analytically, for the fixed points of the map in exercise 11. (That is, find values of xf in terms of s.) Answer

(b) Determine an expression for the slope of the tangent line to the graph of xn+1, for any value of xn. Answer

(c) From your result in (b), what is the slope of the tangent line to the graph of xn+1 at xn = 0 and 1? Answer

(d) From your result in (b), determine the value of xn for which xn+1 is a maximum. Answer

(e) What is the maximum value of s for which iterations of this map remain bounded between 0 and 1? Answer

(f) From your result in (b), comment on the the stability of the fixed points you found in (a). Answer