The Computational Beauty of Nature — Spring 2020

Problem Set 3: Chaos

Due by class time Tuesday, March 3

For these questions, you should write your answers either by hand or using a word-processing program such as Word. Handwritten work is acceptable as long as you write neatly and legibly, but you'll lose points if I have to struggle to read your answers.

Reading

• For next week, begin reading Chapters 3 and 4 (pp. 40-70) of Complexity: A Guided Tour.

Written Questions

1. From Lab 3: Dynamics and Chaos:

   1. Include the plots you created in question 9 for different values of R for the logistic map, as well as your plot of the "time to separation" as a function of R. Briefly describe what happens to the observed sensitivity of the system to initial conditions as R increases.

   2. Include the plots you created in question 10 for different values of R for the sine map, as well as your plot of the "time to separation" as a function of R. Briefly describe what happens to the observed sensitivity of the system to initial conditions as R increases. How does the behavior of the sine map compare to the behavior of the logistic map in this regard?

2. From Lab 4: Bifurcation Diagram:

   • Question 8: What is the observed value of the fixed point when R = 3.0?
   • Question 9: What are the observed values of the two attractor points when R = 3.4?
   • Question 10: What is the observed value of the unstable fixed point when R = 3.4?
   • Question 11: About how many steps does it take to reach the periodic attractor for R = 3.4, starting from an initial value of x = 0.7058823529411764?
   • Question 12: Give values of R for the logistic map that produce the specified long-term behaviors.
   • Question 13: About how many iterations are necessary to reach the period-3 attractor you found in question 12, for each of the specified initial conditions?
   • Question 14: About how many iterations are necessary to reach the period-6 attractor you found in question 12, for each of the specified initial conditions?
   • Question 15: Using the bifurcation diagram, estimate Feigenbaum's constant as accurately as you can. Explain your approach, and show your work clearly.

3. If we set the R "knob" to 2.0 and iterate the logistic map starting from any initial x value between 0 and 1, we will observe that the values quickly converge to a fixed point of 0.5. We can also calculate this fixed point value directly, as follows. If x is a fixed point, plugging x into the logistic equation produces the very same value as output. That is:

   x = 2 x (1 - x)

   Solving this equation for x gives x = 1/2, which agrees with our observations.

   1. In the same way, calculate the fixed point of the logistic map for R = 2.5, showing your work.

   2. Calculate the fixed point of the logistic map for R = 3, showing your work.

   3. Give a general formula for finding a fixed point of the logistic map, in terms of the R parameter.

   4. Using your formula, calculate a fixed point for R = 4. Is this fixed point stable or unstable? Explain.