Cycles and Chaos Practice Problems

MATH 242 practice problems

Learning Goals: Use computation to explore mathematical patterns. Understand how the dynamics of the logistic map changes as the parameter r increases.

Your task: Use computation to estimate the values requested in the problems below.

Due: Friday, October 17, at 5:00pm. Upload your solution as a Mathematica notebook to the Cycles and Chaos Practice Problems assignment on Moodle.

Grading: If you obtain reasonable estimates of the requested quantities, supported by your computational evidence, you will receive a Satisfactory grade for each problem.

Need help? Review the notes from class (which are posed on the course web site), talk with your classmates, email Prof. Wright, or visit the help session.

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Problem 1

Determine values a and b that form the largest interval a<r<b on which the logistic map displays a stable 3-cycle.

Problem 2

Find a value of r for which the logistic map displays a stable 5-cycle. Then find a value of r for which the logistic map displays a stable 7-cycle.

Problem 3

Choose real numbers (Subscript[x, 0]) and (Subscript[y, 0]) between 0 and 1 whose difference is 0.01. That is, |(Subscript[x, 0])-(Subscript[y, 0])|=0.01. How many iterations of the logistic map with r=4 are required until (Subscript[x, n]) and (Subscript[y, n]) differ by at least 0.99? Repeat this at least ten times and estimate the average number of iterations required.