Feigenbaum Constant 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: Monday, March 23, at 11:59pm. Upload your solution as a Mathematica notebook to the Feigenbaum Constant assignment on Moodle.

Grading: If you follow the instructions and obtain reasonable estimates of the Feigenbaum constant or a reasonable bifurcation diagram, 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 (Exercise 3.79 in the text)

Find a value of r for which the logistic map produces a 5-cycle. Then, to the best of your computational ability, estimate a few period-doubling bifurcation values starting with the 5-cycle. Use the ratios of the differences of your bifurcation values to estimate the Feigenbaum constant. Discuss how your estimate compares with the actual value δ≈4.669... of the Feigenbaum constant.

Problem 2

Let (Subscript[g, r])(x)=r sin(π x) on the domain 0<=x<=1, with parameter 0<=r<=1. Plot a bifurcation diagram. Discuss how this bifurcation diagram compares with the logistic map bifurcation diagram that we’ve seen in class.

Problem 3

As in problem 2, let (Subscript[g, r])(x)=r sin(π x) on the domain 0<=x<=1, with parameter 0<=r<=1. Focus on the initial period-doubling cascade, where convergent trajectories bifurcate into 2-cycles, then 4-cycles, and so on. Estimate at least four period-doubling bifurcation values in this period-doubling cascade and use the ratios of the differences of your bifurcation values to estimate the Feigenbaum constant. Discuss how your estimate compares with the actual value δ≈4.669... of the Feigenbaum constant.