Bifurcations 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: In the previous homework, you estimated the bifurcation value (Subscript[r, 2]) such that logistic map trajectories exhibit 2-cycles for r slightly less than (Subscript[r, 2]) but 4-cycles for r slightly greater than (Subscript[r, 2]). Today you will estimate the next two bifurcation values.

Due: Wednesday, March 18, at midnight. Upload your solution as a Mathematica notebook to the Bifurcations Practice Problems assignment (Moodle link).

Grading: If you estimate the bifurcation values, 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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What assistance did you receive and what resources did you use (if any) while working on this assignment?

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Problem 1: Exercise 3.55 in the text

Recall that a fixed point (SuperStar[x]) of a function h(x) is an attractor if \[LeftBracketingBar]h'((SuperStar[x]))\[RightBracketingBar]<1. A fixed point (SuperStar[x]) of  h(x) is a repeller if \[LeftBracketingBar]h'((SuperStar[x]))\[RightBracketingBar]>1. (See pages 126 – 133 in the text.)

(a) Use the derivative condition above to confirm that for r=2.8, the single fixed point of (Subscript[f, r])(x) is an attractor.

(b) Similarly, with r=3.3, confirm that (SuperStar[x])≈0.479 and (x^(⋆⋆⋆))≈0.824 are attractors of (Subscript[f, r]^((2)))(x), while (x^(⋆⋆))≈0.697 is a repeller.

Note that this problem is asking you to compute derivatives. You can either do this by hand or you can use Mathematica’s D function (you may wish to refer to Wolfram's documentation and examples for D).

Problem 2

Estimate the parameter value (Subscript[r, 3]) at which the 4-cycle bifurcates into an 8-cycle. In other words, the value (Subscript[r, 3]) is such that:

If r is between (Subscript[r, 2]) and (Subscript[r, 3]), then the logistic map trajectories enter a stable 4-cycle.

If r is slightly larger than (Subscript[r, 3]), then the logistic map trajectories enter a stable 8-cycle.

Try to find the value of (Subscript[r, 3]) correct to three decimal places.

Problem 3

Estimate the parameter value (Subscript[r, 4]) at which the 8-cycle bifurcates into an 16-cycle. In other words, the value (Subscript[r, 4]) is such that:

If r is between (Subscript[r, 3]) and (Subscript[r, 4]), then the logistic map trajectories enter a stable 8-cycle.

If r is slightly larger than (Subscript[r, 4]), then the logistic map trajectories enter a stable 16-cycle.

Try to find the value of (Subscript[r, 4]) correct to three decimal places.

Optional (not for a grade)

How many more bifurcation values can you find? Can you estimate the value where 32-cycles emerge? How about 64-cycles or 128-cycles? How far can you go?