Logistic Map Practice Problems

Math 242 practice problems

Name: _________________

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

Your task: Complete the three exercises below.

Due: Friday, March 17, at 5:00pm. Upload your solution as a Mathematica notebook to the Logistic Map Practice Problems assignment (Moodle link).

Grading: For Problems 1 and 2, computational evidence in support of the stated claim will earn a Satisfactory grade. Problem 3 requires you to approximate a parameter value, supported by computational evidence.

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.

Problem 1

Produce your own computational evidence to show that for 0<=r<=1, all logistic map trajectories converge to a fixed point (SuperStar[x])=0. (Demonstrate this for various parameters r and starting values (Subscript[x, 0]).)

Problem 2

Produce your own computational evidence to show that for 1<r<3, the logistic map has a single stable fixed point. (Demonstrate this for various parameters r and starting values (Subscript[x, 0]).)

Problem 3

Let (Subscript[r, 1])=3, the parameter value at which the logistic map fixed point is no longer stable. For parameters r slightly larger than (Subscript[r, 1]), the logistic map trajectories settle into oscillation between two values in the long run. Now estimate (two decimal places are sufficient) the largest value (Subscript[r, 2]) such that for (Subscript[r, 1])<r<(Subscript[r, 2]), the logistic map has an attracting 2-cycle. In other words, approximate the parameter value (Subscript[r, 2]) such that you observe 2-cycles for r<(Subscript[r, 2]) but not for r>(Subscript[r, 2]).