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 increaes.

Your task: Complete the two exercises from the text, reprinted below.

Due: Wednesday, March 23, at 5:00pm. Upload your solution as a Mathematica notebook to the Logistic Map Practice Problems assignment (Moodle link).

Grading: If you complete each problem below, you will receive a Satisfactory grade.

Need help? Review the notes from class, which are posed on the course web site. If you still need help, email Prof. Wright.

1. Exercise 3.56: In class, we found that for 0<=r<3, logistic map has a single attracting fixed point. Let (Subscript[r, 1])=3, the parameter value at which the fixed point is no longer stable. 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.

2. Exercise 3.58: For the logistic map (Subscript[f, r])(x)=r x(1-x), estimate the bifurcation values (Subscript[r, 3]), (Subscript[r, 4]), (Subscript[r, 5]) such that for any value r between (Subscript[r, k]) and (Subscript[r, k+1]), there is a single attracting (2^k)-cycle. 
That is, (Subscript[r, 3]) is the largest value r at which you find a 4-cycle, (Subscript[r, 4]) is the largst value r at which you find a 8-cycle, and (Subscript[r, 5]) is the largest value r at which you find a 16-cycle.