Solution to Question #1

Questions on One-Dimensional Dynamical Systems
Questions on Part 3:

ITERATES VERSUS TIME

Question 4
- Use the Java Applet Iterates versus time for the Logistic map to see how the plot changes as lambda varies. Describe what you see.
  
  aaaaaaaaaaaaaarrrrrrrrrgggggggghhhhhhhhhhh!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
  (more to come...what follows is a template for the rest of this document)
  p = 0,1,-1.
- What is the relation between the fixed points of the square function and the intersection points of the graph of the square function with the special lines y = x and y = -x?
  
  The square function and the special lines y=x and y=-x intersect at the fixed points of the square function.

Question 2
- Consider the square function on your calculator. Pick a point "near" (distance less than 0.1 is near enough) a fixed point. Keep iterating (pushing the squaring button) until you see a pattern for the long term behavior. Try again with other numbers near each of the fixed points. Write down what happens for each choice.
  
  Any number that was greater than 1 or less than -1 tended towards infinity or negative infinity, respectively, for the long term behavior. Any number greater than -1 but less than 1 (that is, between -1 and 1) converged to 0.
- Does it matter if the number you pick is less than or greater than the fixed point?
  
  Yes. As stated above, any number less than the fixed point -1 will diverge, as will any number greater than the fixed point 1. Any number in between the open interval of -1 and 1 will converge to the fixed point 0.

Question 3
- Which of the fixed points for the square function are attracting? Which are repelling?
  Attracting: p=0
  Repelling: p=1,-1