Chaos The Iteration of Functions

1
Chaos The Iteration of Functions

• Let f(x) sqrt(x)
• Let the initial value x0 256
• Define xn1 sqrt(xn)
• Applying this iterative definition, what is
  happening to the answers?
• Let x0 .5. Applying the iterative definition,
  what is happening now?

2
Chaos The Iteration of Functions

• Let f(x) cos(x)
• Let the initial value x0 1
• Define xn1 cos(xn)
• Applying this iterative definition, what is
  happening to the answers?

3
Chaos The Iteration of Functions

• The list of answers during the iterative
  process is called the orbit. If the orbit never
  changes from the initial value, then that value
  is called a fixed point.
• A fixed point for sqrt(x) is 1
• A fixed point for cos(x) is .739085

4
Chaos The Iteration of Functions

• Using the Logistic Model from Ecology
• Pn1 cPn(1 Pn),
• generate the 1st 27 iterations for c 3.2 and
  using any initial P0, 0lt P0lt 1
• What do you notice?

For c 3.2, this has an orbit of period 2
5
Chaos The Iteration of Functions

• Using the Logistic Model from Ecology
• Pn1 cPn(1 Pn)
• Generate the 1st 50 iterations for the following
  c values
• C .5, .8, 1, 1.5, 2, 3, 3.5, 3.55, 3.83, 4, 5
• using any initial P0, 0lt P0lt 1
• Record and turn in your findings