Chapter 6, Application 9
Use the graph below of the function \(x_{n+1} = f_a(x_n) = ax_n(1 − x_n)\) for values of \(a \in (1,3)\). Use this graph to repeat applications 2-8 in Chapter 6 of the text. The questions are repeated here.
- What are the fixed points and what are their stabilities?
- Describe the orbits of all initial conditions that are less than the smallest of the two fixed points.
- Describe the orbits of all initial conditions that are between the two fixed points.
- On your graph, locate the two points that are eventually fixed points. There is one that lands on each of the two fixed points.
- Describe the orbits of all initial conditions that are between the fixed point \(x=0\) and the eventually fixed point \(x=1.\)
- Describe the orbits of all initial conditions that are between the eventually fixed point \(x=1\) and the other eventually fixed point.
- Describe the orbits of all initial conditions that are greater than the largest of the two eventually fixed points.
a: