Chapter 3, Exploration 19

Explore the dynamics of the system \(x_{n+1} = f_a(x) = ax_n(1-x_n)\). Use the slider along the horizontal axis to change the initial condition. You can manipulate the value of \(a\) with the dropdown menu. Determine the values of \(a\) for which the fixed point at \(x = \frac{a-1}{a}\) is attracting, repelling, or neither. What is the “slope” of \(f_a(x)\) at the fixed point when the fixed point is just to the left of the critical point? What is the “slope” of \(f_a(x)\) at the fixed point when the fixed point is just to the right of the critical point?