Chapter 5, Exploration 28

The exploration here shows the the logistic family of functions for values of \(a \in [2.5, 3.3]\). Use the tool to describe what happens to the dynamics as \(a\) increases from \(a = 2.5\) to \(a = 3.3.\) In particular, what happens to the stability of the fixed point and why? Describe the dynamics near the fixed point for values of \(a\) less than 3 and for values of \(a\) greater than 3. The initial condition for this exploration is set to \(x_0 = \frac{1}{2}\).