What Happens As Increases?

First, display an applet that graphs and it's iterates by clicking the "Logistic Map" button (to the left). Then press the "Plot" button to get a graph of for = 2.5.

Observe that when is between 1 and 3 there is one unique attracting fixed point of period 1. But as increases to 3 and beyond, there are periodic points with period higher than 1. For example, try to find a periodic point of period 2 with < 3, then try x = 0.5 and = 3.2 and notice that this converges to an approximately 0.512, 0.799 period very quickly.

Now, for = 3.2, enter a 2 in the "n =" text field and look at the graph. Notice that x = 0.5 converges to a fixed point of F2 very quickly. So, increasing through 3 has given birth to a period 2 attracting point.

Next: Follow the "Graph of F2" link.