For determining the final, steady-state behavior, it is sometimes
interesting to compare several initial values. However, for *
verification* of the precise steady state, you should always use an initial
of 1/2. This is because is a ``critical value'' of the map,
or a point at which . An orbit arising from a critical value
must lie in the ``basin of attraction'' [2]
for an * attracting periodic point* [4]
(provided that it exists).

Then, a definite change in behavior occurs for the following **a** values:

(The very adventurous may wish to examine .)
You are now looking at the famous * period-doubling cascade to chaos*.
Do you see why the period of the population doubles for each new case?
Each of the above **a** values is in the * center* of a
period-multiplying region; how would you characterize each region?
The * boundaries* of these regions are specified in
Figure 1.