The last case is for the largest **a** value allowed; do you see why? How
does the distribution of points differ now from the case (near
the * beginning* of the chaotic region)? By considering two initial
points very close together (e.g., and , see if
the ``Butterfly effect'' manifests itself. (The Butterfly effect is
sensitivity to initial conditions.) If the two initial points are brought
even closer together, how many iterations are required to see differences
in the final patterns? Note: for , do * not* use .
(For , , and **0** is an * unstable*
fixed point.)

The results for all of these studies are summarized in Figure 1--3 [5].

Figure 2 shows typical plots
of vs. **n** for various values of **a**, while

Figure 3 gives a plot of the final **x** value (or values!) for a wide range of the
parameter **a**.

Figure 2 Population vs. time for the logistic map. The diagrams were created by first executing a fortran program with in (a), in (b), in (c), and in (d); a value of was used in all four. Thirty iterations were performed in each case and the results plotted with xmgr.

Figure 2: Population vs. time for the logistic map .

Click here to calculate and plot the Logistic map.

Figure 3 Bifurcation diagram for the logistic map.
The diagram was created
by first executing a fortran program with x_1 set to 0.01, the value of
**a** computed to fall within the range of 1.0 to 4.0, and approximately
900,000 points were computed and plotted with xmgr.

Figure 3: Bifurcation diagram for the logistic map.