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 .
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.