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2 Simple Maps     continued...

Again, how many iterations are required, and is there ``sensitivity to initial conditions''? Check the convergence if you perform the calculation closer to a boundary (where a ``bifurcation'' takes place); the behavior here should make it clear what occurs at the bifurcation points. (Note: close to the boundary, it is especially important to use .)

Figure 1: Major bifurcation points and chaotic boundaries for the logistic map.

Next, consider the case:

which is in the chaotic region. How does this case compare with the previous ones and what can you say about chaos here? Notice, however, that the points are distributed only within certain regions. Also, what can you say about sensitivity to initial conditions? Within the chaotic region, there are very short ``windows'' of regular behavior (periodicity) which open up. The largest of these, a period-tripling region, begins at and ends at . Thus, examine the cases:

How would you characterize the behavior for these cases and how are they related? You might want to see if you can find the very small regular, period = 5 window for . Finally, consider the two cases: