Elements of Mathematics: General Topology, Pt.2 by Nicolas Bourbaki

By Nicolas Bourbaki

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In order for the sequence of iterates to stay within the range, f(x) :::; 1, the maximum value possible for a is 4. As a approaches 4, this surprising, chaotic phenomenon occurs. 2. 11A The programmable calculator provides a quick way to numerically iterate the function f(x) = ax(1 - x). In this activity, we use a program to study the issues of predictable and unpredictable behavior in the iteration. We will see how small errors in the graphical or numerical evaluation of the function mayor may not lead to chaos.

Thus, the small drawing errors in the function f(x) = ax(1 - x) will be most severe when the parameter a is the greatest. In order for the sequence of iterates to stay within the range, f(x) :::; 1, the maximum value possible for a is 4. As a approaches 4, this surprising, chaotic phenomenon occurs. 2. 11A The programmable calculator provides a quick way to numerically iterate the function f(x) = ax(1 - x). In this activity, we use a program to study the issues of predictable and unpredictable behavior in the iteration.

Other results occur when a < 1 or a > 4. 1. 5. Where is the attractor? y : I : : -----t---~--1--- . 5 --L----i---- : I I I 2. __-l+- x for all the points in the interval 1 < Xo < 2· -1 essentially the same as that shown? • What about the pOints in the two intervals, -1 < Xo < 0 and 0 < Xo < 1? What are . . the iteration behaviors for the individual points -1, 0, and 1? y 3. 9. Describe the long term behavior. Is there an attractor or can a point in the interval o S Xo S 1 escape to negative infinity?

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