Part III. A study of the Feigenbaum diagrams for the logistic family.

1)
a) Determine the period-2 bifurcation value for the logistic map, call this a2.
b) Determine the period-4 bifurcation value for the logistic map, call this a4.

2) The curves that you see in the bifurcation diagram are the attracting fixed and periodic points as a function of the parameter.
a) Find and graph the curve of fixed points for a in [0, 1].
b) Find and graph the curve of fixed points for a in [1, a2].
c) Find and graph the curves of period-2 points for a in [a2, a4].
d) Display the graphs of (a) -- (c) together on [0, a4] and compare them to the Feigenbaum diagram graphed on [0, a4].

3) By using the bifurcation diagram for the logistic map, estimate the value for the period-8 bifurcation point. Try to find the next period doubling bifurcation value after that!

Period 8 appears to be around a8 = 3.545, period-16 is about a16 = 3.565.

4) Suppose a point in an orbit hits very VERY close to a repelling fixed point. It will be repelled, but it may take many iterations before it can escape from that fixed point. Explain, in light of this fact, what causes the dark bands running through the Feigenbaum diagram.

When very close to a fixed point, f(x) is approximately x meaning that the output is close to the input. So, it takes a while to be repelled. Since iterates close to fixed points remain nearby for a while, the dark bands in the Feigenbaum diagram must represent points very close to repelling periodic and fixed points.

5) The curves that you graphed in problem 2 correspond to attracting fixed and periodic points. While they are not attracting outside of the intervals graphed in the problem, the fixed and periodic points don't just disappear. Graph all of the fixed, period-2, and period-4 points of the logistic map for a in [0, 4]. Compare this graph to the Feigenbaum diagram for the logistic map on [0, 4].

6) While the formulas for the periodic points of the logistic map may be quite messy, you should be able to tell that the the curves are well defined for arbitrarily large values of a. In particular, while they are not attracting outside of the intervals graphed in problem 2, the fixed and periodic points still remain for a > 4. Verify this by plotting the curves from the last problem on the interval [0, 5].

7) In light of the result in the last problem, any guesses as to why the Feigenbaum diagram cannot be plotted for a > 4? (You should verify that indeed the Feigenbaum diagram appears empty for all a > 4.)

While there are infinitely many repelling fixed points, most (an uncountable set of points of measure 0 remains!) of the other points diverge. So, even if you are lucky enough to choose a periodic point when a > 4, error in floating point calculations will ultimately take you off into a diverging orbit.