Part IV. A study of the Feigenbaum diagrams for the quadratic family.
Part IV. A study of the Feigenbaum diagrams for the quadratic family.
1) Write a program that draws a Feigenbaum diagram for the quadratic family Q(x) = x^2 + b. Draw the diagram for -2 <= b <= 1/4 and -2 <= x <= 2.
2)
3) The curves that you see in the bifurcation diagram are the attracting fixed and periodic points as a function of the parameter.
4) The curves that you graphed in problem 3 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 and period-2 points of the quadratic map for b in [-2, 1/4]. Compare this graph to the Feigenbaum diagram for the quadratic map on [-2, 1/4].
5) While the formulas for the periodic points of the quadratic map may be quite messy, you should be able to tell that the the curves are well defined for arbitrarily large values of b. In particular, while they are not attracting outside of the intervals graphed in problem 3, the fixed and periodic points still remain for b < -2. Verify this by plotting the curves from the last problem on the interval [-3, 1/4].
6) In light of the result in number 5, any guesses as to why the Feigenbaum diagram cannot be plotted for b < -2? (You should verify that indeed the Feigenbaum diagram appears empty for all b < -2.)
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 b < -2, error in floating point calculations will ultimately take you off into a diverging orbit.