Extra Credit Problem Set L (logistics equation)

The systems administrator has written a program which will iterate the
logistics equation

       u(n+1) = (\rho) u(n)(1 - u(n)).

Here \rho is the Greek letter rho which is commonly used as a parameter (as
in your text).

To see the output at \rho = 3.2, type

logistic1(3.2)

in the command window.  To see the bifurcation diagram (a very inelegant
version) type

logistic

.


Experiment and find the bifurcation points:

from one stable limit point to a periodic point of period two

from the points of period two to four

from four to eight

from eight to sixteen

and give an estimate for the onset of chaos, where the set of limit points
is which appear for large n is no longer a set of points of order 2^n.

Print out an example of a choice of \rho for which the set of limit points is
eight points of period eight, and write your estimates for the befurcation
points on this single sheet of paper.