Goal:
Using the orbit diagram, we have seen in the previous lab
that the quadratic function
Qc(x) = x2 + c and
the logistic function Fc(x) = cx(1-x) both
undergo a sequence of period-doubling bifurcations as the parameter
tends to the chaotic regime. We have also seen that magnifications of
the orbit diagram tend to look "the same." In this experiment, we
will see that there really is some truth to this.
Procedure: In this experiment you will work with
both of the above families of functions.
We first need a definition:
Definition. Suppose x0 is a critical point for a function F, that is, F'(x0) = 0. If x0 is also a periodic point of F with period n, then the orbit of x0 is called superstable . Note that (Fn )' (x0) = 0.
Recall that the critical point for Qc is 0. Similarly, the critical point of Fc is 1/2.
In this experiment you will first determine the c-values at which Qc and Fc. have superstable cycles of periods 1, 2, 4, 8, 16, 32, and 64. The easiest way to do this is to use the Orbit Diagram applet to determine experimentally the c-values at which the function has a superstable point of the given period. This can be done by zooming in on the appropriate branch of the curve representing cycles with the right period. By zooming in over and over and then using the ``Current Location'' feature of the lab, you should be able to determine the parameter value for which there is a superstable cycle. You should try to determine this value to five or six digits of accuracy. Recall that the parameter values are plotted horizontally on the screen and the x-coordinates along the orbit are plotted vertically. Once you have found the appropriate parameter value, you may check your results using the list feature of the Function Iterator or Lab 1. Simply use the critical point as the initial seed for the iteration and enter the desired number of iterations. If your parameter value is correct, you should see that the critical point is very close to being periodic with the right prime period. You can try to get better accuracy by changing the parameter slightly to try to make the critical point exactly periodic (up to the accuracy of the computer). Note that the software stores more digits for the parameter value than it displays. After finding the seven c-values for your function, record these numbers in tabular form:
c0 = ____________________ = c-value for period 20
c1 = ____________________ = c-value for period 21
c2 = ____________________ = c-value for period 22
c3 = ____________________ = c-value for period 23
c4 = ____________________ = c-value for period 24
c5 = ____________________ = c-value for period 25
c6 = ____________________ = c-value for period 26
Now use a calculator to compute the following ratios:
f0 = (c0 - c1)/(c1 - c2) = ______________________
f1 = (c1 - c2)/(c2 - c3) = ______________________
f2 = (c2 - c3)/(c3 - c4) = ______________________
f3 = (c3 - c4)/(c4 - c5) = ______________________
f4 = (c4 - c5)/(c5 - c6 ) = ______________________
List these numbers in tabular form, too. Do you notice any convergence? You should, at least if you have carried out the above search to enough decimal places.