Math 457, first prep assignment
April 2, 2007
In order to gain some experience with dynamical systems before plunging into the theory, each person
in class will look at iterating the following functions, first on their own and then in groups at the
beginning of class. If your last name begins with A-E, work on function(s) listed in 1 below; F-J on
2; K-O on 3; P-T on 4; U-Z on 5. Some of these will be facilitated by using the āIteration Appletā,
but donāt worry if you have to do one which requires work āby handā. The expectations and grading
(which will be fairly generous for preparation work) will be uniform.
1. Use the āIteration Appletā to look at the functions x2+c(so for example, f(x) = x2ā1.8)
and cx(1 āx) for at least five different values of cand at least three different seed values x(0).
Make the values of cvary over the whole range which the applet allows. Try to characterize the
eventual behavior in each case - e.g. āapproaching a fixed point of x=Ā·Ā·Ā·.ā
2. Use the āIteration Appletā to look at the functions ccos(x) (so for example, f(x) = 1 Ā·cos(x) =
cos(x) itself) and csin(x) for at least five different values of cand at least three different seed
values x(0). Make the values of cvary over the whole range which the applet allows. Try to
characterize the eventual behavior in each case - e.g. āapproaching a fixed point of x=Ā·Ā·Ā·.ā
3. Consider the function covered at the end of Mondayās class: f(x)=2xif x < 1
2or 2xā1 if
xā„1
2. This function is called ā2xmod 1 in the āIteration Applet.ā Try at least 12 different
values of x(0). Also try at least five different see values āby handā, including numbers like Ļā3
and ā2ā1. Finally, even though 0 is a fixed point, is it ever the case that numbers close to 0
are āattractedā by it?
4. Consider the function of positive integers defined by f(n) = n
2if nis even or 3n+ 1 if nis
odd. Determine the behavior of iterating this function for at least twenty different seed values,
iterating each at least a dozen times (if not more). Try to include at least one seed value over
100, and one over 1000. Also try a few negative numbers (like ā5).
5. Iterate the functions 1
1+xand x
2+1
x, with at least three different seed values and a dozen
iterations. What patters do you notice? Next make up at least two of your own examples of
functions of the form ax+b
cx+dand iterate them. Calculate the fixed points and try to make some
kind of general statement about iterating these functions.
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