MATHEMATICS
CHAOS AND CALCULUSSYLLABUSASSIGNMENTS
MATHEMATICS
CHAOS AND CALCULUSSYLLABUSASSIGNMENTS
Turn in individual solutions. You may work together in solving the problems, or get help from others, but the final write-up should be your own. Do not turn in anything you cannot explain completely.
Calculus p. 95 #32, 34, 37, 40, 42, 44, 47, 57, 67 (you do not
need a graphing calculator)
p100 #4, 12
p
106 #15
1. Using the website
Bifurcation Diagram Applet, locate a 5-cycle windows in
the
bifurcation diagram of the logistic map. Write down the
s value where the 5-cycle seems to
begin, and the s value where 5-cycle appears to break down
into complete chaos. Then find
a 7-cycle window and a 9-cycle window (try looking to the
left of your 5-cycle window), and
do the same for them.
Work in pairs on this assignment. Turn in one solution for you
and your partner. Make sure both of you contribute and
understand the solutions.
Calculus p. 126 #25 (do you know the significance of this for
statistics?), 26c, 27, 31, 33
p. 139 #13b, 13h
1. A slight variation of the logistic map is given by xn+1=sxn(1-xn2)
a. Is the graph of xn+1as a function of xn
symmetric about xn=1/2 (try some values on
opposite sides of 1/2 to test this)?
b. For what value of xn is
xn+1 maximum?
c. Using your answer to b to determine the maximum
value s can have and still have
the dynamics bounded between 0 and
1.
d. Make plots of xn+1as a function of xn
for x=1/2, 1, and 2, for xn between 0 and 1.
e. For what values of s does this map have a single
fixed point? What is the fixed point?
Determine whether the fixed point is
stable or unstable, and explain why.
f. For what values of s does the map have more than one
fixed point? Solve for all fixed points
as a function of s. Determine
when these fixed points are stable.
2. Play Derivative puzzles 1, 2 and 3 at the Differentiation
game website, and turn in rough sketches of the answer
grids.