Homework 2
Math 323, Fall 2012
Due Date: Thursday, September 20
For the following problems, let f : [0, 1] [0, 1] be the logistic map f (x) = cx(1 x), where
c is a constant in the interval [0, 4].
1. Write Mathematica code that draws cobweb plots for the
Homework 1
Math 323, Fall 2012
The goal of this homework assignment is just to get you started with using the Mathematica
computer algebra system to analyze dynamical systems.
Due Date: Thursday, September 13
Instructions: Feel free to work together with
Homework 3
Math 323, Fall 2012
Due Date: Thursday, September 27
1. Let f : [0, 1] [0, 1] be the function f (x) = 4x(1 x).
(a) Find the endpoints of the LRR interval (shown in Figure 1.12 of the textbook).
Your answers must be correct to four decimal place
Homework 5
Math 323, Fall 2012
Due Date: Thursday, October 25
Let f : [0, 8] [0, 8] be the following piecewise-linear function:
8
7
6
5
4
f (x) =
if 0 x < 1
if 1 x < 2
if 2 x < 4
12 x if 4 x < 6
30 4x if 6 x < 7
16 2x if 7 x 8.
3
2
1
0
2x
4x 2
x + 4
0
1
2
Takehome Final
Math 323, Fall 2012
Due Date: Friday, December 21
Rules: This is a nal exam, not a homework assignment. You must solve the problems
entirely on your own, and you should not discuss the problems with any other students
in the class, or with
Takehome Midterm
Math 323, Fall 2012
Due Date: Saturday, October 20
Rules: This is a midterm exam, not a homework assignment. You must solve the problems
entirely on your own, and you should not discuss the problems with any other students
in the class, o
Homework 7
Math 323, Fall 2012
Due Date: Thursday, November 8
1. Let f : [0, 1] [0, 1] be the function f (x) = cx(1 x), where c is the smallest positive
value for which 1/2 is periodic with period 7.
(a) Determine the value of c. Your answer must be corre
Homework 6
Math 323, Fall 2012
Due Date: Thursday, November 1
1. Let f : [0, 1] [0, 1] be the following piecewise-linear function:
3x
if 0 x 1/3,
f (x) = 2 3x if 1/3 < x 2/3,
3x 2 if 2/3 < x 1.
Find a cubic polynomial g : [0, 1] [0, 1] and a homeomorphism
Homework 9
Math 323, Fall 2012
Due Date: Thursday, November 29
1. Consider the following iterated function system on the square [0, 1]2 :
PP
P
P
P
Find formulas for the four maps of this system, and use Mathematica to plot 10,000
points in a typical orbit
Homework 4
Math 323, Fall 2012
Due Date: Thursday, October 4
1. Use Mathematica to draw a bifurcation picture for the functions f (x) = sin(kx), where
2 k . I recommend the following approach:
Start by creating a list of values for k in the interval [2,
Homework 8
Math 323, Fall 2012
Due Date: Thursday, November 15
The following pictures show the rst four steps in the construction of a self-similar fractal
curve C in the plane:
0,0
1,0
Stage 1
Stage 2
Stage 3
Stage 4
Note: The six marked points are (0, 0
Homework 10
Math 323, Fall 2012
Due Date: Thursday, December 6
Consider the fractal subset of [0, 1]2 dened by the following iterated function system:
F1 (x, y) =
1 1
x, y
2 2
F2 (x, y) =
1 1
1
x+ , y
2
2 2
F3 (x, y) =
1 1
1
x, y +
2 2
2
F4 (x, y) =
1 1
1