University of Toronto
MAT335 TERM TEST 1
Friday Oct. 9, 2015
Duration: 100 minutes
Aids allowed: non programmable calculators, and a ruler
Instructions: Please answer all questions in the spaces provided (if you use back of a sheet
please clearly specify
Start with any triangle in a plane (any closed, bounded region in the plane will
actually work). The canonical Sierpinski triangle uses an equilateral triangle with a
base parallel to the horizontal axis (rst image).
Shrink the triangle to height and widt
University of Toronto, Faculty of Arts and Sciences
MAT335H1F - Chaos, Fractals, and Dynamics
Final Exam - December 9, 2013
Examiner: Bernardo Galvao-Sousa
Duration: 180 minutes.
Aids permitted: None.
Full Name:
Last
First
Student ID:
Instructions
This t
UNIVERSITY OF TORONTO
Faculty of Arts and Science
DECEMBER 2012 EXAMINATIONS
MAT335H1F
Chaos, Fractals and Dynamics
Examiner: D. Burbulla
Duration - 3 hours
Examination Aids: A Scientic Hand Calculator
Name: Student Number:
INSTRUCTIONS: All six questions
UNIVERSITY OF TORONTO
Faculty of Arts and Science
DECEMBER 2012 EXAMINATIONS
MAT335H1F Solutions
Chaos, Fractals and Dynamics
Examiner: D. Burbulla
Duration - 3 hours
Examination Aids: A Scientic Hand Calculator
INSTRUCTIONS: All six questions have equal
UNIVERSITY OF TORONTO
Faculty of Arts and Science
DECEMBER 201 1 EXAMINATIONS
MAT335H1F
Chaos, Fractals and Dynamics
Examiner: D. Burbulla
Duration 3 hours
Examination Aids: A Scientic Hand Calculator
Name: Student Numb er:
INSTRUCTIONS: All six questio
MAT335 - Chaos, Fractals, and Dynamics - Fall 2014
Solution of Term Test 1 - October 21, 2014
Time allotted: 60 minutes.
1.
Consider the function F : R R dened by
x
2(x 1)
F (x) = 0
2(x + 1)
x
Aids permitted: None.
if x
2
if 1 < x < 2
if 1
x
1
if 2 < x <
Homework Assignment #5
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
Chapter 9.5.
1
If s satises d[s, 0] = 2 , then
i=0
1
si
= ,
i
2
2
so s0 = 0. We then have two cases:
Case 1.
If s1 = 1, then si = 0, for all i = 2, 3, . . .
Case 2.
If
Homework Assignment #4
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
Chapter 7.15.
In this solution, I use the notation x (0.a1 a2 a3 . . .) to write x has the ternary expansion
(0.a1 a2 a3 . . .).
Let x = K, which has ternary expansion
Homework Assignment #2
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
5.5.
Assume that F (x0 ) = x0 , F (x0 ) = 1, and F (x0 ) > 0.
Then the graph of the function near x0 is the following.
y = F (x)
y=x
x0
From here we can conclude that
Homework Assignment #3
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
Extra Question.
(a) Let p < x0 < p+ . We prove by induction that xn > p . Assume that xn > p . Then
xn+1 = x2 + c > p2 + c = p .
n
By induction, we deduce that xn > p
Homework Assignment #6
MAT 335 Chaos, Fractals, and Dynamics Fall 2014
Partial Solution
Chapter 11.8.
Part 1 .
This question has two parts.
Show that this function has a 6-cycle.
First the function can be written as
2 + 2x
F (x) = 6 2x
4 x
Then take x0 =
What is the perimeter of the Koch snowake?
Step 0: 3 segments length 1, total perimeter=3
Step 1: 3*4 segments of length 1/3, total perimeter=3*(4/3)
Step 2: 3*4^2 segments of length 1/3^2, total perimeter=3*(4/3)^2
.
Step k: 3*4^k segments of length 1/3^
Homework Assignment #5
MAT 335 Chaos, Fractals, and Dynamics Fall 2013
Due: November 18 at 12:15pm
Late homework assignments will not be graded.
Submit your solutions to the following problems from the textbook.
Chapter 9.
1, 3, 5, 9
Chapter 10.
Justify
Homework Assignment #6
MAT 335 Chaos, Fractals, and Dynamics Fall 2013
Due: December 2 at 12:15pm
Homework assignments handed in at 12:16pm will not be graded.
Submit your solutions to the following problems from the textbook.
Chapter 11.
1, 2, 4, 6
Chapt
Homework Assignment #4
MAT 335 Chaos, Fractals, and Dynamics Fall 2013
Due: October 30
Submit your solutions to the following problems from the textbook.
Chapter 7.
7, 9, 10, 11, 12, 13, 14, 15
Chapter 8.
5 (using x0 = 0), 10 x0 =
2+ 2
4
1