MAT335 - Chaos, Fractals and Dynamics
Department of Mathematics, University of Toronto
R. Pyke (office: at the back of the math aid centre, SS1071B; email: email@example.com)
Lectures: Monday and Wednesday, 8:45-10:00 a.m. in SS 2127.
Office hours: Monday and Wednesday; 10:10-11:30 a.m. in SS1071 (and by appointment).
Text: Chaos and Fractals: New Frontiers of Science,
H.-O. Peitgen, H. Jurgens, D. Saupe. Springer-Verlag, 1992.
Course web page: http://www.math.toronto.edu/courses/335
Please check the course web page regularly:
You will find comments about the course material, homework problems,
computer programs that we will use during the course.
References (The following books are on reserve in the
Gerstein Science Information
Centre, short term loan section):
You may also want to browse through sections Q 172.5, QA 447,
and QA 614.8 in
library, which contain books on the topics
of fractals and chaos, as well as applications.
- Chaos and Fractals: New Frontiers of Science, H.-O. Peitgen,
H. Jurgens, D. Saupe. (A copy of this book is also on reserve in the
mathematics library, SS622)
- The following three books are standard introductory texts for
courses such as this one. Refer to them if you want to see a more
mathematical and concise treatment than is given in
our text, as well as more examples
and a supply of exercises to help you think about the material.
- A First Course in Chaotic Dynamical Systems, R. Devaney.
- Exploring Chaos, B. Davies.
- Encounters with Chaos, D. Gulick.
Some popular books
For a general discussion about the history behind 'chaos theory', the
scientific ideas and the people involved, I would recommend the following
- Chaos: Making a New Science, J. Gleick. (A journalist's
lucid description of the main discoveries and contributors to
'chaos theory'. No mathematics here though.)
- Does God Play Dice?, I. Stewart. (A mathematician's
(entertaining) survey of the historical development of the ideas that
lead to 'chaos theory', along with accessible explanations of the
The Audio Visual library has several
videos about chaos and fractals (just type the key words 'chaos' or 'fractal'
on the on-line catalogue). We will watch one video, Fractals: An
Animated Discussion, call #002948, at some point in the course.
There are many web sites devoted to chaos and fractals, so there is much
to explore here (eg., 'text book' descriptions, pictures,
applications in science, fractal music, etc).
Note that many universities have web sites describing the research
of 'dynamical systems' or 'nonlinear dynamics'
groups working in mathematics, physics, chemistry,
biology, computer science, and medicine.
You can start with the links listed in the resources
section of the course
web page (the 'Internet Resources' appendix of the Hypertextbook
on Chaos website lists many, many links).
Students should have second year calculus (can be taken
concurrently) and a course in linear algebra. Differential equations and
complex numbers will come up, but you are not expected to have
studied these before.
Students may be able to do a term project in lieu of writing the final exam.
Those interested in doing a project
should talk to me about possible topics
before the end of February and present to me a written outline of the project by
I will then discuss the project with the student and once we have reached a
satisfactory outline of the
content and goal of the project, the student
may proceed with the project.
The project should consist of a written report of 10-15 pages. Students
doing the term project will also have to make a short (10-15 minutes) presentation to myself.
could also contain
a computer program demonstrating the topic - the computer program is
not a necessary component of the project, however.
The project could be, for example, a topic in the text that we didn't
cover in class, an 'experiment' using a computer program, or some topic
related to your studies in another course (some possible topics are listed
in the "suggestions for term projects"
link on the course webpage; see
the "student projects" link on the course webpage
for previous year's projects).
The due date for the term project is the same date as the final exam for the course.
- Final exam : 40% of final mark
- 2 term tests (50 min. each) : 20% of final mark. (Tentatively: February 26 and March 26)
- Homework : 40% of final mark
Homework Homework questions will be posted incrementally on the web page (new
questions will be added to the homework as the course progresses; check for updates
regularly). They will be handed in for marking approximately every two weeks.
Students will be allowed to work individually or in
groups of 2 for the homework, but otherwise students are expected to
work independently. Students who are working on the homework together will submit one
solution set with the names of the group members written on it. Due to limits on the
marker's time, only some of the homework questions will be marked (the ones to be marked will
not be announced). Students will, however, receive credit for attempting the questions
that were not marked.
- Part I - Fractals
- Self-similarity, examples of classical fractals (Ch 2,3)
- Fractal dimension (Ch 4)
- Drawing fractals: Iterated Function Systems (Ch 5) and
the Chaos Game (Ch 6)
- Fractals in nature: plants, landscapes, random fractals (Ch 7,9)
- Part II - Dynamics and Chaos
- Discrete dynamical systems (iteration of functions) in one and
two dimensions: random vs deterministic systems, graphical analysis,
invariant sets, attracting sets, fixed points, periodic points,
analyzing deterministic chaos (Ch 10)
- Symbolic dynamics
- Charkovsky's Theorem. Bifurcations and the route to chaos.
Fiegenbaum final state diagram. (Ch 11)
- Strange attractors, continuous dynamical systems (differential
equations) (Ch 12)
- Part III - Complex Dynamics
- Complex numbers (section 13.2)
- Julia sets (Ch 13)
- The Mandelbrot set (Ch 14)