MA1251 Chaos & Fractals
2002/2003, Semester 2, Wks: 21-26
Convenor: Dr Ruslan L Davidchack
Official
module description
MOTIVATION
: Occasionally he [the scientist] would straighten up and say things
like 'Hurrah, I've discovered Boyle's Third Law.' And everyone knew
where they stood. But the trouble was that ignorance became more
interesting, especially big fascinating ignorance about huge and important
things like matter and creation, and people stopped patiently building
their little houses of rational sticks in the chaos of the universe and
started getting interested in the chaos itself - partly because it was a
lot easier to be an expert on chaos, but mostly because it made really good
patterns that you could put on a t-shirt. - Terry Pratchett
from The Chaos Hypertextbook by Glenn Elert
TIMETABLE:
| WHAT |
WHEN |
WHERE |
| Lecture |
Mon 15.30 - 16.30 |
BEN LT4 |
| Computer Lab |
Tue 9.30 - 11.30 |
BEN F75b |
| Problem Class |
Thu 11.30 - 12.30 |
PHY LTC |
LECTURES:
Lecture 1: Dynamical systems. Orbits. Fixed and periodic points.
Lecture 2: Graphical analysis. The logistic function. Stable and unstable
orbits.
Lecture 3: Chaotic orbits. Sensitive dependence on initial conditions.
Lyapunov exponent.
Note: Problem
Set 1 is due on Tuesday, May 6th, 11:30am
Lecture 4: Period doubling bifurcation. Bifurcation diagrams.
Note: Problem
Set 2 is due on Tuesday, May 13th, 11:30am
Lecture 5: Self-similarity and fractals. Cantor sets. Constructing fractals.
Fractal dimension.
Note: Problem Set 3
is due on Tuesday, May 20th, 11:30am
Lecture 6: Julia sets. Parameter spaces and Mandelbrot sets.
Fractal basins of the Newton-Raphson method.
LABORATORIES:
Lab 1: Introduction to Matlab Due: Thursday, March 20th, 11:30am
Lab 2: Maps and Orbits
Due: Thursday, May 1st, 11:30am
Lab 3: Bifurcation diagrams Due: Thursday, May 8th, 11:30am
Lab 4: Lyapunov exponent Due: Thursday, May 15th, 11:30am
Lab 5: Fractal basins Due:
Thursday, May 22th, 11:30am
ASSESSMENT:
There is no examination for this module. The module assessment
will be as follows:
- 30% for coursework problems: there will be 3 problem sets,
testing student's understanding of the basic concepts and ability to
analyse simple cases of dynamical systems, where computer is not necessary.
The solutions to the problem sets will be discussed during the Problem
Class.
- 70% for lab reports: Lab reports will usually consist
of the Matlab programs computing various objects used in the study of
dynamical systems together with the output of these programs.
TEXTS:
- H.-O. Peitgen, P.H. Richter, The beauty of fractals:
images of complex dynamical systems
- Heinz-Otto Peitgen, Hartmut Jurgens, Dietmar Saupe, Chaos
and fractals : new frontiers of science
- Robert L. Devaney, Chaos, fractals and dynamics: computer
experiments in mathematics
- Jens Feder, Fractals
- Paul S. Addison, Fractals and chaos: an illustrated
course
- Michael Barnsley, Fractals everywhere
- Kathleen T. Alligood, Tim D. Sauer, James A. Yorke, Chaos:
an introduction to dynamical systems
- Joseph L. McCauley, An introduction to nonlinear dynamics
and chaos theory
WWW:
- B. Fraser, The Non-Linear Lab, http://www.apmaths.uwo.ca/~bfraser/version1/nonlinearlab.html
- Glenn Elert, The Chaos Hypertextbook, http://hypertextbook.com/chaos/
- Larry D. Bradley, Chaos and Fractals, http://www.pha.jhu.edu/~ldb/seminar/
- Manus J. Donahue III, Chaos Theory, http://www.duke.edu/~mjd/chaos/chaos.html
- The MathWorks, Matlab Documentation, http://www.mathworks.com/access/helpdesk/help/techdoc/matlab.shtml