Mount Holyoke College
Math 125: Seminar on Fractals and Chaos
The course was established by Prof. Alan Durfee. See
his Home Page.
Instructor: Sergei Chmutov
- Office: Clapp 423
- Telephone x2720
- email: schmutov
- Office hours: Tuesday, Thursday 6:00-8:00 pm or by appointment
Lectures: Tuesday, Thursday 10:50-12:05 in Cleveland L1.
There are weekly homework assignments.
Basic book: Ian Stewart, Does God Play Dice? (Blackwell 1989).
Other books cited can be found in the bibliography.
Technology: You will need a calculator for this course, one that
does logarithms. We will also be using a computer program called "Chaotic
dynamical systems software". This program runs on the Mac and is
available in the computer labs.
Quizzes, Tests: There will be two tests, a self-scheduled final
examination, and perhaps some quizzes (which will be announced in
Grading: Each test will count 20%, the final examination will count
25%, and the homework will count 35%.
List of students
Useful web site:
Dynamical Systems and Fractals Lecture Notes by
David J. Wright
(Oklahoma State University).
- Lecture: Introduction
- Fractals in nature (pictures from Gleick, Briggs). Sec.11, page
- Mathematical fractals (the Koch curve and other). Sec.11,
- Chaos in nature (the whorling dolphins). Sec.7,page 127-144.
- Mathematical chaos (the Lorenz attractor). Sec.7,page 127-144.
- Reading: Sec. 1 "Chaos from Order", page 5-22.
Lectures 2 and 3
- Lectures: Fractals
- The standard mathematical fractals: The Cantor set (page121-124)
Dancing Cantors by
Curtis T. McMullen ),
the Koch curve and variants (page 217-219),
the Sierpinski triangle and carpet, the
sponge. The Cantor set is of length zero.
The Koch curve is infinitely long.
- Self-similar fractals, fractal dimension. A set is
self-similar if it is made up of N smaller copies of itself, all
reduced by the same scale factor r. The fractal (self-similarity)
dimension is D = (log N)/(log (1/r)). The fractal dimension
intuitively measures how much the set fills up space.
The fractal dimension of the
Koch curve is 1.26... (page 219-220).
- Video: The strange new science of chaos.
- Matheatical references: Peitgen FFC I 2.1, 2.4, 2.2.
- Reading: Briggs pp. 61-72
Lectures 4, 5 and 6
- Lectures: Iteration
- Introduction to iteration. Make table showing initial
value and its eventual behavior; summarize behavior on a number line.
Terminology: initial value, orbit,
fixed point (attracting, repelling, neither), k-cycle (attracting,
repelling, neither). (Sec. "Calculator Chaos", page 17-21.)
- Find fixed points algebraically by solving an equation
(and using the quadratic formula if necessary).
The difference between a mathematically exact value for a repelling fixed point and a decimal
approximation on the calculator (which may eventually get repelled and go elsewhere).
- Three graphical representations of iteration (We will see the second two in the computer
- The number line, with arrows showing where points go
- A graph with the the steps on the horizontal axis and the values of
the function on the vertical axis.
- A histogram
- Iteration of x^2 - 2, and introduction to chaos. Characteristics:
- Points travel all around in no particular pattern
- Sensitive dependence on initial conditions
- The sequence of points is deterministic (not random)
- Reading: From Sec. "Recipe for Chaos", page 155-164.
- Mathematical references: Devaney "First
Course" Ch 3, 5, 7.1, 10.2; FFC 10.2.
Lectures 7 and 8: The Feigenbaum diagram
- Lecture: Period doubling, windows,
almost self-similarity, the Feigenbaum constant, similarity of the
Feigenbaum diagram for different functions.
(Sec. 10 "Fig-trees and Feigenvalues", page 193-208.)
- Sharkovskii Theorem. (Sec. "Order admit Chaos", page 159-161)
- Mathematical references: Devaney Ch. 8, FTC Ch. 11.
Lecture 9: Dectecting whether iteration is regular or chaotic:
- Regular motion is characterized by attracting fixed points or cycles.
These can be dectected by looking at the list of iterates, or by looking
at the histogram, which will have spikes, or the chart, which will look periodic.
- Chaotic iteration is characterized by sensitive dependence on
initial conditions (use the multilist to check what happens to initial
values which are close together), and histograms which have solid areas
(the points move all over).
Video "Chaos, fractals and dynamics"
Lectures 10 and 11: Coastlines and the ruler dimension
- Lecture: Some natural objects like coastlines are jagged over
many scales, but are not self-similar. Measure the jaggedness by
seeing how fast the length L grows as the length c of the ruler
decreases. Assume that L grows as a power d of 1/c. Estimate d by
estimating a slope. The compass or ruler dimension
is by definition 1 + d. For self-similar fractal curves, the compass
dimension is the same as the (self-similarity) fractal dimension.
FRACTALS AND SCALE
- Reading: From Sec. 11, "The Texture of Reality", page 215-234.
Lecture 12: Box dimension. (Mathematical reference: FFC I: p.240)
Introductory computer laboratory
Lecture 13: More on fractals
- Random fractals: Random Koch curve (not self-similar,
but fractal dimension (ruler dimension) can be computed, and is the
same as that of the usual Koch curve); random triangle (not
- Fractal forgeries: Random triangle, after many iterations, looks
like a rock formation (McGuire p. 24). Computer generated mountain
ranges (Briggs p. 84-92); computer generated landscapes (Gleick p.95).
Lectures 14: Continuous dynamical systems and strange attractors
- Discrete and continuous dynamical systems. Phase space. The motion of
a pendulum. Mathematical modelling.
- Atractors of typical 2-dimensional systems. Nodes, saddles, focuses,
- Strange attractors: An attractor in phase space which is not a
point or a cycle and is not easily recognizable is a strange
attractor. Strange attractors have a fractal structure. The motion
of points on a strange attractor is chaotic. Example: The Lorenz
attractor. Demo with the "chaos" program.
- Reading: Sec. "Strange attractors", page 95-125; Briggs "Visualizing
chaos as a strange attractor".
Lecture 15 and 16: Complex numbers.
- Arithmetics: adding, multiplying. Representation
in the plane. Functions in complex variable.
- Reading: From Sec. 11, "The Texture of Reality", page 234.
- Mathematical references: Devaney Ch.15 (Sec. 15.1, 15.2).
Lecture 17: Iteration of functions in complex variable.