Mount Holyoke College

Math 125: Seminar on Fractals and Chaos

(Idealized) Syllabus

Note: Each "lecture" is 75 minutes long. The books cited can be found in the bibliography.

There are weekly homework assignments. There are also computer laboratories using the software designed by Devaney.

Lecture 1

• Lecture: Introduction
  • Mathematical fractals (the Koch curve, the Mandelbrot set)
  • Fractals in nature (pictures from Gleick, Briggs)
  • Mathematical chaos (the Lorenz attractor)
  • Chaos in nature (the whorling dolphins)
• Reading: Gleick: "Prologue" and "The butterfly effect".

Lectures 2 and 3

• Lectures: Fractals
  • The standard mathematical fractals: The Koch curve and variants, the Sierpinski triangle and carpet, the sponge. The Koch curve is infinitely long. The carpet can be picked up by one corner.
  • 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...
  • The Cantor set (Gleick p. 93)
• Matheatical references: Peitgen FFC I 2.4, 2.2.
• Handouts: Picture of tetrahedron, picture of pulpit designs in Ravello cathedral.
• 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).
  • 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 program):
    • 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: Gleick "Life's ups and downs". (This chapter has a lot of material; we'll get to most of it over the course of the semester.)
• Mathematical references: Devaney "First Course" Ch 3, 5, 7.1, 10.2; FFC 10.2.

Lectures 7 and 8: 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 as in Homework 4. 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.
• Reading: Gleick, "A geometry of nature"
• Handout: Richardson's measurement of the length of the coast of Great Britain.

Introductory computer laboratory

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"

Lecture 10: Box dimension. (Mathematical reference: FFC I: p.240)

Lecture 11: 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 self-similar).
• 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 12 and 13: The Feigenbaum diagram

• Lecture: Period doubling, windows, almost self-similarity, the Feigenbaum number, similarity of the Feigenbaum diagram for different functions.
• Mathematical references: Devaney Ch. 8, FTC Ch. 11.

Lectures 14 and 15: Continuous dynamical systems and strange attractors

• Lecture:
  • Discrete and continuous dynamical systems. Phase space. The motion of a weight on a spring. Mathematical modelling.
  • 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.
  • The "flying dolphins toy": This is a very simple mechanical system, but the motion appears to be irregular. Hence it is probably chaotic motion. If we could analyse it mathematically, and make a phase space, we would probably see a strange attractor. We tried testing for sensitive dependence on initial conditions by proping the rotator up with a floppy disk and letting it go.
• Reading: Gleick "Strange attractors"; Briggs "Visualizing chaos as a strange attractor".
• Handout: The Roessler attractor.

Lecture 16 and 17: Julia sets.

• Complex numbers (adding, multiplying, representation in the plane), iteration of complex numbers.
• Some new terminology for iteration:
  • The basin of attraction of an attracting fixed point or cycle is the set of points which get attracted to the fixed point or cycle.
  • The points which go off to infinity are called escapees; they are in the basin of attraction for infinity. Points which escape are usually colored red, or some bright color which depends on how fast they escape.
  • Points which don't escape are called prisoners . The prisoners are usually colored black. Most of the black points are in some basin of attraction, and get attracted to a fixed point or cycle. The set of prisoners is also called the filled-in Julia set.
  • The set of points on the boundary of the red and black sets is called the Julia set . The points of the Julia set move around chaotically (but we won't be able to see this).
• Various shapes for (filled-in) Julia sets for the quadratic functions x^2 + c:
  • A disk. This only occurs for c = 0. The origin is an attracting fixed point. The points in the disk go to the origin. The points outside the disk escape. The points on the boundary of the disk move around chaotically.
  • A fractal version of a disk. (Example: c = -.5 + .5i). The boundary is infinitely jagged, and has some properties of self-similarity.
  • An infinite number of fractal disks joined together. (Example: c = -1).
  • Dust. (Example: c = .5) There are no attracting fixed points, just repelling ones. The points on the dust move around chaotically. The dust looks self-similar and fractal-like.
  • Dendrite. (Example: c = i) This looks like lightning. It also looks self-similar and fractal-like.
  • A line segment. This only occurs for c = -2.
• Reading: Gleick "Images of chaos"
• Mathematical references: Devaney Ch. 16; FCC Ch. 13.

Lecture 18: The Mandelbrot set

• Geometric features of the Mandelbrot set:
  • The boundary is a fractal, and probably has fractal dimension two.
  • It is connected (It is possible to walk on it from any point to any other point.)
  • It has some properties of self-similarity: the qualitative aspects of features are repeated over and over.
  • It is not exactly self-similar, since every point looks different. For example, the number of spokes on the tentacles on the buds varies (If one bud has p spokes and another has q spokes, then the largest bud between them has p+q spokes).
  • There are little copies of itself ("Mandelbroties") everywhere. These are more hairy than the main Mandelbrot set.
• The Mandelbrot set is a "roadmap of Julia sets": Each point of the plane with the Mandelbrot set corresponds to a Julia set. (The point c of the Mandelbrot set corresponds to the Julia set of the function z^2 + c.) The Julia set often resembles the Mandelbrot set near the point c (see handout).
  • Julia sets from the main cardioid of the Mandelbrot set are funny disks.
  • Julia sets from the buds on the Mandelbrot set are collections of disks. (See Homework 12)
  • Julia sets from the tentacles of the Mandelbrot set are dendrites.
  • Julia sets from a point in a "Mandelbrotie" are fat dendrities.
  • Julia sets froma point outside the Mandelbrot set are dust.
• Many times a Julia set can look like the point in the Mandelbrot set it came from (handout).
• The Mandelbrot set is the set of c such that the corresponding Julia set is connected. Equivalently: A point c is in the Mandelbrot set exactly when 0 is a prisoner for the function x^2 + c.
• Mathematical references: Devaney Ch. 17; FCC Ch. 14.

Lectures 19 and 20: Odds and ends

• Guest lecture by Lewis Thayne from the Development Office at MHC. Fractals and chaos as organizational principles. Examples: The Stop and Shop Supermarket, and the network of the Development Office.
• Pendulum and magnets demonstration. If the pendulum starts at some point, it ends up at a magnet. The starting point is in the basin of attraction for that magnet. The Chaos program has a computer simulation of the same thing. The basins of attraction are fractals.
• Basins of attraction are usually fractals. Examples:
  • Julia sets
  • The pendulum and magnets toy
  • Newton's method for finding roots of equations (Briggs p. 149-150)
• EEG recordings of brain waves are possibly strange attractors; the fractal dimension may have a use here (handout).
• Ecologists find that in addition to the "balance of nature" there is also a certain amount of chaos (handout on grass experiment).
• Ian Malcolm claims that Hammond's island will fail because of chaos theory (Jurassic park handout).
• Video of Julia sets and zooms into the Mandelbrot set.