Mount Holyoke College

Math 125: Seminar on Fractals and Chaos

Syllabus

Lecture 1

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

Lecture 2

• Lecture: Fractals
  • The standard mathematical fractals: The Cantor set (page 93) ( Dancing Cantors by Curtis T. McMullen ), the Koch curve and variants (page 99), 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 100-102).
• Matheatical references: Peitgen FFC I 2.1, 2.4, 2.2.
• Reading: Briggs pp. 61-72; Gleick "A Geometry of Nature".

Lectures 3 and 4

• Lectures: L-systems
  • Basic definition, examples. Fibonacci L-system.
  • Turtle graphics as a programming language. Creating fractals with turtle graphics.
  • Bracketed L-systems and modeling of biological growth.
• Video: The strange new science of chaos.
• Matheatical references: Peitgen FFC II, 8.1, 8.2,8.3,8.4, 8.6; L-systems, a chapter of Dynamical Systems and Fractals Lecture Notes by David J. Wright (Oklahoma State University).

Lectures 5 and 6: 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
• Fractal Coastlines
• Reading: Gleick "A Geometry of Nature".

Lecture 7: Box dimension. (Mathematical reference: FFC I: p.240)

• BOX DIMENSION

Lecture 8: 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. 84-89); computer generated landscapes (Gleick p.95).

Lectures 9, 10 and 11

• 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). (Devaney "First Course" Ch. 3, 5.)
  • 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".
• Mathematical references: Devaney "First Course" Ch 3, 5, 7.1, 10.2; FFC 10.2.

Lectures 12 and 13: The Feigenbaum diagram

• Lecture: Period doubling, windows, almost self-similarity, the Feigenbaum constant, similarity of the Feigenbaum diagram for different functions.
• Sharkovskii Theorem. ((Devaney "First Course" Ch. 11.)
• Mathematical references: Devaney Ch. 8, FTC Ch. 11.

Lecture 14: 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"

Introductory computer laboratory

Lectures 15: 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, limit cycles.
• 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. ???; Briggs "Visualizing chaos as a strange attractor".

Lecture 16 and 17: Complex numbers.

• Arithmetics: adding, multiplying. Representation in the plane. Functions in complex variable.
• Reading: From Sec. ???.
• Mathematical references: Devaney Ch.15 (Sec. 15.1, 15.2).

Lecture 18: Iteration of functions in complex variable.

• Iteration of a linear function. Classification of orbits.
• Iteration of the squaring function.
• Reading: From Sec ???.
• Mathematical references: Devaney Ch.15 (Sec. 15.3) and Ch.16 (Sec. 16.1). <\UL>

  Lecture 19: Julia sets.

  • 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.
  • An example of the Julia set
  • Reading: From Sec. ???.
  • Mathematical references: Devaney Ch. 16.

  Lecture 20, 21 and 22: The Mandelbrot set

• Video: Transition to chaos (The orbit diagram and the Mandelbrot set).
• The Mandelbrot set and the Feigenbaum diagram.
• 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.
• 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.
  • 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.
  • 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.
• Java Julia Set Generator
• Mandelbrot and Julia Set Explorer by David E. Joyce

• Geometric features of the Mandelbrot set:
  • The boundary is a fractal, and probably has fractal dimension two.
  • 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.

• Bulbs
  • The Farey addition. The Farey tree.
  • B_p/q bulbs and their relarions to Farey tree and Julia sets.

• Reading: From Sec. ???.
• Mathematical references: Devaney Ch. 17; FCC Ch. 14.
• Video
• Guest lecture by Emily Willard