Mount Holyoke College
Math 125: Seminar on Fractals and Chaos
- 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".
- 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
- 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
- Reading: Gleick "A Geometry of Nature".
Lecture 7: Box dimension. (Mathematical reference: FFC I: p.240)
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
- 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
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
- 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,
- 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.