Prerequisites: Three or four years of high-school mathematics.
This course is different from currently-published textbooks in that it covers both the mathematical aspects of fractals and chaos as well as real-world applications. (Developed by AD 1994-97)
Here are four famous images associated with the subject of fractals and chaos. One of the main goals of the course is to understand these images.
The Koch curve is one of the simplest fractals. It is
actually the limit of an infinite process (Start with a line segment,
remove the middle third, replace it by the top two sides of a triangle,
remove the middle thirds of each of those four line segments, replace
them by the top two sides of a triangle, etc.)
It is self-similar in the sense that any piece of it looks like
the whole. It has infinite length.
Its fractal dimension
is 1.26..., which says that it is between a curve (dimension 1) and a
surface (dimension 2).
The Feigenbaum diagram
shows the long-term behavior of iterating the simple
function x2+c for various values of c.
(This is the same function whose graph is a parabola!)
The Feigenbaum diagram is a fractal.
It and the Feigenbaum number 4.66... occur in many situations
in the natural sciences.
(Image from Devaney)
The Mandelbrot set has been called one of the most complicated
images in mathematics, yet it also is generated using the function
x2+c.
Despite its complexity, though, it also has a fair amount of
regularity, as can been seen upon close examination.
There are many unanswered questions about the Mandelbrot set that are
the subject of current mathematical research.
(Image from Devaney)
The Lorenz attractor is an image of chaos.
It is the trace of a point travelling in three-space around two lobes,
seemingly changing sides at random.
It occurs in a simple model for the weather and offers an explaination
(the butterfly effect )
as to why the weather is so hard to predict.
It is one of the simplest strange attractors.
(Image from Stewart)