A Primer on Chaos and Fractals Bruce Kessler Western Kentucky University as a prelude to Arcadia at Lipscomb University

The ideas of chaos theory and fractal geometry are some of the newest ideas in all of mathematics. Their study has been enabled by the use of the computer, and better computing power has just increased our understanding of these topics. A full coverage of these topics would take longer than an hour to provide.

Let’s start with a simple construction. Take a triangle, bisect each side, and divide it into four congruent triangles. Then, remove the middle triangle. Then comes the only hard part of the construction... repeat an infinite number of times. Sierpinski Triangle

Another way to think of the construction is as the union of three contraction mappings. This union, called an iterated function system, is itself a contraction mapping, which must have a fixed point – the Sierpinski triangle! This is true regardless of the non- zero set of points with which you begin. I’ll stop here, because the next iteration virtually disappears!

So, what happens if we use other carefully- chosen contraction mappings? The union will have a new fixed point, perhaps something that looks a bit more natural. Barnsley Fern

Where does chaos theory enter the discussion? Let’s start by looking at the logistic equation. In a continuous setting, the logistic equation has the form where r is a constant, x is a measure of a population, and c is the carrying capacity, also a constant. This is a differential equation, with the solution The solution, with r = 4, c = 1, and x(0) = 0.01, is shown at right.

Where does chaos theory enter the discussion? In a discrete setting, the logistic equation has the form where again r is a constant, x is a measure of a population, and c is the carrying capacity, also a constant. This is a difference equation. The graph of x 0, x 1, etc. with r = 4, c = 1, and x 0 = 0.01, is shown below. What is going on here? Chaos? Randomness?

Chaos? Yes! Randomness? No! This is considered chaotic behavior, because if we change the initial condition slightly, to x 0 = for example, we get a completely different result.

Chaos? Yes! Randomness? No! This is considered chaotic behavior, because if we change the initial condition slightly, to x 0 = for example, we get a completely different result. However, there is a formula to predict each of the points!

When we iterate this function, we basically take the y-value over to the line y = x, and then go back up/down to the graph. Show the movie “cobweb1.mov”.

Not every initial value gives chaotic results.

Yep, it’s a fractal!

And then there’s randomness, which has nothing to do with chaos or fractals, right? Let’s play a game. Select three points (red). Pick a point (black) from among the first three points. Completely at random, generate another point that is halfway between your point and one of the original three. Repeat. By the way, the name of this game is the Chaos Game.

I can use the Chaos Game to finish my fern drawing.

The Mandelbrot set is found and drawn in the complex plane, by taking each point in the plane z 0 and iterating the function f(z) = z 2 + z 0. Points z 0 that stay in the circle |z| = 2 when iterated are colored black, and other points are colored by how fast they leave the circle.

The Julia set is found and drawn in the complex plane by choosing a constant z 0, and then iterate the function f(z) = z 2 + z 0 for each point in the plane. The black points are points that stay inside the circle |z| = 2, and the rest are colored by how long it takes them to escape. In this particular Julia set, z 0 = –1.22 – 0.15i. Thank you!