Examples of Chaotic Attractors and Their Fractal Dimension

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1 Examples of Chaotic Attractors and Their Fractal Dimension Ulrich A. Hoensch Rocky Mountain College Billings, MT hoenschu February 2005 Abstract We present the Sierpinski Triangle as an example of a chaotic attractor. We define the fractal dimension of a self-similar set, and use this definition to compute the dimension of various other fractal sets. Students are encouraged to generate other attractors using a Java applet, and compute their fractal dimension. Example 1 Pick three vertices (points) in the xy-plane. Also, pick a point at position p 0 somewhere in the xy-plane. vertex 3 p 0 p4 p 3 p 1 p 5 p 2 vertex 1 vertex 2 Now, choose one of the three vertices at random, and move the point halfway (on a straight line) towards the vertex. Let p 1 be its new position. Then, choose another one of the three vertices at random, and move the point halfway towards that vertex, to position p 2. Repeat this process over and over. In this way a sequence of positions p 0,p 1,p 2,p 3,... is obtained. 1

2 Question What does the set consisting of the points p0, p1, p2, p3,... look like? Answer We use the computer to plot these points. A pattern emerges. 100 points 1,000 points 10,000 points The points accumulate on a set Q which is an example of a chaotic attractor. It is an attractor, because for any initial position p0, the orbit p0, p1, p2, p3,... is approaching Q; it is chaotic, because two initial positions in the attractor that are arbitrarily close will have completely different orbits. If we start with three vertices as in the example above, the attractor Q is called Sierpinski s Triangle. Here is a detailed picture of this attractor. We now turn our attention to the geometry of the attractor. Observe that the entire set is identical to each of its smaller sub-triangles, except in scale. In other words, if we magnify each of the smaller triangles, then we obtain the entire set. 2

3 Definition We say that a set S is self-similar, if there is a number k and anumberr, such that k congruent subsets are magnified by the factor r to yield the entire set S. Example For Sierpinski s Triangle, we can choose k =3,andr =2. Definition If S is self-similar, with k and r as above, then the fractal dimension of S is defined to be dim frac = log(k) log(r). A self-similar set is called a fractal, if its fractal dimension is not a whole number. Example The fractal dimension of Sierpinski s Triange is log(3)/ log(2) = Note that Sierpinski s Triangle can also be obtained by the following sequence of removals. Starting with a full triangle, remove the middle triangle; from each of the remaining three triangles, again remove the middle triangle; and so on, ad infinitum. This method gives a general approach to constructing fractal sets. 3

4 Box Fractals Box Fractals are obtained by removing pieces from a square, and removing the same portions from each left-over piece. Example 2 removals. Sierpinski s Carpet is obtained by the following sequence of Since k = 8 pieces have to each be magnified by factor r = 3 to yield the entire figure, Sierpinski s Carpet has fractal dimension log(8)/ log(3) = Both Sierpinski s Triangle and Sierpinski s Carpet have fractal dimension less than 2, because they are generated by an infinite sequence of removals, starting with a set of dimension 2 (a triangle and a square, respectively). It is interesting to note that in each finite step, the dimension of the figure is still 2, but it jumps down at infinity. The next example deals with a set that is obtained through an infinite sequence of additions, starting with a set of dimension 1. Example 3 Koch s Snowflake is generated by starting with an equilateral triangle without interior, and then by replacing the middle third of each side by an outward-facing equilateral triangle. 4

5 To compute the fractal dimension of Koch s Snowflake, we inscribe the first iterate (the hexagram, see figure at left, below). Then we observe that each of the twelve pieces is similar to each of the three pieces, if the original triangle is inscribed. So there are k = 4 parts that are similar to one larger part, and they have to be magnified by factor r = 3. This means that Koch s Snowflake has fractal dimension log(4)/ log(3) = Experiments and Exercises To use the Java applet that produces chaotic attractors, visit the web page hoenschu/fractalgens/chaosgame/chaosgame.html 5

6 Observe what happens if you choose four vertices instead of three. Do you see a pattern? Change the attraction factor to a =0.67 (this means in every step the point is moved 2/3 of the way to the vertex); is there a pattern now? What if you choose five vertices? Six vertices? Experiment with different attraction factors, and different relative positions of the vertices. How do you have to choose the vertices, and which attraction factor do you need to obtain attractors like these: (a) (b) What is the fractal dimension of each of the two attractors above? Show that the total area of both Sierpinski s triangle and Sierpinski s Carpet is zero. Find the total area inside Koch s Snowflake. About Me: I have graduated from Michigan State University in May 2003 with a Ph.D. in Mathematics, and am now an Assistant Professor of Mathematics at Rocky Mountain College. My main interests and field of expertise lie in the area of Dynamical Systems. I find great pleasure in teaching a wide array of mathematics and computer science classes at Rocky, including a class on chaos theory, from which this presentation was derived, and currently a class on cryptology. About the Math Department at Rocky Moutain College: We take great pride in our program, which offers a major and minor in both mathematics and mathematics education. As faculty at a small liberal arts college, we are committed to educating our students at the highest level possible. We care about our students academic progress and about preparing them for their professional future. Small classes, a broad array of course offerings, accessible faculty, and a modern curriculum make the mathematics program at Rocky truly outstanding within the region. Most of our graduates enter graduate school, where we have a 100% placement rate for math majors, or become math teachers. Other career choices for math majors are to work in the financial sectors of banking and insurance, in technology-related fields such as engineering and computer science, or for the Federal Government. 6