Dynamical Systems on the Torus
by Paul Trow
Dynamical systems - systems that changes over time - are central to all the sciences. A swinging pendulum, the molecules of air in a room, a chemical reaction in a test tube - even the solar system itself - are all examples of dynamical systems. Since Galileo first gave formulas for the dynamics of moving objects, scientists have used mathematical models of dynamical systems to explain the physical world.
The pictures below illustrate a mathematical model of a dynamical system on the torus, the mathematician's term for the surface of a doughnut. The pictures - six consecutive snapshots of the torus - show how the system evolves over time, as points on the surface move around. You can see the evolution of the system in the changes to the colored regions of the torus. In the first picture, the top half the torus is colored light yellow and the bottom half is transparent. You can think of the two regions as fluids of different colors that are initially separated. In the next five pictures, the white and transparent regions stretch and wind around the torus, as if the two fluids were being stirred together, until in the last picture you can barely distinguish them
In fact, the behavior of this model is mathematically quite similar to what happens when you mix two fluids - for example, by stirring milk in a cup of coffee. Despite its abstract nature, the model has many properties in common with real, physical systems.
In order to understand this model, you need to visualize where points on the torus move over time. But visualizing how points move on a curved surface can be challenging. Fortunately, there is an easier way to view the model: by cutting the torus and flattening it out, as shown in the following figure.
Starting with the torus in figure 1, cut it along the green curve and straighten it out to form the cylinder shown in figure 2. Then, cut the cylinder along the purple line, unroll it to form the square shown in figure 3.
You can now think of the square as though it were still a torus by mentally joining the two pairs of opposite sides. If you have played a video game that "wraps around," you have already imagined the torus this way: the video screen is like a square in which opposite sides are identified. When objects in the game move off the right side of the screen, they reappears on the left, as if they were going around the torus in a horizontal circle. And when objects move off the top of the screen, they reappear on the bottom, as if they were going around in a vertical circle. If you have played this kind of game, you were playing on the equivalent of a torus.
Now that you have seen how to visualize the torus as a square, here is the rule that determines how points move on the torus. First, identify the torus with a 1-by-1 square shown in the following figure.
The rule is that if a point has coordinates (x, y), then after one unit of time it moves to the point with coordinates (2x+y, x+ y).
For instance, after one unit of time, the point p1 = (.1, .1) moves to the point p2, whose coordinates are
(2(.1) + .1, .1 + .1) = (.3, .2).
After a second unit of time, the point p2 moves to the point p3 = (.8, .5). What happens to p3 in the next unit of time? If we compute the coordinates of the next point, we get (2.1, 1.3), but that point, which is not in the unit square, wraps around to a point in the square. The coordinates of this point are the fractional parts of (2.1, 1.3) - namely, (.1, .3). The preceding figure shows the sequence of points p1, p2, p3, p4, representing the motion of the point p1 in three units of time.
The sequence p1, p2, p3, p4, continued indefinitely, is called the orbit of p1. Here is a picture of the first fifty points in the orbit of p1.
The orbit of p1 appears to be random, like a molecule of coffee bouncing around in a cup. Of course, the orbits in this system cannot be truly random, because they are produced by a deterministic rule; but many of them have the statistical properties of randomness. This is one reason the system can model phyical systems, like mixing fluids, that have random behavior.
Another way to visualize this dynamical system, besides seeing what it does to a single point, is to observe what happens to an entire region of the square over time, just as the pictures at the beginning of this article showed the changes to the white region of the torus. Viewing the torus as a square, we can see what happens to the bottom half of the 1-by-1 square, which is colored blue in the following figure.
The following figure illustrates what happens to the blue rectangle over three units of time. For the purpose of comparison, the pictures in the left column show what happens without wrap around - that is, if you apply the rule in the plane. The pictures in the middle column, on the other hand, show what happens with wrap around - that is, on the surface of the torus. In the left column, the rectangle is stretched, or expanded, in one direction and contracted in another direction over time. (If you have studied linear algebra, you will recognize that the rule is a linear transformation of the plane, and the direction in which the transformation stretches the rectangle is an eigenvector corresponding to an eigenvalue greater than 1.) But in the middle column, all of the blue region wraps around into the original square. The parallel bands in the pictures in the middle are really the single band in the pictures on the left wrapping around the torus.
When you identify the square with the torus, the pictures in the middle column correspond to the pictures of the three-dimensional torus shown in the right column..In each pair of pictures in the middle and right columns, the white region of the square corresponds to the white region of the torus, while the blue region of the square corresponds to the transparent region of the torus. As time goes on, the white band gets longer and longer in the expanding direction, causing it to wrap around the torus many times. At the same time, it becomes narrower in the contracting direction.
We can test whether this system really has the properties of mixing fluids.by seeing what happens to two small regions of the torus over time. The following picture shows 10,000 randomly generated points, colored blue, in the small square at the lower left, and 10,000 random points, colored red, at the upper right. The squares appear to be solid because there are so many points. You can think of the two squares as different fluids.
Now, let's apply the rule to the squares twenty five times in succession The following figure shows the result.
As you can see, the red and blue points are uniformly mixed throughout the torus. In fact, the fraction of the blue (or red) points that lie in any region in the square is approximately equal to the ratio of the area of the region to the entire square.
So far, we have been looking at points in this system whose orbits appear to be random. Despite this chaotic behavior, the system has an underlying order. Many of its points have well-behaved orbits that always return to the same position after a fixed amount of time, like a swinging pendulum. Such points are called periodic. For example, the point (.5, .5) is periodic. If we calculate its orbit, we get
After applying the rule three times, we are back to where we started, so the point has period three. The following picture shows the orbit of (.5, .5).
There are many periodic points in the system: in fact, every point on the torus is arbitrarily close to a periodic point. On the other hand, there are points very close to (0.5, 0.5) whose orbits that look much more random. For example, the following picture shows the orbit of (.501, .501), colored blue, superimposed on the preceding picture.
What makes this system interesting is the richness of its dynamics: it contains both periodic points, whose behavior is completely predictable, along with many other points whose behavior appears to be random. This combination of order and randomness, which occurs in many physical systems, is a typical feature of chaos.
"Deforming the Torus", Matthew Grayson, Bruce Kitchens and George Zettler, Mathematical Intelligencer, Vol. 15, No. 1.
Copyright 2003 by Paul Trow