Dynamical Systems on the Torus
by Paul Trow

Dynamical systems - systems that changes over time - are central to all of science. 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 position and velocity of falling bodies, scientists have used mathematical models of dynamical systems to explain the physical world.

The pictures below illustrate a 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. Imagine that 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

      Pictures from "Deforming the Torus," by Matthew Grayson, Bruce Kitchens and George Zettler. I thank the authors for permission to use them.

In fact, the behavior of this model is mathematically quite similar to what happens when you mix two fluids - for example, by stirring cream in a cup of coffee. Despite its abstract nature, the model has many properties in common with real, physical systems.

Deconstructing the Torus

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 is not so easy. Fortunately, there is an easier way to view the torus: by cutting it and flattening it out into a square, 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 and unroll it to form the square shown in figure 3.

You can think of the square as though it were a torus by mentally joining the two pairs of opposite sides. If you have played a video game that "wraps around," you have already visualized the torus this way. When objects in the game move off the right side of the screen, they reappear on the left, as if they were going around the torus in a horizontal circle.

The Rule for the Dynamical System

Now, imagine the torus as a unit square - the set of all points (x, y) whose coordinates lie in the interval from 0 to 1. Here is the rule that defines the dynamical system shown in the pictures at the beginning of this article. If a point in the square has coordinates (x, y), then after one unit of time it moves to the point with coordinates

(2x + y, x + y)

For example, the point p1 = (0.1, 0.1) moves to the point

p2 = (2(0.1) + 0.1, 0.1 + 0.1) = (0.3, 0.2).

After a second unit of time, the point p2 moves to the point p3 = (0.8, 0.5). But what happens to p3 in the next unit of time? The coordinates of the next point are (2.1, 1.3), which is outside the unit square - so this point wraps around to the point whose coordinates are the fractional parts of (2.1, 1.3) - namely, p4 = (0.1, 0.3).

The picture below 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, and so on, is called the orbit of p1. Here is a picture of the first fifty points in this orbit.

The orbit appears to be random, like a molecule of coffee bouncing around in a cup. Of course, the orbit is not truly random, because it is produced by a deterministic rule. Nevertheless, the orbits of most points in this system have the statistical properties of randomness. This is one reason the sytem can be used to model the random behavior of phyical systems, such as mixing fluids.

Automorphisms of the Torus

The rule for this system is an example of an automorphism of the torus. An automorphism is defined by multiplying points in the torus by a matrix with determinant plus or minus 1 — called a unimodular matrix. The matrix for this example is \[\begin{bmatrix} 2 & 1\\ 1 & 1 \end{bmatrix}\] The unimodular condition ensures that the matrix and its inverse maps every integer point to another integer point. This is essential because the construction of the torus identifies all integer points. As a result, a toral automorphism is a continuous bijection from the torus to itself, which is also a group isomorphism.

You can experiment with plotting the orbits of points under different automorphisms of the torus with this interactive toral automorphism plotter.

The pictures at the beginning of this article show how this dynamical system evolves over time, by showing the orbits of the points in the white and blue regions of the torus. You can imagine the white region of the torus as the top half of the 1-by-1 square, and the blue region as the bottom half, as shown in the following figure.

The sequence of figures below 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. (In the language of linear algebra, the rule is a linear transformation; the direction in which the the rectangle is stretched 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 random 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 squares as two different fluids, which are initially separated.

Now, let's apply the rule to the squares twenty five times in succession The following figure shows the result.

Squares after mixing

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.

Order Versus Disorder

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 periodic orbits that always return to the same position after a fixed amount of time, like the motion of a pendulum. Such points are called periodic. For example, the point q = (0.5, 0.5) is periodic. If you calculate the first three points in its orbit, you get

After applying the rule three times, the point q is back to where it started, so it has period three. The following picture shows the orbit of q.

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 q whose orbits appear to be random. For example, the following picture shows the orbit of (.501, .501), colored blue, superimposed on the orbit of q.

What makes this system so interesting is the richness of its dynamics: it contains both periodic points, whose behavior is completely regular and predictable,and points whose behavior has the properties of randomness. 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