Orbits of Dynamical Systems

A much simpler configuration space, of only four dimensions, arose in the work of two researchers in the Division of Applied Mathematics at Brown. Professors Hüseyin Koçak and Fred Bisshopp generated enormous lists of numbers from their experiments with a pair of pendulums. A computer simulated vast numbers of positions and velocities for the pair of pendulums, depending on different starting points and different ratios between the frequencies of the pendulums. Koçak and Bisshopp wanted to display their data in a visual way that would reveal patterns impossible to see in a mere catalogue of numbers. The applied mathematicians came to our geometry project to see if our methods for visualizing higher-dimensional configurations could help them in their work. As it happened, the fit between our two research projects could not have been better.

Koçak and Bisshopp recorded the position of each pendulum by marking a point on a circle in a two-dimensional plane, so at any given time points on two circles specified the positions of the two pendulums. Since each observation involved two points, each with two coordinates in the plane, the data set was four-dimensional. The two circles could have been graphed side-by-side on a plane, but that representation would not have indicated sufficiently how the motions of the two pendulums were related. A new technique was needed for displaying the relationships more effectively.

We can see the relationship between two variables like height and armspan by graphing them on a plane, thought of as a line of lines, with one vertical line for each point on the horizontal axis. Similarly, we can see the relationship between points on circles by graphing them on a torus, thought of as a circle of circles. We could draw the graph of such a relationship on a torus in three-dimensional space, with one vertical circle for each horizontal ``axis circle.'' But we can get an even more symmetrical graph if we use a four-dimensional configuration space, plotting the positions of the pendulum system on the Clifford torus on the hyper-sphere described in the previous chapter. This more symmetrical representation made it possible to recognize the most significant patterns in the different orbit curves.

The history of the motion of the two pendulums could be represented as a sequence of points tracing out a curve on the torus in the hypersphere. Visualizing the structure of the orbits for different choices of beginning positions and frequency ratios of the two pendulums was a challenge that was well suited to our computer graphics techniques. The way the data points from the physical experiment were arranged in nearby orbits suggested displaying the surface in strips, a device that turned out to be especially effective for the presentation and investigation of other surfaces in three- and four-dimensional space.

When one of the pendulums is stationary, the orbits traced on the torus are circles of latitude or longitude. When both pendulums move in a synchronized way, beating in unison, the orbit curve goes around the torus once each way, hitting each parallel of latitude and each meridian of longitude exactly once. These orbits turn out to be precisely the circles that we obtained in Chapter 3 by slicing a torus obliquely so that the slicing plane is tangent to the torus at two points. The film The Hypersphere: Foliation and Projections, made together with Koçak, Bisshopp, and computer science graduate students David Laidlaw and David Margolis, presents a visualization of all possible orbits of synchronized pendulums, named Hopf circles after the Swiss mathematician Heinz Hopf, who studied their properties in the 1930s. The collection of Hopf circles on the hypersphere is one of the most intriguing higher-dimensional images. More complicated data sets produce orbits of greater complexity, leading to knotted curves and curves that do not close up after a certain amount of time. It is by comparing such complex systems to the collection of Hopf circles that researchers can begin to visualize more subtle relationships in the orbits of dynamical systems.

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