When is a bifurcation diagram not a bifurcation diagram?

When it's an orbit diagram!

All right, so you may not see that exchange in the next edition of 100 Best Party Riddles. But for making a new distinction between orbit and bifurcation diagrams, Chip Ross won a prestigious award in mathematics last year. Associate Professor Ross and co-author Jody Sorensen, a former Bates math department colleague now in Michigan, won the Mathematical Association of America's George Pólya Award for their article in the January 2000 issue of College Mathematics Journal, titled "Will the Real Bifurcation Diagram Please Stand Up!"

Both diagrams are useful in teaching chaotic dynamical systems, Ross's specialty, which he explains as the study of "how small changes in a system may produce big changes later on down the road." (Or, as the cliché goes, "Can the flap of a butterfly's wings in Brazil cause a tornado in Texas?")

For a Short Term unit in the mid-1990s, Ross and Sorensen had students plot a series of values on a new kind of graph. The results were so intriguing that Ross pulled an all-nighter writing software to generate such graphs. Because the graph represented the relative positions of two different kinds of points (the computed values seemed to avoid one kind, the "repelling points," and gravitate toward the other, the "attracting points"), he and Sorensen had drawn what is defined as the bifurcation diagram.

Problem is, that term is often incorrectly applied to a different diagram, the orbit diagram, which shows the attracting points but only a chaotic scramble of dots where the repelling points should be. The beauty of Ross and Sorensen's diagram is that it shows both — but the orbit diagram is often used in its place, perhaps because it's much easier to program a computer to draw it.

"My mission in life now is to explain to everybody that there's really a difference between the two, and the computer can now draw them both. They each show different things," Ross says — and the "real" bifurcation methodology adds an important dimension to teaching about chaotic dynamical systems. To learn more, visit:

http://abacus.bates.edu/~sross/