My pages are an introduction to what I do at Bates, with information on fractals, bifurcation and orbit diagrams, Julia sets and more. Also, check out my galleries of fractals, people, pipe organs, flowers, and other interests. Years listed in red are an indication of the relative newness of the subpages. Newest pages are the bifurcation pages (here at the top) and flowers (scroll down to the bottom).

We'll begin with a note about Bifurcation and Orbit diagrams. Plots of these two diagrams for the family of functions fc(x) = x2 + c appear below. The shadowy curves sweeping up and down through both diagrams are called "Q-curves"; we'll say more about them in a moment.

The orbit diagram of the middle picture will probably look familiar. At each value of c on the horizontal axis, it shows how the orbit of 0 behaves after many iterations and in doing so, shows where fc has attracting cycles and suggests where fc behaves chaotically. This image has appeared in countless books and webpages, and indeed ``has become the most important icon of chaos theory''PJS.

The bifurcation diagram in the left-hand picture may not be so familiar. It shows the locations of all attracting and repelling cycles of fc (up to period 8 in this figure).

Clicking on either picture brings up a page with information about both diagrams, [2004] and the bifurcation diagram in particular, including the algorithm used to draw it. Pictures of bifurcation and orbit diagrams for other functions such as the logistic map and sin(cx) are included.

Now, please be careful! The names I am using for these two diagrams are also used by authors such as Devaney, Strogatz, Peitgen/Jürgens/Saupe , but there are many books, articles, and webpages where the orbit diagram is called ``the bifurcation diagram'', and that which we call the ``bifurcation diagram'' is not discussed. One can also find places where bifurcation diagrams are discussed, but then computer-drawn orbit diagrams are used in the illustrations without any mention of the difference(!). However, I have been unable to find any other sources of detailed, computer-drawn bifurcation diagrams. Our images show that computers can draw excellent pictures of both diagrams, and both are repleat with fascinating features.

One such feature shared by the two diagrams is the illusion of shadowy polynomial curves - Q-curves - sweeping up and down as you look left-to-right across either diagram. These curves are not implicitly drawn in either diagram, and in fact appear as illusions in the two diagrams for nearly opposite reasons. The Q-curves interact with the explicitly drawn curves of the bifurcation diagram in a beautiful way, and this interact is the subject of an article we've called "The Shadow-Curves of the Orbit Diagram Permeate the Bifurcation Diagram, Too" (Download a PDF copy). This paper appeared in the September 2009 issue of the International Journal of Bifurcation and Chaos.